From d82a679985888b703f689833c2b12780a8243382 Mon Sep 17 00:00:00 2001 From: ross Date: Wed, 21 Oct 2026 00:08:31 +1100 Subject: [PATCH 1/2] Prove actual BIT snapshot path floor and price coded caller repairs --- research/bit-copy-path/NOTICE.md | 10 + research/bit-copy-path/PREDECESSOR.json | 55 + research/bit-copy-path/README.md | 156 + research/bit-copy-path/RELATED-WORK.md | 82 + research/bit-copy-path/REPLAY.json | 86 + research/bit-copy-path/SOURCES.json | 342 + research/bit-copy-path/TASK.json | 72 + research/bit-copy-path/bounds/BUDGET.json | 6626 ++++ .../bit-copy-path/bounds/INDEPENDENT.json | 226 + research/bit-copy-path/bounds/PROOF.md | 477 + research/bit-copy-path/bounds/README.md | 46 + research/bit-copy-path/bounds/VALIDATION.json | 39 + research/bit-copy-path/bounds/check_budget.py | 583 + .../bit-copy-path/bounds/check_independent.py | 253 + .../inputs/DIRECT-FLOOR-PRIOR-README.md | 129 + .../bounds/inputs/SELECTED-ASSEMBLY.json | 345 + 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research/bit-copy-path/composition/inputs/C511-PROOF-INTERFACE.md create mode 100644 research/bit-copy-path/composition/inputs/C511-PROOF-OBSERVATION.md create mode 100644 research/bit-copy-path/composition/inputs/CODED-EXPECTED.json create mode 100644 research/bit-copy-path/composition/inputs/CODED-PROOF.md create mode 100644 research/bit-copy-path/composition/inputs/CODED-WORD.json.gz create mode 100644 research/bit-copy-path/composition/inputs/FULL-E8-GEOMETRY.json.gz create mode 100644 research/bit-copy-path/composition/inputs/R429-PHYSICAL-CODE-REFERENCE.py.txt create mode 100644 research/bit-copy-path/composition/inputs/SOURCES.json create mode 100644 research/bit-copy-path/construction/ACQUISITION-RELAXED.json create mode 100644 research/bit-copy-path/construction/DIRECT-SCATTER.json create mode 100644 research/bit-copy-path/construction/LATE-FACTORY-SCREEN.json create mode 100644 research/bit-copy-path/construction/NEWCUT-SCREEN.json create mode 100644 research/bit-copy-path/construction/PROOF.md create mode 100644 research/bit-copy-path/construction/SOURCES.json create mode 100644 research/bit-copy-path/construction/WALK-CERTIFICATE.json create mode 100644 research/bit-copy-path/construction/acquisition_screen.py create mode 100644 research/bit-copy-path/construction/direct_scatter.py create mode 100644 research/bit-copy-path/construction/late_factory_screen.py create mode 100644 research/bit-copy-path/construction/newcut_screen.py create mode 100644 research/bit-copy-path/construction/walk_certificate.py create mode 100644 research/bit-copy-path/independent/REVIEW.md create mode 100644 research/bit-copy-path/independent/VALIDATION.json create mode 100644 research/bit-copy-path/independent/review-result.json create mode 100644 research/bit-copy-path/independent/review_check.py create mode 100644 research/bit-copy-path/inputs/FULL-CALLER-PROOF.md create mode 100644 research/bit-copy-path/inputs/FULL-CALLER-README.md create mode 100644 research/bit-copy-path/inputs/GEOMETRY.json.gz create mode 100644 research/bit-copy-path/inputs/TERMINAL-CHARTS.json create mode 100644 research/bit-copy-path/inputs/WORD.json.gz create mode 100644 research/bit-copy-path/inputs/full-check-screens.py.txt create mode 100644 research/bit-copy-path/inputs/full-exact-interval.py.txt create mode 100644 research/bit-copy-path/inputs/full-invoices.json create mode 100644 research/bit-copy-path/inputs/full-schedule-witness.json create mode 100644 research/bit-copy-path/inputs/global_k.py.txt create mode 100644 research/bit-copy-path/verify.py diff --git a/research/bit-copy-path/NOTICE.md b/research/bit-copy-path/NOTICE.md new file mode 100644 index 0000000000..67c569b8ab --- /dev/null +++ b/research/bit-copy-path/NOTICE.md @@ -0,0 +1,10 @@ +# Notice + +Prepared for Ross Richardson with substantial OpenAI Codex assistance in +mathematical derivation, implementation, exact checks, and independent +agent review. No human or formal-kernel review is claimed. + +This research contribution is provided under the repository's Apache-2.0 +license. Preserved source excerpts retain their original authorship and +provenance; see `RELATED-WORK.md`, `PREDECESSOR.json`, and the component +source manifests. diff --git a/research/bit-copy-path/PREDECESSOR.json b/research/bit-copy-path/PREDECESSOR.json new file mode 100644 index 0000000000..943e734783 --- /dev/null +++ b/research/bit-copy-path/PREDECESSOR.json @@ -0,0 +1,55 @@ +{ + "commit": "b4ec15cbdb017776c04b0bf369a807f2278113cd", + "files": { + "WORD.json.gz": { + "source": "/Users/ross/git/math/.research-automation/runs/bit-full-caller-20261020T1028Z/release/bit-full-caller/inputs/consumer/inputs/retained/boundary/inputs/WORD.json.gz", + "old_relative": "inputs/consumer/inputs/retained/boundary/inputs/WORD.json.gz", + "sha256": "c46ccd7d3eedabc6a7efbb1a91ec13f2928bad8ee0bac57c6e593a092d13d96c" + }, + "GEOMETRY.json.gz": { + "source": "/Users/ross/git/math/.research-automation/runs/bit-full-caller-20261020T1028Z/release/bit-full-caller/inputs/consumer/inputs/retained/boundary/inputs/GEOMETRY.json.gz", + "old_relative": "inputs/consumer/inputs/retained/boundary/inputs/GEOMETRY.json.gz", + "sha256": "a353999bceac4ac957f540adfd3e5f8768297138b27132eeb83332e2ea4b1022" + }, + "TERMINAL-CHARTS.json": { + "source": "/Users/ross/git/math/.research-automation/runs/bit-full-caller-20261020T1028Z/release/bit-full-caller/inputs/consumer/inputs/retained/boundary/inputs/TERMINAL-CHARTS.json", + "old_relative": "inputs/consumer/inputs/retained/boundary/inputs/TERMINAL-CHARTS.json", + "sha256": "489216ea25f47f061675ab7b7e4a17289a531bce6fe77cca166a13265bfcbb17" + }, + "global_k.py.txt": { + "source": "/Users/ross/git/math/.research-automation/runs/bit-full-caller-20261020T1028Z/release/bit-full-caller/inputs/consumer/inputs/retained/source-context/global_k.py", + "old_relative": "inputs/consumer/inputs/retained/source-context/global_k.py", + "sha256": "6368196a393ac25bb2918880e4c4bd31ed733b762c27b6873ab391cbc0c5c3c8" + }, + "FULL-CALLER-PROOF.md": { + "source": "/Users/ross/git/math/.research-automation/runs/bit-full-caller-20261020T1028Z/release/bit-full-caller/bounds/PROOF.md", + "old_relative": "bounds/PROOF.md", + "sha256": "ebdd9283a7762e01f84ee9dde8885d4ba05cc6c93a4430296ff0a71ea7205832" + }, + "FULL-CALLER-README.md": { + "source": "/Users/ross/git/math/.research-automation/runs/bit-full-caller-20261020T1028Z/release/bit-full-caller/README.md", + "old_relative": "README.md", + "sha256": "d8ea32275410a0fc13e3e1a67f3dba134679a67912b4ffc893b88b2ced967823" + }, + "full-check-screens.py.txt": { + "source": "/Users/ross/git/math/.research-automation/runs/bit-full-caller-20261020T1028Z/release/bit-full-caller/bounds/check_screens.py", + "old_relative": "bounds/check_screens.py", + "sha256": "c07d15a1e5edcbbab584fd77b26c2e6787c2baa4ef013de72dbfd52f5d2960e6" + }, + "full-exact-interval.py.txt": { + "source": "/Users/ross/git/math/.research-automation/runs/bit-full-caller-20261020T1028Z/release/bit-full-caller/bounds/exact_interval.py", + "old_relative": "bounds/exact_interval.py", + "sha256": "d8a78c2a2f87a5cf9fff0367d70f21dcdb712a509ba365879aee134b18c4c25d" + }, + "full-schedule-witness.json": { + "source": "/Users/ross/git/math/.research-automation/runs/bit-full-caller-20261020T1028Z/release/bit-full-caller/schedule/WITNESS.json", + "old_relative": "schedule/WITNESS.json", + "sha256": "8420b5c57fd033f05675c7a1d72a1ba1a07c19542042698028b300019295a94b" + }, + "full-invoices.json": { + "source": "/Users/ross/git/math/.research-automation/runs/bit-full-caller-20261020T1028Z/release/bit-full-caller/bounds/INVOICES.json", + "old_relative": "bounds/INVOICES.json", + "sha256": "d0877305ca03e88622586fd76622fc9e2e2da0484de3b112577467c82f386a92" + } + } +} diff --git a/research/bit-copy-path/README.md b/research/bit-copy-path/README.md new file mode 100644 index 0000000000..38a5d29bae --- /dev/null +++ b/research/bit-copy-path/README.md @@ -0,0 +1,156 @@ +# Actual BIT COPY paths and the price of repairing the coded caller + +**Scoped obstruction and complete planning budgets; no new supplier or κ.** + +This investigation changes the proposed query word, rather than applying the +preceding fixed-word allocation ceiling to a different construction. It finds +a stronger obstruction to replacing one of the nine actual centre copies. +It also tests the existing 429-helper coded word in the actual recursive +caller. Neither route supplies a κ improvement. + +## Main result: closed snapshot paths cannot cheapen a centre COPY + +Each retained centre has a seven-dimensional frame \(F_j\). Its required +scatter column \(k_j\), reconstructed from the literal R438 word, reaches +port roots spanning all eight dimensions. + +Allow any finite number of independent copies of the **same unchanged +centre total**. Copy \(i\) may start in any subframe of \(F_j\), costing +\(r_i\), and follow an arbitrary closed path of nested private frame moves, +costing \(m_i\) in both directions. It may add into any compatible public +register during the continuation after the last original COPY. Public +frames remain monotone and the continuation contains no further COPY. +Scalar additions, legal public retiming, and other cleanup are granted free +for this necessary bound. The private copies must be restored for their +guarded ERASEs. + +The written geometric argument proves + +\[ + \sum_i\left(r_i+\frac{m_i}{2}\right)\geq 7. +\] + +Consequently the replacement cannot reduce either the COPY rank mass or +its recursive moment +\(\sum_\ell 64^\beta r_\ell^{\,1-\beta}\), for \(0\leq\beta<1\). +For a single copy of rank \(r<7\), the complete rank cost is at least +\(14-r\geq8\). + +This includes moving the private copy outside the original centre frame. +For example, a rank-one COPY followed by a rank-one promotion and its +rank-one inverse is a valid local transport costing three. It still cannot +deliver the complete actual scatter column. A private slot starts and ends +zero; no arbitrary private dirt is discarded. + +See [the proof](construction/PROOF.md) and +[independent review](independent/REVIEW.md). The argument quantifies over +the stated continuous frame class; its application to these nine centres +uses exact finite source and rational/real geometry checks. It is not a formal-kernel +proof, or a bound on mixtures of different totals, changed public queries, +or a rebuilt factory. + +## Earlier cuts and the recoded caller + +Two finite screens test ways around the late-continuation premise. + +- Source acquisitions at all existing common-frame cuts in the actual + factory cannot replace any deleted centre scatter. Even relaxing the + five partial-helper terminal conditions leaves exact binary separators. +- Copying each completed centre earlier, before the remaining centre + copies, fails across all **254** eligible cuts. The screen includes the + unchanged downstream COPYs and allows an arbitrary real intermediate + frame separately for each receiver. It does not cover a changed + incomplete total or a changed factory graph. + +The existing R429 source-code construction has a real local saving, but +its unmodified substitution into the nine-stage caller fails. The complete +888-column scalar defect has rank 63. An integral ADDRESS impulse in the +last pair produces a nonzero TAIL component that no invertible terminal +decoder made solely of scalar mixing and separate HEAD/TAIL routes can +erase. Restoring the scalar source code locally does not remove the +mismatch between successive fresh blocks. + +[The caller proof](composition/PROOF.md) binds this failure to the actual +order-28 code and native charts. Its priced repair floors cover two +specified interfaces: a minimum nine-reader common-cap repair costs at +least 63 rank units per local call; the previous one-snapshot-per-repaired-row +interface needs all 111 rows. Shared or differently routed repairs remain +outside those floors. + +[The private-transport and wrapper appendix](composition/PRIVATE-MOVE-PROOF.md) +also prices an explicit correction around each coded call. Cancelling and +commuting its full-bank sweeps, while keeping both the calls and individual +code factors in their fixed chronological order, leaves at least nine +sweeps: 999 rank-one requests. That exceeds the allowance below 526 per +nine-call block. Reordering code factors or replacing these sweeps by other +routed queries is outside this particular cancellation argument. + +## Complete budget consequence + +The comparison uses the existing \(m=64\), \(W=28416\), 288-call controller +and includes returned helper-owner obligations and the inherited child +surcharge. The target is the necessary BIT endpoint for a 10% gain over +the selected κ, under the retained outer relation: + +\[ + c_*=\frac{9199631083923049}{4990800368916076951}. +\] + +| Proposal | Remaining allowance at \(c_*\) | +|---|---:| +| Delete one COPY7 in every local call | Less than 4.978 tariff units per call for its replacement | +| Use the existing coded R429 word | Less than 58.383 per call for all added repair children | + +The second row returns all 72 owner rank units per call from the nine +deleted helpers. Its complete rank saving before decoding is 60 per call. +These are planning allowances under the declared owner relaxation, not +constructed chronological schedules. + +The historical P111 label-decoder mass floor of 63 is **prior work**. +Even granting the most favorable direct-reader histogram consistent with +that floor, \(4e_6+e_7+4e_8\), the complete optimistic BIT endpoint is below +`0.001743442083496`. Under the retained outer relation its possible gain +over the selected κ is below **4.050289853587%**. This consequence applies +only to that direct-reader class. It is not a global upper bound on BIT +algorithms. + +[The budget proof](bounds/PROOF.md) and +[exact receipt](bounds/BUDGET.json) give rational intervals, the complete +histograms, and adjacent rejection/pass tests. A separate positive-binomial +calculation checks the numerical signs. The selected complex endpoint +remains binding, so a BIT-only saving would not by itself change the +selected κ. + +## Reproduce + +The package is self-contained and uses only the CPython standard library. +From a checkout or a Git-free export: + +```sh +python3.12 -I -B research/bit-copy-path/verify.py --output /tmp/bit-copy-path-check.json +``` + +The runner authenticates every packaged source, runs the declared checks +in ordinary and optimized Python modes, compares their exact receipts, +and writes all logs and generated evidence outside the package. See +[REPLAY.json](REPLAY.json) for the command inventory. It does not reexecute +the predecessor's complete historical suite. + +## Scope and next mathematical boundary + +A successful construction must change something excluded by the proved +class: the queried totals, their factory or public query events, the +restoration contract, or a shared routed caller repair. It must emit the +complete arbitrary-input word and pay its native queries, helpers, +chronological owners, rounding, storage, and outer costs. Merely moving +the same completed snapshot later, or treating a source code as a free +terminal decoder, does not meet those requirements. + +The source and attribution record is in [RELATED-WORK.md](RELATED-WORK.md), +[PREDECESSOR.json](PREDECESSOR.json), and [SOURCES.json](SOURCES.json). +The changed word is not installed as a physical supplier. Selected +certificates are unchanged. Local verification is distinct from hosted CI, +human review, formal proof, and the full multiplication theorem. + +Apache-2.0. Prepared for Ross Richardson with substantial OpenAI Codex +assistance, including separate construction and independent review agents. diff --git a/research/bit-copy-path/RELATED-WORK.md b/research/bit-copy-path/RELATED-WORK.md new file mode 100644 index 0000000000..06d2a692fe --- /dev/null +++ b/research/bit-copy-path/RELATED-WORK.md @@ -0,0 +1,82 @@ +# Source attribution and the new scope + +The checkout base is `12fa8c5c844be21bfefb38a19503fbd867a11244`, +with research priority revision 7. This package continues the local +full-caller investigation at +`b4ec15cbdb017776c04b0bf369a807f2278113cd`. The exact ten-file dependency +slice is pinned in `PREDECESSOR.json`; the predecessor's full suite is +not claimed as freshly rerun here. + +## Preserved construction and caller premises + +- DreamingOfClouds's PR589 supplier and PR446 Theorem C supply the underlying + BIT word and COPY pricing model. The R438 transport, five partial rank-four + terminals, and their independent geometry checks are retained source + constructions, not discoveries here. Their word and geometry bytes are + copied without alteration. +- Ross Richardson's `research/bit-r438-source-code` supplies the existing + three-pivot repair and 429-helper source-code word. Its scientific source + pin is `20ecf2891d80b5acf18d980b2164fb44ea7faa90`. The local 132-unit + saving and 72-unit owner return were already established there. +- The C511 mutable-source boundary package already distinguishes scalar + closure from physical native closure, and includes the actual order-28 + R429 code and the failed fresh-chart alignment. The frozen source + references and full E8 geometry are in `composition/inputs`. No + order-511 conclusion is transferred to the order-28 code. +- The preceding local full-caller package supplies the complete controller, + ordinary owner inventory, allocation proof, and existing chronological + comparator. Its fixed-word ceiling is not applied to a changed local + histogram. + +The present caller contribution is the actual nine-stage last-pair impulse, +the all-column endpoint audit, and the explicitly bounded paid repair +screens. The general warning about physical noncommutation is prior work. + +## Prior direct-decoder floor + +Russell Seligmann's [PR757](https://github.com/CrocSwap/integer-mult-bounds/pull/757) +(head `e2ec7d7`) and Cody Harris's +[PR761](https://github.com/CrocSwap/integer-mult-bounds/pull/761) +(head `2877587`) established the P111 label-decoder floor of 63. +Joel Pulikkan / GamingPuzzled's +[PR620](https://github.com/CrocSwap/integer-mult-bounds/pull/620) +identified the pairing model and orthogonal-pair opportunities. +`research/bit-orthogonal-pair-decoders` records these premises, the +arbitrary-term floor, other punctures, and the excluded off-pattern class. +Its README is frozen in `bounds/inputs/DIRECT-FLOOR-PRIOR-README.md`. + +The present finite support and basis calculations rebind that floor to +the actual local word. They are a reproduction, not a new decoder-optimality +claim. The new budget consequence is the complete best-histogram +comparison, including returned owners, exact child moments, and its +restricted endpoint ceiling. + +## Actual query changes tested here + +The main new written argument concerns immutable same-total snapshots +following closed private frame paths during a monotone public +continuation. It does not assume one catalogue frame or one private copy. +Exact rank-eight target supports and rank-seven projected supports bind +the argument to all nine actual retained totals. + +The source-acquisition and completed-centre factory-cut screens are +different finite classes. Their exact separators include the actual +downstream scalar dependencies and COPY events. They do not establish +impossibility for altered incomplete totals or a rebuilt factory. + +The entry overlap refresh read all 308 open PRs, 2703 discussion records, +and 5234 local records, with no pagination warnings, as well as discussion +hubs 1163, 1166 and 1167. This is an overlap search, not a novelty proof: +it is not a census of closed remote PRs or every code diff. No public +research claim, discussion post, or PR was made by this run. + +## Evidence boundaries + +The finite independent implementation uses separate binary state and +rational arithmetic code. The continuous frame result is a written +linear-algebra proof checked by separate agents, not a formal-kernel +certificate or independent human review. There is no new physical +compiler, rounded supplier admission, whole-owner construction, or joint +κ claim. + +Apache-2.0. Substantial OpenAI Codex assistance. diff --git a/research/bit-copy-path/REPLAY.json b/research/bit-copy-path/REPLAY.json new file mode 100644 index 0000000000..cc0a8ab9a9 --- /dev/null +++ b/research/bit-copy-path/REPLAY.json @@ -0,0 +1,86 @@ +{ + "format": "bit-copy-path-replay-v1", + "python": "CPython 3.12 or later; standard library only", + "modes": [ + "ordinary", + "optimized" + ], + "checks": [ + { + "name": "walk", + "script": "construction/walk_certificate.py", + "expected": "construction/WALK-CERTIFICATE.json", + "arguments": [], + "output_flag": "--out" + }, + { + "name": "acquisition", + "script": "construction/acquisition_screen.py", + "expected": "construction/ACQUISITION-RELAXED.json", + "arguments": [ + "--all", + "--allow-partial" + ], + "output_flag": "--out" + }, + { + "name": "late-factory", + "script": "construction/late_factory_screen.py", + "expected": "construction/LATE-FACTORY-SCREEN.json", + "arguments": [], + "output_flag": "--out" + }, + { + "name": "postscatter-cuts", + "script": "construction/newcut_screen.py", + "expected": "construction/NEWCUT-SCREEN.json", + "arguments": [], + "output_flag": "--out" + }, + { + "name": "prior-direct-floor", + "script": "construction/direct_scatter.py", + "expected": "construction/DIRECT-SCATTER.json", + "arguments": [], + "output_flag": "--out" + }, + { + "name": "coded-caller", + "script": "composition/check_composition.py", + "expected": "composition/RESULT.json", + "arguments": [], + "output_flag": "--output" + }, + { + "name": "private-transport", + "script": "composition/PRIVATE-MOVE-CHECK.py", + "expected": "composition/PRIVATE-MOVE-RESULT.json", + "arguments": [], + "output_flag": "--output" + }, + { + "name": "budgets", + "script": "bounds/check_budget.py", + "expected": "bounds/BUDGET.json", + "arguments": [], + "output_flag": "--output" + }, + { + "name": "binomial", + "script": "bounds/check_independent.py", + "expected": "bounds/INDEPENDENT.json", + "arguments": [], + 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"5849fd491b27073309a0d1acb632774015e002d984b432cd600f9c556a414764" + } + ], + "format": "bit-copy-path-sources-v1", + "scope": "Complete portable package bytes. This inventory is an integrity check, not a mathematical proof or a signature." +} diff --git a/research/bit-copy-path/TASK.json b/research/bit-copy-path/TASK.json new file mode 100644 index 0000000000..8bc84237eb --- /dev/null +++ b/research/bit-copy-path/TASK.json @@ -0,0 +1,72 @@ +{ + "schema_version": 1, + "task_id": "coverage-law-change", + "priority_revision": 7, + "base_commit": "12fa8c5c844be21bfefb38a19503fbd867a11244", + "original_card": { + "id": "coverage-law-change", + "question": "Can a template or call contract leave two or more unpaid coordinates per live family?", + "starting_points": [ + "docs/research/research-family-status.md", + "research/complex-local-word-ceiling/README.md", + "research/idle-axis-hosting/PROOF.md", + "certificates/selected-result.json" + ], + "methods": [ + "State a replacement for the coverage identity (mass = 72W + 64N) and the contract change that realizes it, with its implementation cost.", + "Find an obstruction: show which data-endpoint obligation forces exactly one unpaid coordinate in the current template." + ], + "acceptance_gate": "An explicit replacement identity, verified on a small complete instance with dirty inputs, priced with ceiling.ceiling(unpaid=...). A scoped local result may be retained without a complete supplier. Expanding it into further construction rounds requires the family-level budget and obligation gate in research-family-status.md; an optimistic passing relaxation alone is not acceptance.", + "first_step": { + "command": [ + "python3", + "-B", + "research/complex-local-word-ceiling/verify.py" + ], + "expected": "PASS_COMPLEX_LOCAL_WORD_CEILING", + "inspect": "Read the selected word's per-call local profile, its packing ceiling and the lever table; run levers.py for the ideal budget of the lever you target.", + "scope": "Positive control and planning budget only; this is not a new construction or full cold replay." + }, + "first_experiment": "Explain where the single unpaid coordinate per family comes from in the hosted template; propose one change and price it. Ideal budget: two unpaid coordinates per family is 3.77x (levers.py).", + "bottleneck_link": "The deficit is one coordinate per family minus the centre charge; doubling unpaid coordinates per family scales it directly.", + "allowed_variations": [ + "One template change or one endpoint-contract change.", + "If the coverage law, template or guard changes, derive an applicable fully charged replacement bound instead of applying the incumbent ceiling outside its scope." + ], + "scope_change_triggers": [ + "Counting a coordinate as unpaid while leaving it unadvanced in the data endpoint.", + "Continuing a partial cost improvement while leaving the same unpriced consumer, restoration or central-readout obligation unresolved; refresh the family expansion decision first. Local product-retiming evidence has a narrow scope and is not a ban on changed graphs or contracts." + ], + "overlap_terms": [ + "coverage identity", + "unpaid coordinate", + "deficit law", + "W - 64N" + ] + }, + "scope_status": "changed", + "actual_scope": "Actual R438 COPY-path changes: immutable same-total closed private walks; source acquisitions and early completed-centre cuts; actual R429 recursive caller defect and explicitly priced repair classes.", + "scope_change_reason": "Direct user request to change the actual query word and escape the preceding fixed-word allocation ceiling. This bounded result proves an independently reviewable obstruction and complete cost consequences; it does not establish a new coverage law or supplier.", + "bottleneck_link": "COPY7 deletion in every call must fund its replacement for less than 4.978 tariff units at the necessary 10-percent BIT target. Same-total B-continuation replacements cost at least the original COPY7. The coded-word repair allowance is below 58.383 per call after every returned owner obligation.", + "budget_minutes": 240, + "stop_rule": "Close this scoped round once exact actual-word screens and the continuous same-total path proof are independently checked, all concrete attempted replacements fail their declared algebra or complete cost gate, and portable evidence is preserved. Do not treat these exclusions as a global obstruction to new factories or mixed queries.", + "outcome": "partial", + "evidence": "README.md; construction/PROOF.md; composition/PROOF.md; bounds/PROOF.md; independent/REVIEW.md; verify.py", + "author": "Ross Richardson, with substantial OpenAI Codex assistance", + "ai_assistance": "Substantial assistance in derivation, implementation and independent agent review; no independent human or formal-kernel review claimed.", + "no_new_kappa": true, + "open_questions": [ + { + "question": "Can a changed factory and genuinely different queried totals share a paid native path?", + "why_it_matters": "The obstruction retains the same completed total and monotone public continuation. The earlier finite screens also keep their declared factory dependencies.", + "first_step": "Emit one complete changed query word with arbitrary dirty-input restoration and actual frames; compare its full child histogram and returned helper owners.", + "blocked_by": "No favorable literal candidate in the classes tested here." + }, + { + "question": "Can an ADDRESS-dependent shared repair beat the coded caller allowance?", + "why_it_matters": "The actual native counterexample excludes a free terminal decoder, while the63 and111 floors cover specified paid interfaces only.", + "first_step": "Derive a full nine-call operator correction with every native query and source observation priced; do not commute a source code through its own port exchange.", + "blocked_by": "No complete legal shared routed repair has been supplied." + } + ] +} diff --git a/research/bit-copy-path/bounds/BUDGET.json b/research/bit-copy-path/bounds/BUDGET.json new file mode 100644 index 0000000000..81dec588cc --- 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LP lower bounds do not assert chronological attainability; adding the inherited surcharge prices the displayed primal, without claiming charged LP optimality. 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+ "precision, rounded cleanup, physical storage/tape and all-size native compilation", + "a compatible improved complex supplier and every joint outer inequality" + ] +} diff --git a/research/bit-copy-path/bounds/INDEPENDENT.json b/research/bit-copy-path/bounds/INDEPENDENT.json new file mode 100644 index 0000000000..d7613ea37c --- /dev/null +++ b/research/bit-copy-path/bounds/INDEPENDENT.json @@ -0,0 +1,226 @@ +{ + "algorithm": "positive binomial recurrence; integer outward rounding; next_term/(1-z) entire-tail bound", + "checks": [ + { + "charged_excess": { + "lower": "3535533957396076187744038403483738685089284890382427006689665161324427/5986310706507378352962293074805895248510699696029696000000000000000", + "upper": "1767766978698038093872019201741869342544642445191430795094051130029539/2993155353253689176481146537402947624255349848014848000000000000000" + }, + "label": "ideal0", + "root": { + "lower": "1743393258899/1000000000000000", + "lower_excess": { + "lower": 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counts", + "rejected": true + } + ], + "no_new_kappa": true, + "primary_receipt_sha256": "68a44707fdd2e66476119680ff7d3291643548b835f44f6384e9441591dac775", + "scope": "Independent moment arithmetic and full inventory specialization; not operator, owner chronology, native compiler, or joint assembly admission", + "status": "PASS_INDEPENDENT_COPY_BUDGET_BINOMIAL" +} diff --git a/research/bit-copy-path/bounds/PROOF.md b/research/bit-copy-path/bounds/PROOF.md new file mode 100644 index 0000000000..eca5e3a143 --- /dev/null +++ b/research/bit-copy-path/bounds/PROOF.md @@ -0,0 +1,477 @@ +# Exact COPY-path planning budgets + +Apache-2.0. Substantial OpenAI Codex assistance. + +The main arithmetic finding is conditional: deleting one rank-seven COPY +query in each of the 288 local invocations leaves more than +`4.977907425421188143929582511024` units of additional query moment per +invocation at the necessary primitive endpoint for a 10% κ improvement. +The existing coded-source word, with every deleted helper's owner +obligation returned, leaves more than +`58.382679011994524685794411232083` units per invocation. Those numbers +include the inherited child surcharge. + +The latter is an allowance for a **paid** decoder and all additional +recursive children, conditional on the stated owner relaxation. It is not +a zero-price decoder, an attained chronological caller, or a new κ. +The currently selected complex endpoint still precludes the 10% target. + +All displayed decimals denote exact terminating rationals. The complete +rational endpoints, adjacent signs and source hashes are in `BUDGET.json`. + +## 1. Fixed inputs and the target + +The preserved predecessor proves the complete ordinary BIT inventory, +its exact fractional allocation optimum and its all-parameter resource +corollary. We reuse those premises, not its obsolete fixed-word ceiling +after changing the word. In particular, + +\[ + m=64,\quad W=28416,\quad N=288,\quad + H=1322e_1+1342e_2+79e_3+39e_4+102e_5+11e_6+18e_7. +\] + +There are 433 whole rank-eight helpers and five whole rank-four helpers +per invocation. The nine private reserve families do not add to `W`. +The source controller's idle sequence is +`14,7,28,7,42,7,42,28,14,0`. Each event has `2*111*32` public families; +the positive-rank bin inventory is + +\[ +(C_7,C_{14},C_{28},C_{42})=(21312,14208,14208,14208). +\] + +The checker derives these bins from the pinned controller's literal +recipe and idle-insertion block, with its guards active under `python -O`. +It reconstructs both local histograms and helper terminal widths from +the actual records and rational frame ranks. This is cost-source +authentication; it does not reprove their arbitrary-input operators. + +Write + +\[ +\Phi_b(r)=r(64/r)^b,\quad +\tau_b=\frac{32\cdot64^2}{10^{16}}64^b,\quad +\chi_b(r)=\Phi_b(r)+\tau_b. +\] + +For the complete positive-rank histogram `G`, the inherited test is + +\[ +B_G(b)=\sum_rG(r)\chi_b(r)<64W=1818624. \tag{1} +\] + +Every local COPY and MOVE and every public idle completion is counted. +The inherited good/bad parameter is `epsilon=10^-16`. The raw bill omits +the positive surcharge; the even more favorable free-bad bill is +`(1-epsilon) sum G(r) Phi_b(r)`. Those variants are explicitly labelled. + +The selected exact value is + +\[ +\kappa_0=\frac{836330098538459}{500000000000000000}. +\] + +The retained outer bound `κ=9`, ranks in `{6,7,8}`, mass at least 63, is + +\[ +4\chi_b(6)+\chi_b(7)+4\chi_b(8). \tag{13} +\] + +Here is a written proof covering every child count. For `n=9` or 10, +an optimizer has mass exactly 63: otherwise a child above rank six can +be reduced by one until that mass is reached. Strict concavity gives +`Phi(6)+Phi(8)<2Phi(7)`, so the number of rank-seven children is +minimized, giving `4e6+e7+4e8` for nine children and `8e6+e7+e8` +for ten. The ten-child raw tariff exceeds (13) by +`4Phi(6)-3Phi(8)>0`, because `(8/6)^(1-b)<4/3`. +It also has one extra surcharge. + +For `n>=11`, its raw tariff is at least `11Phi(6)`. Since +`Phi(7)<(7/6)Phi(6)` and `Phi(8)<(8/6)Phi(6)`, +the right side of (13), without surcharge, is less than +`(4+39/6)Phi(6)=(21/2)Phi(6)<11Phi(6)`. +The surcharge comparison again favors nine children. This proves (13) +throughout `0=11` tail bound. + +Replacing the original nine rank-seven COPY children with the +optimistic profile in (13) changes each local histogram by + +\[ +4e_6-8e_7+4e_8. +\] + +It preserves rank mass and child count. Its raw saving per local call +at `c_*` lies in + +\[ +(0.001059310845595607293914001593,\ + 0.001059310845595607293914001594). +\] + +Even after applying that favorable change to all 288 calls and using +the optimal raw owner relaxation, the raw target excess lies in + +\[ +(590.298056111490115249043858634130,\ + 590.298056111490115249043858634131). +\] + +Granting every exceptional packet free leaves an excess greater than +`590.298056111308193819238247485118`. Therefore this entire direct +factorization class cannot reach the 10% target under the fixed helper, +public-bin and stock premises, even with no new decoder or other costs. +This obstruction is conditional on that class; it does not exclude a +changed factory, source endpoint, common cut, helper contract or caller. + +The receipt includes separate charged and free-bad root brackets. +The free-bad upper endpoint is the favorable necessary ceiling; a +charged displayed-primal root is not mistaken for a universally optimal +charged LP value. + +## 7. Covered domain, validation and unresolved implications + +Run from the checkout root: + +```sh +python3 -B research/bit-copy-path/bounds/check_budget.py +python3 -O -B research/bit-copy-path/bounds/check_budget.py --output /tmp/copy-budget-opt.json +python3 -B research/bit-copy-path/bounds/check_independent.py +``` + +`check_budget.py` uses pinned exact rational logarithm/exponential +enclosures with explicit tails. Favorable signs use upper moment bounds; +rejections use lower bounds. It includes strict adjacent root intervals +for the key displayed profiles, adjacent integer decoder/deletion signs, +all 64 LP patterns, source hashes and mutation controls. The independent +checker uses a positive binomial series and outward-rounded integer +arithmetic, without importing the primary arithmetic or budget checker. +These are exact finite arithmetic certificates and a written LP +specialization, not formal-kernel proofs. + +The selected complex endpoint is exactly +`67018507072324140917/40000000000000000000000`, which is below (2). +Its cap `c/(1+c)` is below the 10% κ target. The currently selected +assembly therefore does not become a 10% improvement through any of +these BIT-only substitutions. + +The construction specialist owns legal COPY removal/sharing, the +coded-source endpoint and a paid decoder at actual terminal caps. +The owner specialist owns chronological, physically disjoint whole +helper placement. Remaining global obligations include source access, +complete ADD/controller costs at their proper recurrence scale, integer +growth, precision and rounded cleanup, physical storage and tape +realization, all-size native compilation, an improved compatible complex +supplier, and every joint outer inequality. + +The prior immutable inputs retain their authorship and Apache-2.0 +notices: the R438 and five-terminal local constructions, the whole-helper +allocation proof, and `global_k.py` (PR386/PR436/PR439 origins and later +odd-stage replication). The coded-source proof and literal word are +copied byte-for-byte into `inputs/`, with their source hashes. This note +adds the changed-word exact budget specialization; it does not replace +those operator or allocation proofs. diff --git a/research/bit-copy-path/bounds/README.md b/research/bit-copy-path/bounds/README.md new file mode 100644 index 0000000000..1b2278c7fa --- /dev/null +++ b/research/bit-copy-path/bounds/README.md @@ -0,0 +1,46 @@ +# COPY-path complete-moment budgets + +This package prices proposed COPY deletions and the existing coded-source +word against the exact primitive target +`9199631083923049/4990800368916076951` necessary for a 10% gain over the +selected κ. It claims no new supplier admission or κ. + +- One ideal rank-seven COPY deletion per local call leaves more than + `4.977907425421188143929582511024` extra moment units per call. + Arbitrary global deletion needs at least 85 of the 2,592 COPY episodes + in the favorable allocation relaxation. +- The actual coded 429-helper word returns all 72 deleted-helper owner + rank units per call. Its net complete rank saving is 60 per call, + leaving more than `58.382679011994524685794411232083` extra moment + units per call. This can pay for 57 extra rank-one children per call; + an actual decoder and compatible caller remain required. +- Under the separately audited direct-reader premise of at least nine + COPY children, ranks six through eight, and mass at least 63, the best + possible COPY histogram is `4e6+e7+4e8`. The complete caller still + fails the target even with free bad packets. Its favorable primitive + ceiling is strictly below `0.001743442083496`, giving an outer gain + ceiling below `4.050289853587%`. The historical P111 mass floor is + credited to PR757/PR761; the complete changed-histogram consequence is + proved here. + +See [PROOF.md](PROOF.md) for the exact domains, all-parameter histogram +argument, source contracts, decoder allowances and unresolved obligations. +[BUDGET.json](BUDGET.json) contains exact rational enclosures and complete +histograms; [INDEPENDENT.json](INDEPENDENT.json) records the separate +positive-binomial arithmetic verification. + +From the repository root: + +```sh +python3 -B research/bit-copy-path/bounds/check_budget.py +python3 -O -B research/bit-copy-path/bounds/check_budget.py --output /tmp/copy-budget-opt.json +python3 -B research/bit-copy-path/bounds/check_independent.py +``` + +The primary and independent checks passed, including rejection controls. +The selected complex supplier remains the bottleneck. Fractional/integral +pattern counts alone do not prove chronological owners, and the coded +source endpoint requires a separately priced decoder. + +Apache-2.0. Substantial OpenAI Codex assistance. Prior source attribution +and immutable pins are preserved in the proof, receipt and `inputs/`. diff --git a/research/bit-copy-path/bounds/VALIDATION.json b/research/bit-copy-path/bounds/VALIDATION.json new file mode 100644 index 0000000000..36e6e0ff6a --- /dev/null +++ b/research/bit-copy-path/bounds/VALIDATION.json @@ -0,0 +1,39 @@ +{ + "commands": [ + { + "command": "python3 -B research/bit-copy-path/bounds/check_budget.py", + "exit_code": 0 + }, + { + "command": "python3 -O -B research/bit-copy-path/bounds/check_budget.py --output /tmp/copy-budget-opt.json", + "exit_code": 0 + }, + { + "command": "python3 -B research/bit-copy-path/bounds/check_independent.py", + "exit_code": 0 + } + ], + "documentation_only_adjustment": { + "change": "Removed one trailing LaTeX spacing escape flagged by git diff --check. No mathematical value, executable source or receipt changed.", + "final_export_replay_required": true, + "prior_proof_sha256": "41d868b71560c809af5a380e0b3457adff51439e2f08e3b7cc5dca2a763d2249" + }, + "head": "12fa8c5c844be21bfefb38a19503fbd867a11244", + "independent_negative_controls": 3, + "independent_target_and_root_rows": 16, + "no_new_kappa": true, + "optimized_receipt_byte_identical": true, + "primary_negative_controls": 6, + "python": "3.14.6", + "scope": "Exact arithmetic and complete inventory specialization only; no new operator, chronological allocation, native compiler, or joint outer admission.", + "sha256": { + "BUDGET.json": "68a44707fdd2e66476119680ff7d3291643548b835f44f6384e9441591dac775", + "INDEPENDENT.json": "f926f2fdd5839430f853d83fb677345055382c41aff9fea80c2056e87bf786ae", + "PROOF.md": "ab6b6c95c57b3a677f6285e888082dc0b92d8a48aa29a10328f88333adfabfa4", + "README.md": "87277e435b6787c29bae3d7a672d5e37f5d95628f508c17636ce458145a35b16", + "check_budget.py": "740337b2e73319a62500854a175417b067ad2898b8c576083b452b7eed2d01ad", + "check_independent.py": "9a8b5f4ab1ff6daa2cde7197372c0a66ca64cc0722d802c7248ded88572ea843" + }, + "status": "PASS_COPY_BUDGET_VALIDATION", + "utc": "2026-10-20T12:31:26Z" +} diff --git a/research/bit-copy-path/bounds/check_budget.py b/research/bit-copy-path/bounds/check_budget.py new file mode 100644 index 0000000000..72d6b3cde3 --- /dev/null +++ b/research/bit-copy-path/bounds/check_budget.py @@ -0,0 +1,583 @@ +#!/usr/bin/env python3 +# SPDX-License-Identifier: Apache-2.0 +"""Exact COPY deletion and source-code planning budgets, not supplier admission. + +Standard library only. All decisions use rational enclosures. Source and +historical allocation proof attribution are recorded in PROOF.md. +Substantial OpenAI Codex assistance. +""" +import argparse +import ast +from collections import Counter +from fractions import Fraction as Q +from functools import lru_cache +import gzip +import hashlib +import json +from math import prod +from pathlib import Path +import sys +import types + +HERE = Path(__file__).resolve().parent +PKG = HERE.parent +M, W, CALLS, REPLICAS = 64, 28416, 288, 32 +EPSILON = Q(1, 10**16) +SURCHARGE = 32 * M**2 * EPSILON +BIN_COUNTS = {7: 21312, 14: 14208, 28: 14208, 42: 14208} +H = {1: 1322, 2: 1342, 3: 79, 4: 39, 5: 102, 6: 11, 7: 18} +H429 = {1: 1298, 2: 1342, 3: 79, 4: 44, 5: 93, 6: 11, 7: 9} +SELECTED_KAPPA = Q(836330098538459, 500000000000000000) +TARGET_KAPPA = Q(11, 10) * SELECTED_KAPPA +TARGET = TARGET_KAPPA / (1 - TARGET_KAPPA) +PINS = { + "../inputs/WORD.json.gz": + "c46ccd7d3eedabc6a7efbb1a91ec13f2928bad8ee0bac57c6e593a092d13d96c", + "../inputs/GEOMETRY.json.gz": + "a353999bceac4ac957f540adfd3e5f8768297138b27132eeb83332e2ea4b1022", + "../inputs/global_k.py.txt": + "6368196a393ac25bb2918880e4c4bd31ed733b762c27b6873ab391cbc0c5c3c8", + "../inputs/FULL-CALLER-PROOF.md": + "ebdd9283a7762e01f84ee9dde8885d4ba05cc6c93a4430296ff0a71ea7205832", + "../inputs/full-check-screens.py.txt": + "c07d15a1e5edcbbab584fd77b26c2e6787c2baa4ef013de72dbfd52f5d2960e6", + "../inputs/full-exact-interval.py.txt": + "d8a78c2a2f87a5cf9fff0367d70f21dcdb712a509ba365879aee134b18c4c25d", + "../inputs/full-invoices.json": + "d0877305ca03e88622586fd76622fc9e2e2da0484de3b112577467c82f386a92", + "inputs/SOURCE-CODE-WORD.json.gz": + "1a6841fccba51fa922993c77b4b296e659c2c538252e5e06113d0438c5edb870", + "inputs/SOURCE-CODE-EXPECTED.json": + "f078222ab4e99aff085262c00ba77520cc62eb02003d8e0582ab6ad7dab7f402", + "inputs/SOURCE-CODE-PROOF.md": + "ebcc91e74938b525b28e46bf03ec09d42d24ab02070dc96f4f0a51cee2cfb83d", + "inputs/SELECTED-ASSEMBLY.json": + "1122cb3a575f8c788345fc1bf4c108ac7c3250fb81a5988671f5eb04b3ab3045", + "inputs/SELECTED-RECORD.json": + "bb2eaa58647b353f88d0aec2eafb9d30df1be53589231b23e12ceef3e4bad129", + "inputs/DIRECT-FLOOR-PRIOR-README.md": + "7c833024359891e15766be310c3d1a4579dd0231fe7ffdbf981fd0814bac2dfb", +} + + +def need(test, message): + if not test: + raise ValueError(message) + + +def digest(path): + return hashlib.sha256(path.read_bytes()).hexdigest() + + +# Compile these pinned source bytes explicitly, ignoring Python bytecode caches. +_interval_path = HERE / "../inputs/full-exact-interval.py.txt" +need(digest(_interval_path) == PINS["../inputs/full-exact-interval.py.txt"], + "pinned interval arithmetic source") +_interval = types.ModuleType("_copy_budget_exact") +exec(compile(_interval_path.read_bytes(), str(_interval_path), "exec", + dont_inherit=True, optimize=0), _interval.__dict__) +add, exact, linear_phi, mul, phi, power, report, sub = ( + getattr(_interval, name) for name in + ("add", "exact", "linear_phi", "mul", "phi", "power", "report", "sub")) + + +def combine(*terms): + result = Counter() + for factor, coefficients in terms: + for rank, count in coefficients.items(): + result[rank] += factor * Q(count) + return {r: Q(n) for r, n in sorted(result.items()) if n} + + +def validate_histogram(hist): + need(all(isinstance(r, int) and 0 < r < M and n >= 0 + for r, n in hist.items()), "contracting nonnegative histogram") + + +def idle_primal(full, partial): + """The prior all-parameter LP theorem, with the resource premises checked.""" + c7, c14, c28, c42 = (BIN_COUNTS[r] for r in (7, 14, 28, 42)) + need(0 <= partial <= c28, "partial helper inventory in LP regime") + need(c14 + 3*c28 <= full <= c14 + 3*c28 + 5*c42, + "full helper inventory in LP regime") + t = Q(full - c14 - 3*c28, 5) + patterns = {(7, 0, 0): Q(c7), (14, 1, 0): Q(c14), + (28, 3, 0): Q(c28-partial), (28, 3, 1): Q(partial), + (42, 5, 0): t, (42, 0, 0): c42-t} + for r, count in BIN_COUNTS.items(): + need(sum(n for (R, _, _), n in patterns.items() if R == r) == count, + "all public bins retained") + need(sum(a*n for (_, a, _), n in patterns.items()) == full, + "all full helpers retained") + need(sum(d*n for (_, _, d), n in patterns.items()) == partial, + "all partial helpers retained") + idle = Counter() + for (r, a, d), count in patterns.items(): + remainder = r - 8*a - 4*d + need(remainder >= 0 and count >= 0, "nonnegative whole pattern") + if remainder: + idle[remainder] += count + need(sum(r*n for r, n in idle.items()) == + sum(r*n for r, n in BIN_COUNTS.items()) - 8*full - 4*partial, + "returned owner rank mass") + return dict(sorted(idle.items())), [ + {"R": r, "full8": a, "partial4": d, "bins": str(n)} + for (r, a, d), n in sorted(patterns.items())] + + +def full_profile(local, full, partial): + idle, patterns = idle_primal(full, partial) + hist = combine((CALLS, local), (1, idle)) + validate_histogram(hist) + return hist, patterns + + +@lru_cache(None) +def surcharge_per_child(b): + return mul(power(M, b), SURCHARGE) + + +def chi(rank, b): + return add(phi(rank, b), surcharge_per_child(b)) + + +def bill(hist, b, charged=True): + result = linear_phi(hist, b) + if charged: + result = add(result, mul(surcharge_per_child(b), sum(hist.values()))) + return result + + +def excess(hist, b=TARGET, charged=True): + return sub(bill(hist, b, charged), exact(M*W)) + + +def quotient(numerator, denominator): + need(denominator[0] > 0, "positive interval denominator") + candidates = [x/y for x in numerator for y in denominator] + return min(candidates), max(candidates) + + +def strict_integer_cap(allowance, cost): + """Largest integer n with n*cost < allowance; adjacent signs are checked.""" + ratio = quotient(allowance, cost) + lo, hi = (x.numerator // x.denominator for x in ratio) + need(lo == hi, "integer cap isolated") + n = lo + need(sub(allowance, mul(cost, n))[0] > 0, "integer cap accepted") + need(sub(allowance, mul(cost, n+1))[1] < 0, "next integer rejected") + return {"maximum": n, "ratio": report(ratio), + "accepted_slack": report(sub(allowance, mul(cost, n))), + "next_slack": report(sub(allowance, mul(cost, n+1)))} + + +def root_bracket(hist, mode="charged"): + """Adjacent 10^-15 rational grid points; explicit opposite moment signs.""" + need(mode in ("charged", "raw", "free_bad"), "root tariff mode") + def evaluate(b): + if mode == "free_bad": + return sub(mul(bill(hist, b, False), 1-EPSILON), exact(M*W)) + return excess(hist, b, mode == "charged") + scale = 10**15 + lo, hi = 0, scale // 50 + need(evaluate(Q(lo, scale))[1] < 0 and + evaluate(Q(hi, scale))[0] > 0, "initial root bracket") + while hi-lo > 1: + mid = (lo+hi)//2 + interval = evaluate(Q(mid, scale)) + if interval[1] < 0: + lo = mid + elif interval[0] > 0: + hi = mid + else: + raise ValueError("grid root not separated by exact intervals") + return {"lower": str(Q(lo, scale)), "upper": str(Q(hi, scale)), + "mode": mode, + "lower_excess": report(evaluate(Q(lo, scale))), + "upper_excess": report(evaluate(Q(hi, scale)))} + + +def screen(hist, patterns=None, roots=False): + raw = excess(hist, charged=False) + charged = excess(hist) + free_bad = sub(mul(bill(hist, TARGET, False), 1-EPSILON), exact(M*W)) + allowance = mul(charged, -1) + row = {"H": {str(r): str(n) for r, n in hist.items()}, + "children": str(sum(hist.values())), + "rank_mass": str(sum(r*n for r, n in hist.items())), + "deficit": str(M*W-sum(r*n for r, n in hist.items())), + "raw_excess": report(raw), "charged_excess": report(charged), + "free_bad_excess": report(free_bad), + "charged_extra_moment_allowance_global": report(allowance), + "charged_extra_moment_allowance_per_call": + report(mul(allowance, Q(1, CALLS))), + "raw_extra_moment_allowance_per_call": + report(mul(raw, -Q(1, CALLS)))} + if patterns is not None: + row["primal_patterns"] = patterns + row["integral_pattern_counts"] = all(Q(p["bins"]).denominator == 1 + for p in patterns) + if allowance[0] > 0: + row["additional_equal_rank_children_per_call"] = { + str(r): strict_integer_cap(allowance, mul(chi(r, TARGET), CALLS)) + for r in (1, 2, 4, 7, 8, 9, 14, 28, 42, 63)} + row["arbitrary_extra_rank_mass_per_call"] = ( + row["additional_equal_rank_children_per_call"]["1"]) + row["one_decoder_per_replica_rank1_children"] = strict_integer_cap( + allowance, mul(chi(1, TARGET), REPLICAS)) + if roots: + row["charged_root_bracket"] = root_bracket(hist) + return row + + +def dual_controls(): + """All 64 feasible whole patterns, plus exact all-parameter endpoint signs.""" + l8 = {2: Q(1, 5), 42: -Q(1, 5)} + l4 = {4: Q(-1)} + mus = {7: {7: 1}, 14: combine((1, {6: 1}), (-1, l8)), + 28: combine((1, {4: 1}), (-3, l8)), 42: {42: 1}} + strict, equal = [], [] + for R in BIN_COUNTS: + for a in range(R//8+1): + for d in range((R-8*a)//4+1): + r = R-8*a-4*d + coeff = combine((1, {r: 1} if r else {}), + (-a, l8), (-d, l4), (-1, mus[R])) + if coeff: + slack = linear_phi(coeff, TARGET) + need(slack[0] > 0, "strict LP dual pattern slack") + strict.append({"pattern": [R, a, d], "slack": report(slack)}) + else: + equal.append([R, a, d]) + need(len(strict)+len(equal) == 64, "all feasible whole patterns") + endpoint_signs = [] + for terms in ([(14, 5), (2, 1), (6, -5), (42, -1)], + [(28, 5), (2, 3), (4, -5), (42, -3)]): + need(sum(c for _, c in terms) == sum(r*c for r, c in terms) == 0, + "four-power endpoint zeros") + def ratio(weights): + return Q(prod(r**c for r, c in weights if c > 0), + prod(r**(-c) for r, c in weights if c < 0)) + q0, q1 = ratio(terms), ratio([(r, r*c) for r, c in terms]) + need(q0 > 1 and q1 < 1, "four-power derivative endpoint signs") + endpoint_signs.append({"terms": terms, "exp_derivative_at_zero": str(q0), + "exp_derivative_at_one": str(q1)}) + return {"patterns": 64, "equalities": equal, "strict": strict, + "all_parameter_endpoint_products": endpoint_signs, + "written_all_parameter_proof": "../inputs/FULL-CALLER-PROOF.md"} + + +def direct_copy_floor(base, patterns, roots): + """Conditional on the construction audit's N>=9, ranks6..8, mass>=63.""" + optimal = {6: 4, 7: 1, 8: 4} + need(sum(optimal.values()) == 9 and sum(r*n for r, n in optimal.items()) == 63, + "direct factorization optimal relaxed COPY profile") + hist = combine((1, base), (CALLS, {6: 4, 7: -8, 8: 4})) + row = screen(hist, patterns, roots) + need(Q(row["free_bad_excess"]["lower"]) > 0, + "every direct mass63 shape fails target even with free bad packets") + competitors = [] + for n in (9, 10): + for n6 in range(n+1): + for n7 in range(n-n6+1): + n8 = n-n6-n7 + candidate = {r: c for r, c in ((6, n6), (7, n7), (8, n8)) if c} + if sum(r*c for r, c in candidate.items()) < 63: + continue + diff = combine((1, candidate), (-1, optimal)) + if not diff: + continue + slack = add(linear_phi(diff, TARGET), + mul(surcharge_per_child(TARGET), n-9)) + need(slack[0] > 0, "every nine/ten-child competitor dearer") + competitors.append({"H": candidate, "charged_slack": report(slack)}) + tail = sub(mul(chi(6, TARGET), 11), + add(linear_phi(optimal, TARGET), mul(surcharge_per_child(TARGET), 9))) + need(tail[0] > 0, "all eleven-or-more children dearer") + row["construction_premise"] = { + "minimum_child_count": 9, "ranks": [6, 7, 8], "minimum_rank_mass": 63, + "source": "separate actual source-query/direct-reader factorization audit", + "prior_63_floor_credit": "Russell Seligmann PR757 and Cody Harris PR761; " + "see inputs/DIRECT-FLOOR-PRIOR-README.md for the historical model", + "not_reproved_by_this_cost_checker": True} + row["optimal_COPY_histogram"] = optimal + row["all_nine_ten_child_competitors"] = competitors + row["eleven_or_more_lower_bound_slack"] = report(tail) + row["raw_saving_per_local_call"] = report( + linear_phi({6: -4, 7: 8, 8: -4}, TARGET)) + if roots: + row["free_bad_root_bracket"] = root_bracket(hist, "free_bad") + ceiling = Q(row["free_bad_root_bracket"]["upper"]) + row["strict_kappa_gain_ceiling_percent"] = str( + 100*(ceiling/(1+ceiling)/SELECTED_KAPPA-1)) + return row + + +def actual_bins(): + """Reuse the literal source recipe and idle block with active source guards.""" + tree = ast.parse((PKG/"inputs/global_k.py.txt").read_bytes()) + recipe = next(n for n in tree.body if isinstance(n, ast.FunctionDef) + and n.name == "recipe") + blocks = [n for n in ast.walk(tree) if isinstance(n, ast.If) and + ast.unparse(n.test) == "s % 2 == 1 and s <= 2 * K - 3"] + need(len(blocks) == 1, "unique source idle block") + class Guards(ast.NodeTransformer): + def visit_Assert(self, node): + return ast.copy_location(ast.Expr(value=ast.Call( + func=ast.Name(id="need", ctx=ast.Load()), + args=[node.test, ast.Constant(value="source controller guard")], + keywords=[])), node) + env = {"__builtins__": {"range": range, "dict": dict, "reversed": reversed}} + exec(compile(ast.fix_missing_locations(ast.Module( + body=[recipe], type_ignores=[])), "source-recipe", "exec", + dont_inherit=True, optimize=0), env) + block = ast.fix_missing_locations(Guards().visit(ast.Module( + body=[blocks[0]], type_ignores=[]))) + code = compile(block, "source-idle", "exec", dont_inherit=True, optimize=0) + levels, schedule, calls = [-1]*4, [], 0 + for event in env["recipe"](4): + if event["kind"] == "local": + s, p = event["stage"], event["pair"] + need(levels[p] == s-1, "source local entrance") + levels[p] = s + calls += REPLICAS + exec(code, {"__builtins__": {"range": range, "dict": dict}, + "need": need, "K": 4, "s": s, "levels": levels, + "schedule": schedule}) + elif event["kind"] == "bridge": + need(levels[event["dst"]] == levels[event["src"]], + "source bridge entrance") + else: + need(event["kind"] == "exchange" and levels == [8]*4, + "source terminal entrance") + ranks = [e["rank"] for e in schedule] + bins = Counter() + for r in ranks: + if r: + bins[r] += 2*111*REPLICAS + need(dict(bins) == BIN_COUNTS and calls == CALLS, "actual bins and calls") + need(W == 2*4*111*REPLICAS, "actual public stock, excluding private reserve") + return {"idle_ranks": ranks, "bins": dict(bins), "calls": calls, + "W": W, "private_reserve": 9, + "extraction_only_not_changed_source_admission": True} + + +def source_controls(): + for name, pin in PINS.items(): + need(digest(HERE/name) == pin, "source SHA-256: "+name) + geo = json.loads(gzip.decompress((PKG/"inputs/GEOMETRY.json.gz").read_bytes())) + need((geo["h"], geo["v"], geo["R"]) == (8, 111, 438), "source geometry") + @lru_cache(None) + def dim(f): + pivots = {} + for label in geo["frames"][f]: + row = list(map(Q, geo["lines"][label])) + for p, basis in sorted(pivots.items()): + a = row[p] + if a: + row = [x-a*y for x, y in zip(row, basis)] + pivot = next((j for j, x in enumerate(row) if x), None) + if pivot is not None: + a = row[pivot] + pivots[pivot] = [x/a for x in row] + if len(pivots) == 8: + break + return len(pivots) + out = {} + for name, path, want_h, widths, copy_ranks in ( + ("five_terminals", PKG/"inputs/WORD.json.gz", H, {4: 5, 8: 433}, [7]*9), + ("coded429", HERE/"inputs/SOURCE-CODE-WORD.json.gz", H429, {8: 429}, [1]*3)): + word = json.loads(gzip.decompress(path.read_bytes())) + need(all(f == 0 for f in word["start"][222:]), "zero-start helper frames") + got_widths = Counter(dim(f) for f in word["final"][222:]) + need(got_widths == widths, "literal terminal helper width inventory") + paid, copies, adds = Counter(), [], 0 + for op, dst, src, c, f, z in word["records"]: + if op == 0: + need(dim(c)-dim(src) == f >= 0, "actual MOVE rank") + if f: + paid[f] += 1 + elif op == 1: + adds += 1 + elif op == 2: + rank = dim(c)-dim(f) + need(rank > 0 and src == len(word["start"]), "actual COPY rank") + paid[rank] += 1 + copies.append(rank) + else: + need(op == 3, "record opcode") + need(dict(paid) == want_h and copies == copy_ranks, + "complete literal local query histogram") + out[name] = {"H": dict(sorted(paid.items())), "helper_widths": dict(got_widths), + "COPY_ranks": copies, "ADDs": adds, + "children": sum(paid.values()), + "rank_mass": sum(r*n for r, n in paid.items())} + expected = json.loads((HERE/"inputs/SOURCE-CODE-EXPECTED.json").read_bytes()) + need(expected["dirty_helpers"] == 429 and expected["rank_mass"] == 4989, + "coded historical expected contract") + need(expected["deleted_original_families"] == list(range(471, 480)), + "nine deleted historical helper families") + return {"pins": PINS, "literal_profiles": out, "controller": actual_bins(), + "does_not_reprove_arbitrary_input_operator_or_restoration": True} + + +def rejection(label, action): + try: + action() + except ValueError: + return {"control": label, "rejected": True} + raise ValueError("negative control escaped: "+label) + + +def main(): + ap = argparse.ArgumentParser() + ap.add_argument("--output", type=Path, default=HERE/"BUDGET.json") + ap.add_argument("--skip-roots", action="store_true") + args = ap.parse_args() + sources = source_controls() + need(TARGET == Q(9199631083923049, 4990800368916076951), + "exact necessary 10 percent primitive target") + base, patterns = full_profile(H, 433*CALLS, 5*CALLS) + need(sum(r*n for r, n in base.items()) == 1808352, "baseline full mass") + need(excess(base, charged=False)[0] > 0, "zero deletion rejects 10 percent") + ideal = {} + for k in range(10): + hist = combine((1, base), (-k*CALLS, {7: 1})) + validate_histogram(hist) + ideal[str(k)] = screen(hist, patterns, not args.skip_roots and k in (0, 1, 9)) + need((excess(hist)[1] < 0) == (k >= 1), "uniform deletion threshold") + # Arbitrary deletion locations, before legal sharing/chronology are proved. + deletion_ratio = quotient(excess(base), chi(7, TARGET)) + minimum = deletion_ratio[1].numerator // deletion_ratio[1].denominator + 1 + need(minimum == 85, "total COPY deletion threshold") + arbitrary, arbitrary_raw, arbitrary_free_bad = {}, {}, {} + for count in (minimum-1, minimum): + hist = combine((1, base), (-count, {7: 1})) + e = excess(hist) + need(e[0] > 0 if count == minimum-1 else e[1] < 0, + "adjacent total COPY deletion signs") + arbitrary[str(count)] = report(e) + raw = excess(hist, charged=False) + free_bad = sub(mul(bill(hist, TARGET, False), 1-EPSILON), exact(M*W)) + arbitrary_raw[str(count)] = report(raw) + arbitrary_free_bad[str(count)] = report(free_bad) + if count == minimum-1: + need(free_bad[0] > 0, "84 deletions fail even the raw LP free-bad lower bound") + # Existing ordinary chronological invoice, held fixed under ideal deletions. + invoices = json.loads((PKG/"inputs/full-invoices.json").read_bytes()) + literal = {int(r): Q(n) for r, n in + invoices["profiles"]["reuse/reference"]["H"].items()} + need(sum(r*n for r, n in literal.items()) == 1808352, "literal reference mass") + literal_one = screen(combine((1, literal), (-CALLS, {7: 1}))) + need(excess(combine((1, literal), (-CALLS, {7: 1})))[1] < 0, + "one deletion on fixed constructed owner invoice") + # Coded word: 429 full helpers; do not pretend five are already partial. + coded, coded_patterns = full_profile(H429, 429*CALLS, 0) + old_allfull, _ = full_profile(combine((1, H), (5, {4: 1})), 438*CALLS, 0) + old_idle, _ = idle_primal(438*CALLS, 0) + coded_idle, _ = idle_primal(429*CALLS, 0) + owner_return = combine((1, coded_idle), (-1, old_idle)) + need(owner_return == {2: -Q(2592, 5), 42: Q(2592, 5)}, + "every removed helper returns its entire owner obligation") + need(sum(r*n for r, n in owner_return.items()) == 72*CALLS, + "72 owner rank units returned per call") + need(sum(r*n for r, n in coded.items()) == 1791072 and + sum(r*n for r, n in old_allfull.items()) == 1808352, + "net coded saving exactly 60 per call") + combined, combined_patterns = full_profile( + combine((1, H429), (-5, {4: 1})), 424*CALLS, 5*CALLS) + need(sum(r*n for r, n in combined.items()) == 1791072, + "hypothetical combined profile retains same complete mass") + coded_rows = { + "coded429_full": screen(coded, coded_patterns, not args.skip_roots), + "coded424_full5_partial_if_terminal_adaptation_proved": + screen(combined, combined_patterns, not args.skip_roots)} + improvement = sub(bill(coded, TARGET), bill(combined, TARGET)) + need(improvement[0] > 0, "conditional terminal adaptation favorable") + # Pure deletions only: mass efficiency ranges from chi(63)/63 to chi(1). + mass_best = quotient(excess(base), chi(1, TARGET)) + mass_worst = quotient(excess(base), mul(chi(63, TARGET), Q(1, 63))) + need(mass_best[0] > 586 and mass_best[1] < 587, "best pure mass threshold") + need(mass_worst[0] > 590 and mass_worst[1] < 591, "uniform pure mass threshold") + need(excess(combine((1, base), (-587, {1: 1})))[1] < 0, + "587 rank-one deletions meet the arithmetic screen") + selected = json.loads((HERE/"inputs/SELECTED-ASSEMBLY.json").read_bytes()) + need(Q(selected["kappa"]) == SELECTED_KAPPA and selected["binding"] == "complex", + "selected assembly exact binding") + complex_c = Q(selected["complex_lower"]) + need(complex_c < TARGET and complex_c/(1+complex_c) < TARGET_KAPPA, + "selected complex endpoint cannot support target") + controls = [ + rejection("missing_owner_return72", lambda: need( + sum(r*n for r, n in combine((CALLS, H429), (1, old_idle)).items()) + == 1791072, "owner return required")), + rejection("429_full_plus5_partial_without_inventory_change", + lambda: need(429+5 == expected_helpers(), "physical helper count")), + rejection("partial_progress_without_removing5_rank4_moves", + lambda: need(sum(r*n for r, n in full_profile( + H429, 424*CALLS, 5*CALLS)[0].items()) == 1791072, + "changed terminal local tariff")), + rejection("rank_mass_only_substitution", + lambda: need(linear_phi({1: 1, 2: -2, 3: 1}, TARGET)[1] >= 0, + "equal mass need not have equal moment")), + rejection("extra_private_reserve_output_credit", + lambda: need(W+9 == 2*4*111*REPLICAS, "private reserve is not public stock")), + rejection("out_of_regime_helper_inventory", + lambda: idle_primal(0, 0)), + ] + receipt = { + "status": "PASS_EXACT_COPY_PATH_BUDGETS", "no_new_kappa": True, + "scientific_scope": "Conditional complete query-moment planning. LP lower " + "bounds do not assert chronological attainability; adding the inherited " + "surcharge prices the displayed primal, without claiming charged LP " + "optimality. Coded source needs a new caller/decoder contract.", + "sources": sources, "target_primitive": str(TARGET), + "target_kappa": str(TARGET_KAPPA), + "selected_kappa": str(SELECTED_KAPPA), + "inherited_per_child_surcharge_coefficient": str(SURCHARGE), + "LP_dual": dual_controls(), "ideal_uniform_COPY7_deletions": ideal, + "direct_query_mass63_lower_bound": direct_copy_floor( + base, patterns, not args.skip_roots), + "minimum_uniform_deletions_per_call": 1, + "minimum_arbitrary_global_COPY7_deletions": minimum, + "arbitrary_global_deletion_ratio": report(deletion_ratio), + "adjacent_global_deletion_excess": arbitrary, + "adjacent_global_deletion_raw_excess": arbitrary_raw, + "adjacent_global_deletion_free_bad_excess": arbitrary_free_bad, + "fixed_constructed_owner_invoice_one_deletion": literal_one, + "pure_deletion_mass": { + "premise": "Only paid children removed, all ranks 1..63, no added costs", + "necessary_best_efficiency_mass": report(mass_best), + "sufficient_all_rank_shapes_mass": report(mass_worst), + "minimum_attainable_mass_in_relaxation": 587, + "uniform_sufficient_integer_mass": 591, + "COPY7_only_integer_mass": minimum*7, + "warning": "Net mass with both additions and deletions does not determine moment"}, + "coded_owner_return_histogram": {str(r): str(n) for r, n in owner_return.items()}, + "coded_owner_return_mass_per_call": 72, + "coded_net_complete_mass_saving_per_call": 60, + "coded_profiles": coded_rows, + "conditional_five_terminal_charged_improvement_global": report(improvement), + "selected_complex_still_blocks_10_percent": True, + "selected_complex_exact_endpoint": str(complex_c), + "negative_controls": controls, + "unresolved": [ + "legal sharing/deletion word with arbitrary dirty inputs and literal restoration", + "coded-source caller and a paid terminal decoder at actual caps", + "chronological integral whole-helper owners and changed source access", + "precision, rounded cleanup, physical storage/tape and all-size native compilation", + "a compatible improved complex supplier and every joint outer inequality"], + } + args.output.parent.mkdir(parents=True, exist_ok=True) + args.output.write_text(json.dumps(receipt, indent=2, sort_keys=True)+"\n") + print("PASS_EXACT_COPY_PATH_BUDGETS no_new_kappa") + print("Minimum ideal uniform COPY7 deletions: 1; arbitrary total deletions: 85") + print("Coded owner return: 72 rank units/call; net complete saving: 60/call") + + +def expected_helpers(): + return 429 + + +if __name__ == "__main__": + main() diff --git a/research/bit-copy-path/bounds/check_independent.py b/research/bit-copy-path/bounds/check_independent.py new file mode 100644 index 0000000000..2e922b9fe7 --- /dev/null +++ b/research/bit-copy-path/bounds/check_independent.py @@ -0,0 +1,253 @@ +#!/usr/bin/env python3 +# SPDX-License-Identifier: Apache-2.0 +"""Independent integer-rounded binomial checks for the COPY planning receipt. + +No import of check_budget.py or the logarithm/exponential interval engine. +For 0= 0 else (x[1]*c, x[0]*c) + + +def charged_excess(hist, b): + total = (Q(0), Q(0)) + for rank, count in hist.items(): + total = arithmetic_add(total, arithmetic_scale( + power_64_over_rank(rank, b), rank*count)) + total = arithmetic_add(total, arithmetic_scale( + power_64_over_rank(1, b), SURCHARGE*sum(hist.values()))) + return total[0]-WIDTH*STOCK, total[1]-WIDTH*STOCK + + +def free_bad_excess(hist, b): + total = (Q(0), Q(0)) + for rank, count in hist.items(): + total = arithmetic_add(total, arithmetic_scale( + power_64_over_rank(rank, b), rank*count)) + total = arithmetic_scale(total, 1-Q(1, 10**16)) + return total[0]-WIDTH*STOCK, total[1]-WIDTH*STOCK + + +def expected_profile(local, full_per_call, partial_per_call): + """Reconstruct all resource totals independently of the receipt.""" + F, D = full_per_call*CALLS, partial_per_call*CALLS + t = Q(F-56832, 5) + require(0 <= D <= 14208 and 0 <= t <= 14208, "resource regime") + out = {r: Q(n*CALLS) for r, n in local.items()} + for rank, count in ((7, 21312), (6, 14208), (4, 14208-D), + (2, t), (42, 14208-t)): + out[rank] = out.get(rank, 0)+count + return {r: n for r, n in out.items() if n} + + +def subtract_copies(hist, number): + result = hist.copy() + result[7] -= number + require(result[7] >= 0, "COPY inventory") + return result + + +def parse_hist(row): + return {int(r): Q(n) for r, n in row["H"].items()} + + +def intervals_meet(x, serialized): + y = Q(serialized["lower"]), Q(serialized["upper"]) + require(x[0] <= y[1] and y[0] <= x[1], "independent interval intersects primary") + + +def fraction_pair(x): + return {"lower": str(x[0]), "upper": str(x[1])} + + +def verify_row(label, row, expected): + require(parse_hist(row) == expected, label+": full histogram") + require(Q(row["children"]) == sum(expected.values()), label+": child count") + require(Q(row["rank_mass"]) == sum(r*n for r, n in expected.items()), + label+": rank mass") + interval = charged_excess(expected, TARGET) + intervals_meet(interval, row["charged_excess"]) + favorable = interval[1] < 0 + require(favorable or interval[0] > 0, label+": strict target sign") + roots = row.get("charged_root_bracket") + result = {"label": label, "charged_excess": fraction_pair(interval), + "target_favorable": favorable} + if roots: + low, high = Q(roots["lower"]), Q(roots["upper"]) + require(high-low == Q(1, 10**15), label+": adjacent root grid") + lower_sign, upper_sign = charged_excess(expected, low), charged_excess(expected, high) + require(lower_sign[1] < 0 < upper_sign[0], label+": independent root signs") + intervals_meet(lower_sign, roots["lower_excess"]) + intervals_meet(upper_sign, roots["upper_excess"]) + result["root"] = {"lower": str(low), "upper": str(high), + "lower_excess": fraction_pair(lower_sign), + "upper_excess": fraction_pair(upper_sign)} + if favorable: + extra = row["additional_equal_rank_children_per_call"] + for rank_str, cap in extra.items(): + rank, n = int(rank_str), cap["maximum"] + for count, accept in ((n, True), (n+1, False)): + augmented = expected.copy() + augmented[rank] = augmented.get(rank, 0)+CALLS*count + e = charged_excess(augmented, TARGET) + require(e[1] < 0 if accept else e[0] > 0, + label+": independent adjacent decoder count") + return result + + +def verify(receipt): + require(Q(receipt["target_primitive"]) == TARGET, "same exact target") + require(receipt["minimum_uniform_deletions_per_call"] == 1 and + receipt["minimum_arbitrary_global_COPY7_deletions"] == 85, + "declared minimum counts") + base = expected_profile(LOCAL, 433, 5) + out = [] + for k in range(10): + row = receipt["ideal_uniform_COPY7_deletions"][str(k)] + result = verify_row("ideal"+str(k), row, subtract_copies(base, CALLS*k)) + require(result["target_favorable"] == (k > 0), "uniform deletion boundary") + out.append(result) + for t, accept in ((84, False), (85, True)): + interval = charged_excess(subtract_copies(base, t), TARGET) + require(interval[1] < 0 if accept else interval[0] > 0, + "global deletion boundary") + intervals_meet(interval, receipt["adjacent_global_deletion_excess"][str(t)]) + free_bad = free_bad_excess(subtract_copies(base, t), TARGET) + intervals_meet(free_bad, receipt["adjacent_global_deletion_free_bad_excess"][str(t)]) + if t == 84: + require(free_bad[0] > 0, "84 deletions fail the favorable LP lower bound") + out.append({"label": "global_deleted"+str(t), "charged_excess": fraction_pair(interval)}) + coded = expected_profile(CODED, 429, 0) + out.append(verify_row("coded429", receipt["coded_profiles"]["coded429_full"], coded)) + mixed_local = CODED.copy() + mixed_local[4] -= 5 + mixed = expected_profile(mixed_local, 424, 5) + out.append(verify_row("coded424plus5", receipt["coded_profiles"][ + "coded424_full5_partial_if_terminal_adaptation_proved"], mixed)) + require(sum(r*n for r, n in coded.items()) == + sum(r*n for r, n in mixed.items()) == 1791072, + "returned owner obligations included") + direct = base.copy() + for r, n in ((6, 4), (7, -8), (8, 4)): + direct[r] = direct.get(r, 0)+CALLS*n + direct_row = receipt["direct_query_mass63_lower_bound"] + out.append(verify_row("direct_mass63_best_shape", direct_row, direct)) + free_bad = free_bad_excess(direct, TARGET) + require(free_bad[0] > 0, "direct COPY class fails even with free bad packets") + intervals_meet(free_bad, direct_row["free_bad_excess"]) + roots = direct_row.get("free_bad_root_bracket") + if roots: + lo, hi = Q(roots["lower"]), Q(roots["upper"]) + ls, hs = free_bad_excess(direct, lo), free_bad_excess(direct, hi) + require(hi-lo == Q(1, 10**15) and ls[1] < 0 < hs[0], + "direct free-bad ceiling interval") + intervals_meet(ls, roots["lower_excess"]) + intervals_meet(hs, roots["upper_excess"]) + # Fixed constructed owner invoice is independently read from its pinned source. + invoice_path = HERE/"../inputs/full-invoices.json" + require(hashlib.sha256(invoice_path.read_bytes()).hexdigest() == + "d0877305ca03e88622586fd76622fc9e2e2da0484de3b112577467c82f386a92", + "existing full invoice source") + invoices = json.loads(invoice_path.read_bytes()) + literal = {int(r): Q(n) for r, n in + invoices["profiles"]["reuse/reference"]["H"].items()} + out.append(verify_row("literal_owner_one_deletion", + receipt["fixed_constructed_owner_invoice_one_deletion"], + subtract_copies(literal, CALLS))) + return out + + +def main(): + ap = argparse.ArgumentParser() + ap.add_argument("--input", type=Path, default=HERE/"BUDGET.json") + ap.add_argument("--output", type=Path, default=HERE/"INDEPENDENT.json") + args = ap.parse_args() + receipt = json.loads(args.input.read_bytes()) + checks = verify(receipt) + controls = [] + # Real receipt corruptions: lost owner mass, a falsely free next decoder, + # and pretending the 84-deletion rejection already passes. + mutants = [] + x = json.loads(args.input.read_bytes()) + x["coded_profiles"]["coded429_full"]["H"]["42"] = "1728/5" + mutants.append(("lost_returned_owner_mass", x)) + x = json.loads(args.input.read_bytes()) + x["coded_profiles"]["coded429_full"]["additional_equal_rank_children_per_call"]["1"]["maximum"] = 58 + mutants.append(("unaffordable_rank1_decoder", x)) + x = json.loads(args.input.read_bytes()) + x["minimum_arbitrary_global_COPY7_deletions"] = 84 + mutants.append(("one_fewer_COPY_deletion", x)) + for label, mutant in mutants: + try: + verify(mutant) + except ValueError as error: + controls.append({"mutation": label, "rejected": True, "reason": str(error)}) + else: + raise ValueError("mutation escaped: "+label) + output = {"status": "PASS_INDEPENDENT_COPY_BUDGET_BINOMIAL", + "no_new_kappa": True, + "primary_receipt_sha256": hashlib.sha256(args.input.read_bytes()).hexdigest(), + "algorithm": "positive binomial recurrence; integer outward rounding; " + "next_term/(1-z) entire-tail bound", + "dyadic_bits": 192, "included_terms_after_constant": TERMS, + "checks": checks, "negative_controls": controls, + "scope": "Independent moment arithmetic and full inventory specialization; " + "not operator, owner chronology, native compiler, or joint assembly admission"} + args.output.write_text(json.dumps(output, indent=2, sort_keys=True)+"\n") + print("PASS_INDEPENDENT_COPY_BUDGET_BINOMIAL no_new_kappa") + + +if __name__ == "__main__": + main() diff --git a/research/bit-copy-path/bounds/inputs/DIRECT-FLOOR-PRIOR-README.md b/research/bit-copy-path/bounds/inputs/DIRECT-FLOOR-PRIOR-README.md new file mode 100644 index 0000000000..24a571009d --- /dev/null +++ b/research/bit-copy-path/bounds/inputs/DIRECT-FLOOR-PRIOR-README.md @@ -0,0 +1,129 @@ +# Bit-side orthogonal-pair decoders: pairing cost 54 on 102 ports and a 9/17 floor (no new κ) + +**Status: research note with an exact decoder construction and a scoped lower bound. No κ claim and no +new supplier.** The selected κ = 0.001661960547713947 (PR942, T37) and every existing file are unchanged. + +## What this establishes + +The bit unit's copy charge is 63 on 111 ports (x = 63/111). PR757 and PR761 prove that 63 is optimal +for the shipped P111 puncture in the label-decoder model priced by PR446 Theorem C (rational pairing +rank). PR620 searched deeper punctures only under star pricing, and PR595 lists a bit unit with +x = 54/102 as one that "no word realises". This package works in the same Theorem C model on other +punctures. + +1. **Mechanism.** A term (λ_a, λ_b) whose reader functional is the label of punctured line a and whose + source functional is the label of punctured line b costs 6 when a ⊥ b, and 7 when a = b or + ⟨a, b⟩ = ±1. Its reader lines span a^⊥ and its source lines span b^⊥, and the pairing between those + hyperplanes has rank 7 − [a ⊥ b]. In F2 terms, the labels must form dual bases: B0[a_j, b_k] = δ_jk, + which means a_j ⊥ b_j and ⟨a_j, b_k⟩ = ±1 for j ≠ k. +2. **Construction.** Nine such pairs exist (`WITNESS.json`). Puncturing their 18 lines gives an exact + decoder of cost **54 on 102 ports, x = 9/17 = 0.5294**, below 63/111 = 0.5676 and below the complex + frame-pair ratio 7/13 of PR968. The decoder identity is checked on all 102 × 102 live pairs. +3. **Floor (finite census plus written proof).** For every puncture of at most 25 lines and every label + decoder with any number of terms, cst/v ≥ 9/17. Equality holds only for p = 18 with nine + orthogonal punctured pairs. Nine-term decoders satisfy cst ≥ max(72 − p, 54), and this is attained + at every p ≤ 25. The proof uses three thresholds: a reader span drops to a hyperplane with 1 puncture + (its own root) or 14 (a D7 normal), and to rank 6 only with 26. +4. **The selected bit supplier.** Keeping the nine reader functionals fixed, exactly one source dual + is a root label. Puncturing it gives **P110 with cost 62** (8 × 7 + 6). The necessary capacity + inequality is 439 ≤ 440 at v = 110, not a verified owner map. Other reader bases on the same + puncture can contain multiple cost-6 terms: a checked two-pair completion costs 68 overall on P111, + so it does not improve the cost. At v = 102 the fixed R439 inventory exceeds 408 slots. +5. **Favorable pricing.** With the engines pinned by the selected record: + - lowering the copy children 7 → 6 on all 288 calls is worth +2.8017% c per unit and +25.2108% c for + nine units, with ports fixed; + - PR717 records −2.13% to −2.16% c per deleted port (not reproduced here), so one pair on the shipped + puncture is worth roughly +0.6% c; + - κ is complex-bound, so a bit-only gain leaves it unchanged; + - PR968's P104 complex budget is capped by PR589 at κ = 0.001730427469823602. Lifted above that cap, + the same budget gives **κ = 0.001753391564497665** (+5.50%), which requires the bit c to rise by at + least **1.3294%**. One shipped pair falls short of that in this estimate. + +## Reproduce + +From the repository root, with Python 3.10+ and the standard library only (assertions enabled; the +verifier refuses `-O` because the pinned engines signal rejection with `assert`): + +```sh +python3 -I -B research/bit-orthogonal-pair-decoders/verify.py +python3 -I -B research/bit-orthogonal-pair-decoders/independent_check.py +python3 -I -B research/bit-orthogonal-pair-decoders/test_package.py +``` + +Expected final lines (about 3 s, 2 s and 15 s): + +```text +PASS_BIT_ORTHOGONAL_PAIR_DECODERS: code 1+255z^56+255z^64+z^120; thresholds 1/14/theta6=26; witness P102 cst 54 (9/17); chain p=0..25; shipped P111 -> P110 cst 62 (1 dual root, wall 440>=439); copy knob 9 units 25.21084% c; P104 uncapped kappa 0.001753391564497665; controls 6; no new kappa +PASS_BIT_PAIR_INDEPENDENT: membership roots, basis-coordinate functionals, Bareiss ranks; chain p=0..25; thresholds 1/14/26; shipped P110 cst 62 +PASS_PACKAGE_TESTS 8 +``` + +`verify.py --out NEW.json` writes the receipt; `RESULTS.json` is that receipt, and the package test +checks that it regenerates byte for byte. + +## What the checks cover + +| section | content | +|---|---| +| A | 120 lines; B0 = (1 + G) mod 2 has F2 rank 9; reader code 1 + 255z^56 + 255z^64 + z^120; root functionals; all 120 reflections preserve the code | +| B | 8 hyperplane types × 511 functionals; all 120 root hyperplanes directly; all 1080 D7 directions (unique 14-line minimizer, next ≥ 34, pairwise unions ≥ 21); 28 standard rank-6 subspaces (θ6 = 26) | +| C | for one root and all 119 partners: cost 6 (63 orthogonal), 7 (56 odd), 7 (self) | +| D | the witness: F2 duality, entrywise decoder identity on 102 live lines, spans and costs | +| E | explicit decoders for p = 0..25 with cst = max(72 − p, 54) | +| F | the integer inequalities of the ratio theorem and the comparisons with 63/111 and 7/13 | +| G | the PR589 word (pinned file and content hashes): 44 orthogonal ordered pairs, one dual root with readers fixed, P111 cost 63, P110 cost 62, necessary capacity screen; a changed-reader two-pair control costing 68/67 | +| H | pricing pins; PR589 c reproduced exactly; copy knob with two moment engines; bit-only κ unchanged; PR968 P104 capped and uncapped | +| I | controls: swapped partners, a restored partner (cost 55), an odd partner, θ6 sharpness at 26, a D7 reduction, a corrupted B0 pair | +| X | `independent_check.py`: membership-filter roots, basis-coordinate functionals, Bareiss integer ranks, its own simple system, D7 as norm-4 lattice vectors, the shipped P110 | + +The written argument is in [PROOF.md](PROOF.md). It cites one classical premise: every root-spanned +subspace is W-conjugate to a standard parabolic one, as in PR939, PR954 and PR968. + +## Scope and inherited obligations + +- **Covered.** Label decoders of the fixed matrix B0 on the live lines, any number of terms, punctures of + at most 25 lines, cost = rational pairing rank. +- **Not covered.** Decoders that differ from B0 on orthogonal pairs, non-label factors, punctures of 26 or + more lines, non-root ports. For n = 10 the bounds are 64 at p = 7, 62 at p = 8, and 60 for p = 9..11. +- **Not a construction of a word.** A cost-6 term needs an asymmetric copy from the span b^⊥ to the span + a^⊥. PR620 lists the open realization questions: reader frame chains, a nonzero terminal frame with + restoration, and the common cut. The pricing is a favorable budget; the port price is a recorded PR717 + number. A new owner map must be supplied; the aggregate capacity test alone does not prove hosting. +- **Inherited conditions.** All conditions of the selected record remain: all-size compiler, common-label + and mutable-source realization, whole-call hosting, precision/tape, analytic transfer and the bit + supplier's H1-H5 interface. +- **Search scope.** The witness is the first system reached by backtracking with one ordered orthogonal + pair fixed (W(E8) is transitive on such pairs). A full count of nine-pair systems was started and + stopped after 15 minutes without a result; no census of these systems is claimed. +- Finite replay is not a formal proof of the multiplication theorem, and the written proof has not had + independent human review. + +## Provenance, credit and overlap + +- **Base and assignment.** Base commit `2921874911956718fcaaaf9396fe8267b3cfa9f0`. Sprint track 40-49 + (structural decoder research), assignment seed **40** from OS randomness. See [CLAIM.md](CLAIM.md); no + pinned coordination issue existed, so the claim is local. +- **Model.** Joel Pulikkan / GamingPuzzled (PR620, preliminary content retained as `research/e8-decoder-floors-and-cold14-notes`) + defined the Theorem C label-decoder floor on the bit side and observed cost-6 orthogonal punctured + pairs at P111. DreamingOfClouds (PR446 Theorem C; PR595 format limits, deficit identity and hosting + wall; PR717 P2 bit invoice notes at head `4a97676`; PR589 bit supplier) supplied the copy-cost premise, + the accounting and the priced word. +- **P111 floor.** Russell Seligmann (PR757, head `e2ec7d7`) and Cody Harris (PR761, head `2877587`) prove + 63 at P111 in this class; part (b) of the theorem reproduces their nine-term value at p = 9 by a + different, puncture-independent argument, and their ten-term bound is stronger at P111. +- **Complex analogue.** themilkman216 (PR968, `research/complex-frame-pair-decoders`) gave the complex + frame-pair floor 7/13 and the P104 knob scenario re-implemented in section H. The violator census of + section B has the same numbers as PR968's table, since |R(λ) \ ω^⊥| = |Viol(λ) ∩ C(ω)|. +- **Star model.** Chi Zhao (PR631), Ross Richardson (PR939, PR954) and Philipp (@mightyOil, PR875) bound + the star-priced ratio; star pricing charges 7 for both self and orthogonal pairs, so their results and + this one concern different charges. +- **Selected chain.** Jonathan Capalbo (PR942), Chafik Boukhalfa (PR890), Jeremy Kothe (PR869/870), Hao + Yan (PR858), DreamingOfClouds (PR589) and eumemic (PR555); this package only reads their pinned inputs. +- **Overlap check.** The full refresh covered 757 open PRs through #1062, including #1060 (a finite + complex P111 COPY construction with its own extra costs, not a bit supplier), #1020 (complex + F2 label-pairing ranks at 26 punctures), #1008 (tensor shears on fixed P111), #998 (complex single + lifts), #966 (prescribed star triples), #912 (Hurwitz routing) and #757/#761 (fixed P111). These + scopes differ from rational pairing ranks on other bit-side punctures. +- **AI assistance and license.** Prepared for @navinor with substantial AI assistance + through OpenCode in research, proof, both implementations and documentation; resumed review and + packaging used GPT-6.1 Sol (OpenAI). Apache-2.0. 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admission, moment and outer checks; PR559 exact-cover recurrence correction only, bit bad-residue surcharge and H1-H5 retained. The new source-branch gauge and retired-basis completion, the gauge44-barch U8 local word and all 72 cold helper ends, and explicit all144 whole-owner dispatch with a two-copy remainder trade are composed with the cited unchanged merged PR589 bit supplier (PR555 punctured R439 word; derived bit deficit T(2Kv-L*cst)), replacing the PR543 supplier of PR635; This is a new T16 composition of the accepted PR799 and PR589 suppliers. All-size compiler, common-label/mutable-source realization, whole-call hosting, precision/tape and analytic transfer remain assumptions. No unconditional theorem or full Lean certification." +} diff --git a/research/bit-copy-path/bounds/inputs/SELECTED-RECORD.json b/research/bit-copy-path/bounds/inputs/SELECTED-RECORD.json new file mode 100644 index 0000000000..b1fea0c944 --- /dev/null +++ b/research/bit-copy-path/bounds/inputs/SELECTED-RECORD.json @@ -0,0 +1,222 @@ +{ + "status": "Maintainer-reviewed conditional finite construction; see the integration receipt for publication and PR resolution status", + "kappa": "836330098538459/500000000000000000", + "decimal": "0.001672660197076918", + "source_prs": [ + 1094, + 1087, + 1082, + 1074, + 1068, + 1065, + 1063, + 1061, + 1056, + 1055, + 1053, + 1050, + 1042, + 1031, + 1028, + 1022, + 1018, + 1005, + 1004, + 987, + 942, + 890, + 870, + 869, + 858, + 799, + 741, + 699, + 685, + 676, + 664, + 660, + 659, + 656, + 655, + 653, + 658, + 649, + 645, + 635, + 589, + 555, + 545, + 626, + 623, + 622, + 620, + 617, + 615, + 604, + 601, + 596, + 592, + 591, + 593, + 599, + 588, + 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Not a full Lean multiplication theorem.", + "previous_kappa": "1672188119205421/1000000000000000000", + "previous_record": "certificates/checkpoints/frontier1042.json", + "audited_source_head": "0fc329d0bec93ca7cd288eaea1295172e8789810", + "reviewed_dependency_heads": { + "588": "6d0e53b66c29abd17981881a09eaff41ac862d85", + "543": "2065fe4f72a914d907f1046603f5a36f17018535", + "568": "71c99ac935a33bb7b59a1b4a7af8725a939b0b90", + "569": "1424da81263a5305b35e4cf74ef686c275f03f3f", + "571": "b7bdc5a8de18fcf418cfb9c7d83a2b1097a5e4a3", + "575": "eac6f7e378ab622dac50a2fd4b7ef6eab014de44", + "583": "89be768f0157f5a3d8e4db69cf1748768605eac9", + "586": "4846fee5b06af770e9b3a10c36200f673d656725", + "591": "f622547244bf6d895a571a7b035142ba66279806", + "592": "1f471c445c06a45a32e2c47e8d74c8264b68356b", + "593": "fdb6dcc6d7bc29764c5b6b4d32b7fa050acf74a4", + "596": "7732dd5e46863ff294b5e443661cb9d99abf4e01", + "599": "bc13c8a3e000442606bd2dbe8583a17649ba6c32", + "601": "529ab7dcb59d5ea7a5798e9e06cf5b3aee8a1550", + "604": "1b795c5d7dea746ba310572585c0ef067574b4a6", + "615": "bf8a49a34350599763cced8592db81dc62c2c1c0", + "617": "904d62855723c00f8acef9a287a10768598fdeb2", + "622": "f69477f4f6d3c7eb1f740b166881bffed4ffa9f6", + "623": "e39a615ca601ce5d62abeb24f3fcdb8644ca1005", + "626": "660de893d27bb7abd1d5b01b1640207528a5c851", + "589": "a5afc8c6169ea61ba7c51dfd3cade02cc7b13b46", + "741": "e516b1463606840a15f80aac1995d64ce0092e89", + "799": "80e6463157cfc80b85af09e7e2f14748fe2dac60", + "858": "ae2e25fee91540c83ceee5a60e3b07422ce21c7d", + "869": "73e86f69bf9a391e3c334999f7bbe3825dc42759", + "870": "b734850dc348800942f0ffd1f0723fdec7539b2b", + "890": "4b3bd8cbe8e3e81c44161017c9b72c905a8040a8", + "942": "1960cfa919a0945d0ca4cb14ea96237000037903", + "1004": "1974578f82a30158dd5fcdcdf226b83df8e7196f", + "1005": "779e80aef469bc3a28b19f1adde199871a9a22e3", + "1018": "eac7547cb191f875d5f5ea5f9bbb1d2b8c51c4ef", + "1022": "58618bcbfe88a252467755fc283c04c3db5ca4fd", + "1028": "be4bd89cb4483dcee43257200cd27601f938f13d", + "1031": "e900534f0beda892c726723524203b97471fb9a3", + "1042": "7e8ef087c02a77038a28a337a7ee3a77039cac0d", + "1094": "0fc329d0bec93ca7cd288eaea1295172e8789810", + "1087": "589d1cedf5ef5f231182f4ec86408616311a8af0", + "1082": "0c2e294ec204e83846d27344f829d15ed9ee7cc6", + "1074": "052afc128954f7b6a2b3c409eb4d72248e4020f2", + "1068": "cc71375eae4e8754da0f5d940cb519d66625fb2e", + "1065": "387d935e354a931d636e2c5e6ee63a6910f78a1b", + "1063": "d9377d32e0c97e41338e704c3960fd6f2fa73806", + "1061": "223f581afb1d63d5f8476a69040617e11bcd50e4", + "1056": "d0201efbd68c4acc9832b84bcc986c8781b83c4f", + "1055": "a8ea308976f1709eef0a328463f010cf51469277", + "1053": "6f92ce8884929c9fd4730c86a28e2e3735de02af", + "1050": "c87242f846cfab8c0b97fd889b5fecb0eacef939", + "987": "5142511af0437572aca71b35d1987cd2b47bb5e3" + } +} diff --git a/research/bit-copy-path/bounds/inputs/SOURCE-CODE-EXPECTED.json b/research/bit-copy-path/bounds/inputs/SOURCE-CODE-EXPECTED.json new file mode 100644 index 0000000000..1ad8df8143 --- /dev/null +++ b/research/bit-copy-path/bounds/inputs/SOURCE-CODE-EXPECTED.json @@ -0,0 +1,54 @@ +{ + "source_word": "research/bit-r438-transport-rehost/words/r438-t154.json.gz", + "source_sha256": "a353999bceac4ac957f540adfd3e5f8768297138b27132eeb83332e2ea4b1022", + "canonical_word_sha256": "ebc33ee086f610c1fd540fa56bb99b2b96a768e4aed4a18defcb0e1f507fca89", + "frozen_gzip_sha256": "1a6841fccba51fa922993c77b4b296e659c2c538252e5e06113d0438c5edb870", + "semantic_word_sha256": "e5fa1dbcb80f61c7c728bad7cc166f368ec73c673828273bea091caba8708892", + "six_copy_semantic_sha256": "f44b26376769580295293b1e782568d81ed297647106450a13bd82ecf086dd1b", + "unsigned_section_semantic_sha256": "a50fcbbec51830f84f7368bc6153fe3340ef345849a6cf9ea287d624a753cea4", + "original_six_copy_canonical_sha256": "6448fb2b201c6be722087fc547064ec7c3135af5a72ed363434822212a9b2afc", + "normalized_fields": [ + "source_word", + "signed_from", + "parent_word_gzip_sha256" + ], + "normalization_contract": "The final word omits only checked source location and validated same-run parent compressed identity in semantic comparison. Signed ancestry also normalizes its validated same-run parent hash. All schemas, frames, signed records, deleted physical indices, register counts and original source hashes are compared completely. Immutable frozen inputs retain exact byte hashes.", + "profile": { + "1": 1298, + "2": 1342, + "3": 79, + "4": 44, + "5": 93, + "6": 11, + "7": 9 + }, + "counts": { + "0": 2873, + "1": 13223, + "2": 3, + "3": 3 + }, + "rank_mass": 4989, + "rank_saving": 132, + "source_matrix_sha256": "0ab35dade762a3050b6fc15f653041134f8d5a8015cbdbdc049be0528659240b", + "source_code_ADDs": 1499, + "prefix_row_l1": { + "forward": 34285, + "inverse": 5839860 + }, + "support_semantic_sha256": "c281439e7e78c74c8223e5adfebf34c2be945f308b0f7e53b928c700151cbd74", + "support_canonical_sha256": "3541eab8057c8cebadbd46775ff2d8e759fd592d8a1b9aebde834821d96c051b", + "public_registers": 651, + "dirty_helpers": 429, + "deleted_original_families": [ + 471, + 472, + 473, + 474, + 475, + 476, + 477, + 478, + 479 + ] +} diff --git a/research/bit-copy-path/bounds/inputs/SOURCE-CODE-PROOF.md b/research/bit-copy-path/bounds/inputs/SOURCE-CODE-PROOF.md new file mode 100644 index 0000000000..1fbd2cad0e --- /dev/null +++ b/research/bit-copy-path/bounds/inputs/SOURCE-CODE-PROOF.md @@ -0,0 +1,202 @@ +# Three copied source pivots for the actual R438 local word + +This is a finite, source-pinned local construction over F2, an exact signed +integer lift, and a conditional native cap calculation. It is neither a formal +kernel proof nor a complete recursive supplier or multiplication theorem. + +## Registers, frames and endpoint + +The final word has 111 sources x, 111 arbitrary targets y, 429 arbitrary dirty +helpers h and one owned private copied field. All 651 public input columns are +independent. Its construction first uses the 438-helper, 660-column source +word and then deletes nine literally unread physical families as proved below. The private field is zero outside its immutable snapshot +lifetimes. Sources enter at their actual port lines P_p, targets and helpers at +frame zero. Sources and helpers finish at FULL; target p finishes at its actual +cap K_p=P_p-perp. These are the original R438 boundary frames. + +For the explicit invertible binary matrix A constructed below, the new forward +endpoint is `(x,y,h) -> (Ax,y+Ax,h)`. Its literal inverse is +`(x,y,h) -> (A^-1 x,y+x,h)`. The source endpoint is changed. Original theorems +requiring an unchanged source cannot be applied without proving the new caller +contract. Complemented physical backward-call interpretation and the complete +compiler remain separate obligations. + +## Delete centres and rebuild the zero-frame compensation + +Replay D then A of the actual pinned word. The nine retained helper totals are +`R x+T h`, with R of shape 9-by-111. Let K be the actual 111-by-9 scatter matrix. +No original gate reads a target. The complete row masks of M=RK are +`507,135,238,349,185,437,205,503,425`. Both M and R have rank 9, while +N=I+M has rank 6. + +Delete the nine centre COPY/ERASE episodes and their target scatters. Over F2, +this changes the original target `y+x` by `KR x+KT h`. Replace its original +zero-frame compensation by `D_new=D_old+KT`. Duplicate helper-to-target gates +cancel at frame zero: 5,758 ADDs replace 18,030. Every original public frame +visit is retained, including empty-source visits. The resulting clean-centre +word has endpoint `(x,y,h) -> (x,y+Jx,h)`, where J=I+KR and rank(J)=108. +The verifier derives R,K,J from the literal original word and checks every +public input column; it does not assume a model scatter matrix. + +## An invertible source code with three actual repaired target rows + +Let C0 be a basis of the left nullspace of N, with emitted row masks +`39,193,272`. Let P select physical target coordinates 0,1,2 and S=RP. +The 3-by-3 matrix C0 S is invertible. Define + +``` +B0 = (C0 S)^-1 C0. +``` + +Thus B0 S=I and B0 N=0. Let D be rows 0,1,2 of K. Delete these rows +from K to form W=K+PD, and define + +``` +Y = N + S D, A = I + W R, Q = P B0 R. +``` + +The actual row masks of Y are `10,373,282,165,176,405,356,145,166`, of rank 9. +A fixes kernel(R) and induces the invertible map Y on the quotient identified +by R: RA=YR. Thus A is invertible. Equivalently, if Ax=0 then YRx=0, so +Rx=0 and Ax=x=0. Both matrix inverse identities are also checked finitely. +Rows 0,1,2 of A are exactly their identity rows. Since B0Y=D, + +``` +QA = P B0 R A = P B0 Y R = P D R, +(I+Q)A = I + (K+PD)R + PDR = I+KR = J. +``` + +After the full-frame source code, z=Ax and the target is y+Jx. Adding Qz +therefore gives y+Ax over F2. Q has exactly three nonzero physical rows, +0,1,2. The actual source-code rank is rank(A-I)=9; small target support does +not imply small source-code rank. + +Three repaired target rows are minimal in this declared class. For any +invertible A, rank(A+J) >= rank(A)-rank(J)=3. A correction supported on r +physical target rows has rank at most r. This excludes r<3 for this fixed +clean-centre J and invertible source recoding, not arbitrary suppliers. + +## Literal source code and one snapshot per repaired row + +Normalize R on actual source coordinates `0,1,2,4,8,9,14,30,59`. Commuting +CNOTs replace those coordinates by a query basis. In this basis A has an +invertible 9-by-9 query block and fanout to the other source coordinates. +Gaussian elimination synthesizes the small block with 47 CNOTs. The ordered +code uses `474 prepare + 504 fanout + 47 quotient + 474 inverse prepare`, +or 1,499 ADDs, all at FULL. It uses no extra helper or private field. + +For each nonzero row q=Q[p], choose a source index t with q[t]=1. All source +fields are at FULL. Add each z_i with i!=t and q[i]=1 to z_t. The prepared +field is exactly `sum_i q[i] z_i` over the integers: its original z_t is one +of the required terms, not an unknown extra offset. Copy this actual source +into the cleared private field at target p's cap, add it to target p, and +ERASE the copy while the physical source is unchanged. Undo the preparation +in reverse order with negated ADD coefficients. Every source is restored +exactly to its value before this episode. + +The chosen pivots are 0,1,2. Their preparations use 59,53,60 ADDs; their inverse +preparations use the same counts. Including three scatters gives 347 repair +ADDs. Only one private snapshot is live at a time. No dirty helper is touched +by these three repair episodes. The six-copy dirty-helper variant is retained +only as explicit reconstruction ancestry; it is not the final construction. + +## Integer and native boundaries + +The signed word negates the actual factory cleanup, source unpreparation and +repair unpreparation ADDs. Literal reversal negates every ADD coefficient. +The complete 651-column integer replay establishes source `A_Z x`, every +helper exactly h and the private field zero. The target is +`y+T_Z x+E_Z h`, where `T_Z mod2=A` and E_Z is even. Target correctness is +asserted modulo 2, not as integer equality to y+A_Z x. The reverse source is +`A_Z^-1 x`. Full integer roundtrip is checked separately; A is not assumed +involutive. Wrong unpreparation signs preserve parity but change the integer +source, and are rejected independently. + +For each actual rational E8 line vector u with squared norm 4, the checker +constructs P=uu^t/4 and K=I-P. It binds the actual entrance frame to P and +target frame to K, including all 222 bindings. For +`D_U=[[I-U,U],[U,I-U]]`, exact rational multiplication proves +`D_I D_P=D_K` and `D_P^2=I` on all 16 columns, for all 111 ports (3,552 checked +columns). Hence the FULL-to-cap copied-total residual is D_P of rank one, +under the inherited native ADDRESS/copied-total interface. Source and target +port labels need not agree: the physical source is FULL at the cut. + +The independent ordered-word audit checks common frames for every ADD, +monotone containment and rank for every MOVE, actual source COPY, immutable +source throughout each snapshot, exact ERASE and endpoint frames, in both +directions. These local identities do not establish a full physical native +compiler, rounded implementation or recursive hosting theorem. + +## Delete nine literally unread physical families + +In the signed three-copy word, precisely the old helper indices 471 through +479 are never read as ADD sources or COPY/ERASE sources. The independent +checker discovers this set by scanning every record. The only operations on +these fields are 72 ADD destinations and 36 MOVEs, with paid profile +`27 e1+9 e5`. The same no-read property holds in the literal inverse. + +Delete these nine physical input/output families, discard their destination +operations, and renumber every remaining operand in order, including the +private field from 660 to 651. No remaining value or frame depends on a +removed family. Induction through the literal word therefore identifies the +new execution with projection of the original execution, forward and inverse. +The constructor also checks exact integer endpoint projection, and the +independent checker compares the entire projected record list and both +boundary arrays before replaying all 651 independent columns. Every one of +the remaining 429 arbitrary helpers is restored exactly. The deleted families +are absent from the contract; they are not retained with unpaid endpoints. +A mutant that reads a discarded field is rejected before projection. + +This removes another 72 local rank units and 72 ADDs. It does not establish a +72-unit hosted saving: under the incumbent width-8 whole-family accounting, +returning nine families to idle-owner children adds `9*8=72` rank units per +local call at beta=0 (648 across nine calls). Thus that returned idle obligation +cancels the pruning reduction at this mass level. Changed chronological owners, +stock and complete higher moments remain outside this package. The local +132-unit reduction below must not be read as a net global gain. + +## Complete local invoice + +| Item | Pinned R438 | This word | +|---|---:|---:| +| Sources / targets / dirty helpers | 111 / 111 / 438 | 111 / 111 / 429 | +| Owned private fields / maximum live copies | 1 / 1 | 1 / 1 | +| ADDs | 24,248 | 13,223 | +| MOVE children | 2,909 | 2,873 | +| COPY / ERASE pairs | 9 | 3 | +| COPY ranks | 9 times 7 | 3 times 1 | +| Paid children | 2,918 | 2,876 | +| Rank mass | 5,121 | 4,989 | + +The complete profile is `{1:1298,2:1342,3:79,4:44,5:93,6:11,7:9}`. +Its exact change is `-9 e7-24 e1-9 e5`, saving 132 rank units per local invocation. +The ADD invoice is `5758 compensation + 5619 clean factory/cleanup + 1499 +source code + 347 repair`. No chronological owners, native allocation, +private reservation, code restoration or final decoder cost is removed from +a global bill by this local calculation. + +The exact signed prefix row norms are 34,285 forward and 5,839,860 inverse. +They are checked local arithmetic facts only. A new global perturbation/rounding +proof and precision reserve are required before using them in a supplier. + +## Verification and remaining conditions + +`reconstruct.py` regenerates the complete ordered word from the two pinned live +sources through the declared section and signed ancestry. The registered +verifier compares the full decompressed payload with `WORD.json.gz`. Only +validated source-location metadata and validated same-run compressed-parent +identities in signed/pruned ancestry are normalized. Every record, sign, frame, schema and register count +is compared. Immutable source and frozen word files retain exact byte hashes. + +`check.py` imports no producer. Its separate sparse integer interpreter checks +all 651 columns in both directions and the roundtrip, actual R/K/J, source +inverse, all dirty restoration, all target parity, complete profile and exact +rational native caps. Eleven ancestry fault checks and seven independent controls +cover missing compensation/unpreparation, mutable snapshots, wrong inverse +code, wrong integer signs, FULL-copy rank, a corrupt section query and a read of a deleted family. + +Open global obligations include composition of the changed source interface, +terminal code removal at the actual caps, physical helper/owner/access and +storage schedules, rounded cleanup and precision, recursive caller contracts, +and a full joint complex/BIT outer transfer. This local result supplies none +of those admissions and claims no new kappa. diff --git a/research/bit-copy-path/bounds/inputs/SOURCE-CODE-WORD.json.gz b/research/bit-copy-path/bounds/inputs/SOURCE-CODE-WORD.json.gz new file mode 100644 index 0000000000000000000000000000000000000000..672c41534608d0d812b8020e496177d7c41c8527 GIT binary patch literal 59514 zcmZ6y2|QHM8$WC}wrLo+WhqJ4Y!y)m@6qq~`~Tnft@*h3KKGn+&pqck&v~Bj^SuXBAS`?T*mf76j`c^k zqa zSkB6;&ElAiZS&ook18>{+jYCE=DSZ~{_fn_{d00>;lS067sWAK565>lJ{Ir(`Mvvd zd}j%?@lz#c^Y^ozk4`aL-3NA7W4_s3{bo}av$Q!>7ZbSsZlTy=w`=Y4&XCj2kB{GD zHoGq;Ni6LaNgQ}Mp%Sxw-8Z{NISa*(yNjF0YGYQ#yo(pc*B<__nOytU zsnJ72$}OS2Yo}|S0=Kgka_yf@Z5|p@X}S6zQDCpb+L6HC-a~S^h4z8}=Om){+o3zT zg^q#$pA*mAPw$~qKHP!L>@B0Q_DlHMP^7JP#$D-Thrr1!soYPGrn(LtQ*H_9O*?(b z{y(B8Q!|H-skDUmzCL})F>p2OR_-St`g`hR@q$y!&ie(Y)}6J5V*A}+o1R0@0=8SH z7>8_?TLOD?PS@H8PG=p?wQu>4D5CfM=~~CYwXD0jpVm(N+T0%zz3*vTlH)kB?mKPh z_rOBr+<(P4R-Wx{jqh%M-`#n(`@KMSXXPQd0&_Bx?zFo;JHGQ*<=Jno-RbYoc5m!H 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The finite orbit tests reject finite-order shortcuts, not all callers. +Apache-2.0. Prepared with substantial OpenAI Codex assistance. +""" +from fractions import Fraction as Q +from hashlib import sha256 +from pathlib import Path +import argparse +import gzip +import json + +HERE=Path(__file__).resolve().parent +BASE=Path('/Users/ross/git/math') +RAW=BASE/'integer-mult-bounds/research/bit-r438-transport-rehost/words/r438-t154.json.gz' +FULL=BASE/'integer-mult-bounds/research/e8-bit-carrier-skeleton/data/bit/bitE8-r865.json.gz' +GEOMETRY=FULL.parents[2]/'code/nine/geometry.py' +OLD=BASE/'.research-automation/runs/kappa-major-20261018T1330Z' +COMP=OLD/'bit-endpoint/BIT-FULL-COMPENSATION.json' +CODE=BASE/'.research-automation/runs/kappa-3605-successor-20261018T0940Z/major-frontier/root-pruned-code/BIT-PRUNED-WORD.json.gz' + +def need(b,m): + if not b:raise ValueError(m) +def eye(n):return [[Q(i==j)for j in range(n)]for i in range(n)] +def tr(a):return list(map(list,zip(*a))) +def mm(a,b):return [[sum(x*y for x,y in zip(r,c))for c in zip(*b)]for r in a] +def serial(a):return [[str(x)for x in r]for r in a] +def bits(x): + while x: + z=x&-x;yield z.bit_length()-1;x^=z +def bxor(xs): + z=0 + for x in xs:z^=x + return z +def mul2(a,b):return [bxor(b[j]for j in bits(x))for x in a] +def rank2(rows): + b={} + for x in rows: + while x: + j=x.bit_length()-1 + if j in b:x^=b[j] + else:b[j]=x;break + return len(b) + +def actual_routes(w): + full=json.loads(gzip.decompress(FULL.read_bytes())) + roots=full['lines'];I=eye(8);gs={0:I};todo=[0] + while todo: + more=[] + for p in todo: + for q in range(120): + ip=sum(x*y for x,y in zip(roots[p],roots[q]))//2 + if q in gs or abs(ip)!=1:continue + d=[Q(x-y if ip==1 else x+y)for x,y in zip(roots[p],roots[q])] + norm=sum(x*x for x in d) + ref=[[Q(i==j)-2*d[i]*d[j]/norm for j in range(8)]for i in range(8)] + gs[q]=mm(ref,gs[p]);more.append(q) + todo=more + need(len(gs)==120,'actual upstream Weyl route graph') + rhos=[];projectors=[] + for p,label in enumerate(w['ports_geom']): + g=gs[label];A=[r[1:]for r in g];q=w['lines'][p] + P=[[Q(x*y,4)for y in q]for x in q] + K=[[I[i][j]-P[i][j]for j in range(8)]for i in range(8)] + need(mm(A,tr(A))==K and mm(tr(A),A)==eye(7),'actual cap isometry') + rho=[P[i]+A[i]for i in range(8)]+[r+[Q(0)]*7 for r in tr(A)] + need(tr(rho)==rho and mm(rho,rho)==eye(15),'actual rho involution') + rhos.append(rho);projectors.append(P) + return rhos,projectors,full + +def embed(rho,stage): + # common8, old fresh7, new fresh7; stage0/1 chooses the fresh block. + axes=list(range(8))+list(range(8+7*stage,15+7*stage)) + out=eye(22) + for i,a in enumerate(axes): + for j,b in enumerate(axes):out[a][b]=rho[i][j] + return out + +def commutant_rank(rhos,edges,prime=101): + basis={};n=15;progress=[] + for p,q in edges: + M=mm(rhos[p],rhos[q]) + M=[[x.numerator*pow(x.denominator,-1,prime)%prime for x in row]for row in M] + for i in range(n): + for j in range(n): + v=[0]*(n*n) + for k in range(n): + v[i*n+k]=(v[i*n+k]+M[k][j])%prime + v[k*n+j]=(v[k*n+j]-M[i][k])%prime + for k,old in sorted(basis.items()): + f=v[k] + if f:v=[(x-f*y)%prime for x,y in zip(v,old)] + k=next((k for k,x in enumerate(v)if x),None) + if k is not None: + f=pow(v[k],-1,prime);basis[k]=[(x*f)%prime for x in v] + progress.append(len(basis)) + return len(basis),progress + +def orbit_screen(A,w,full): + roots=[tuple(sign*x for x in q)for q in full['lines']for sign in(1,-1)] + indices={q:i for i,q in enumerate(roots)};perms=[] + for q in w['lines']: + perm=[] + for r in roots: + dot=sum(x*y for x,y in zip(q,r));need(not dot%2,'root integral pairing') + rr=tuple(x-(dot//2)*y for x,y in zip(r,q)) + need(rr in indices,'signed root orbit reflection closure') + perm.append(indices[rr]) + need(all(perm[perm[i]]==i for i in range(240)),'actual native N involution') + perms.append(perm) + def N(v): + return [bxor(1<>q&1 for p,q in edges),'actual source-code commutant edges') + cr,cp=commutant_rank(rhos,edges) + need(cr==224,'mod101 rank224 gives rational commutant dimension1') + # The identity is in the rational commutant, so its dimension is exactly1, + # not merely at most1. A common native projector commuting these couplings + # is therefore scalar, and idempotence leaves onlyZERO/FULL. + identity_coeff=[int(i==j)for i in range(15)for j in range(15)] + need(sum(identity_coeff)==15,'nonzero exact identity commutant control') + Hhash=[];coverage=[] + for p,rho in enumerate(rhos): + rs,rt=embed(rho,0),embed(rho,1);H=mm(rt,rs) + # H maps every old fresh basis column into the new fresh subspace. + need(all(H[i][j]==int(i==j+7)for i in range(22)for j in range(8,15)), + 'actual H sends completed old fresh exactly to future fresh') + P0=[[projectors[p][i][j]if i<8 and j<8 else Q(0)for j in range(22)]for i in range(22)] + need(mm(H,P0)==mm(P0,H),'actual H commutes common N_p projector') + Hhash.append(sha256(json.dumps(serial(H),separators=(',',':')).encode()).hexdigest()) + for p,q in edges: + rs_p,rs_q=embed(rhos[p],0),embed(rhos[q],0) + rt_p,rt_q=embed(rhos[p],1),embed(rhos[q],1) + hp,hq=mm(rt_p,rs_p),mm(rt_q,rs_q) + need(mm(hp,mm(rs_p,rs_q))==mm(mm(rt_p,rt_q),hq), + 'actual all-input ADDRESS intertwiner coefficient identity') + M=mm(rhos[3],rhos[0]) + witness=next((i,j,M[i][j])for i in range(8,15)for j in range(8)if M[i][j]) + return {'status':'PASS_PHYSICAL_REPAIRED_A_INTERTWINER_AND_EXPLICIT_COVERAGE_DEBT', + 'scientific_pin':'20ecf2891d80b5acf18d980b2164fb44ea7faa90','no_new_kappa':True, + 'sources':{str(p):sha256(p.read_bytes()).hexdigest()for p in(RAW,FULL,GEOMETRY,COMP,CODE)}, + 'R':R,'K':K,'W':W,'A':A,'J_rank':108,'A_rank':111,'literal_source_code_ADDs':1499, + 'A_order':28,'power_diagnostics':powers, + 'native_factor':'Let F be actual local FULL, N=diag(D_Pp), G=F N, a=N A N. Since F commutes scalar A and N, repaired local Fphys=diag(G,G) F_N diag(a,I). It has source A G and cross G A N. This statement uses the same actual rho conjugacy on all factors at later stages.', + 'routed_source_code':'A_s[p,q]=A[p,q] times VALUE push(rho_s,p rho_s,q). These exact rho_s,p are reconstructed from upstream geometry.py Weyl BFS, not chosen fresh gauges.', + 'all111_actual_rho15':[[[str(x)for x in row]for row in rho]for rho in rhos], + 'intertwiner':'H_p=rho_t,p rho_s,p: H A_s H^-1=A_t, H commutes N_p, and H(E_s)=E_t. H is an orthogonal ADDRESS transform; all its access and head movement remain paid.', + 'all111_intertwiner_H_sha256':Hhash, + 'coverage_debt':'H^-1 D_Et H D_Es=I on arbitrary VALUE functions, while the desired D_(Es+Et) has native rank14. The naive alignment/query/unwind word cancels its two rank7 fresh advances. This prices that word only.', + 'uniform_source_frame_commutant':{'active_dimension':15,'matrix_unknowns':225, + 'field_for_rank_lower_certificate':101,'actual_A_edges':edges, + 'modular_constraint_rank':224,'rank_progress':cp,'rational_commutant_dimension':1, + 'consequence':'A common orthogonal projector Q commuting the actual source-code coupling maps is scalar, hence Q=0 orI15. The routed source code cannot be treated as a code preserving an unchanged intermediate common native frame.'}, + 'actual_future_fiber_witness':{'output_port':3,'input_port':0,'row':witness[0], + 'column':witness[1],'coefficient':str(witness[2])}, + 'AN_native_orbit_screen':orbit_screen(A,w,full), + 'exact_commutator_fact':'For a scalar port code T, [T,N]_(p,q)=T_pq(D_Pq-D_Pp). Distinct actual E8 lines give distinct native permutations. Every A^k,1<=k<28, has an offdiagonal entry, so none commutes N. Low rank(A^14-I)=2 does not make its routed native coefficient a rank2 query.', + 'open_next_question':'Construct a caller using genuinely changing/port-dependent native frames or shared operator-valued readers so H coverage loss is compensated inside the actual word. No such positive complete word is supplied here.', + 'limits':['No general obstruction for arbitrary operator-valued or nonlinear callers.', + 'No chronological helper owners, rounded errors, source/tape bill, full recurrence or new conditional exponent.', + 'The AN test is a finite exact counterexample to periods<=224, not a proof of infinite order or a universal word lower bound.']} + +def main(): + ap=argparse.ArgumentParser(description=__doc__);ap.add_argument('--write',action='store_true');args=ap.parse_args() + out=json.loads(json.dumps(receipt()));path=HERE/'PHYSICAL-CODE.json' + if args.write:path.write_text(json.dumps(out,sort_keys=True,indent=2)+'\n') + else:need(json.loads(path.read_bytes())==out,'stored physical code receipt differs') + print(out['status']);print('literal A order28; actual rho intertwiner; native coverage debt14; commutant dimension1; AN periods<=224 rejected') +if __name__=='__main__':main() diff --git a/research/bit-copy-path/composition/inputs/SOURCES.json b/research/bit-copy-path/composition/inputs/SOURCES.json new file mode 100644 index 0000000000..2432f680f4 --- /dev/null +++ b/research/bit-copy-path/composition/inputs/SOURCES.json @@ -0,0 +1,48 @@ +{ + "source_scientific_pin": "20ecf2891d80b5acf18d980b2164fb44ea7faa90", + "current_checkout_head": "12fa8c5c844be21bfefb38a19503fbd867a11244", + "files": [ + { + "path": "CODED-WORD.json.gz", + "original_path": "/Users/ross/git/math/worktrees/bit-copy-path-20261020T1204Z/research/bit-r438-source-code/WORD.json.gz", + "sha256": "1a6841fccba51fa922993c77b4b296e659c2c538252e5e06113d0438c5edb870", + "scope": "Frozen source or credited proof reference; no imported proof checker." + }, + { + "path": "CODED-EXPECTED.json", + "original_path": "/Users/ross/git/math/worktrees/bit-copy-path-20261020T1204Z/research/bit-r438-source-code/EXPECTED.json", + "sha256": "f078222ab4e99aff085262c00ba77520cc62eb02003d8e0582ab6ad7dab7f402", + "scope": "Frozen source or credited proof reference; no imported proof checker." + }, + { + "path": "CODED-PROOF.md", + "original_path": "/Users/ross/git/math/worktrees/bit-copy-path-20261020T1204Z/research/bit-r438-source-code/PROOF.md", + "sha256": "ebcc91e74938b525b28e46bf03ec09d42d24ab02070dc96f4f0a51cee2cfb83d", + "scope": "Frozen source or credited proof reference; no imported proof checker." + }, + { + "path": "C511-PROOF-INTERFACE.md", + "original_path": "/Users/ross/git/math/integer-mult-bounds/research/c511-mutable-source-boundary/PROOF-INTERFACE.md", + "sha256": "5e72423f336f85ac2c554e889bca3f67670a6cec509f0a719919c682b98d3c09", + "scope": "Frozen source or credited proof reference; no imported proof checker." + }, + { + "path": "C511-PROOF-OBSERVATION.md", + "original_path": "/Users/ross/git/math/integer-mult-bounds/research/c511-mutable-source-boundary/PROOF-OBSERVATION.md", + "sha256": "038fdc9f47584f22ab7b74de82603bcf7cb1623a145be7e51c887487bbc33a5c", + "scope": "Frozen source or 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"distinct_influences": 1, + "influence_rank": 1, + "insertion_options": null, + "option_set_sha256": "6880be685a8fb8c3381cc1850f2f53020fb0e707fc73dcd53ad13f83a0d1e41e", + "retiming": false, + "separator": "0x1000000000000000000000000000", + "solvable": false, + "source": 479 + }, + { + "centre": 8, + "copy_frame": 421, + "copy_rank": 1, + "distinct_influences": 1, + "influence_rank": 1, + "insertion_options": null, + "option_set_sha256": "6880be685a8fb8c3381cc1850f2f53020fb0e707fc73dcd53ad13f83a0d1e41e", + "retiming": true, + "separator": "0x1000000000000000000000000000", + "solvable": false, + "source": 479 + }, + { + "centre": 8, + "copy_frame": 508, + "copy_rank": 3, + "distinct_influences": 45, + "influence_rank": 10, + "insertion_options": null, + "option_set_sha256": "a057a44fd7f116fc144553eb41992af7a34f590df7819975f7378f5cd19e2352", + "retiming": false, + "separator": "0x1000000000000000000000000000", + "solvable": false, + "source": 479 + }, + { + "centre": 8, + "copy_frame": 508, + "copy_rank": 3, + "distinct_influences": 45, + "influence_rank": 10, + "insertion_options": null, + "option_set_sha256": "da0d837c63656a0306db1d1d55354dbe68bc04fdf235a2e57a5f1337240bac30", + "retiming": true, + "separator": "0x1000000000000000000000000000", + "solvable": false, + "source": 479 + } + ], + "scope": "one unchanged retained total; one catalogue lower frame for entire B; original B gates; legal early MOVE splitting", + "source_snapshot_rows": [ + "6097165137335922326917182089439777940915230747392521779021790936768304177382354726797472857545882756171536977435848104956203018055978438412905", + "12194330274671844653834364178879555881830461494785043558043581873536608354764709453594945715091765512343073952285799577491687322379384064510569", + "24388660549343689307668728357759111763660922989570087116087163747073216709529418907189891430183531024686147900424459134929465303231705914112153", + "48777321098687378615337456715518223527321845979140174232174327494146433419058837814379782860367062049372295800329630711430744268584040099383193", + "97554642197374757230674913431036447054643691958280348464348654988292866838117675628759565720734124098744591598586472817176222477268965012386319", + "195109284394749514461349826862072894109287383916560696928697309976585733676235351257519131441468248197489183197530681299081038910903100725512176", + "390218568789499028922699653724145788218574767833121393857394619953171467352470702515038262882936496394978366392616866225052113471582150954529382", + "780437137578998057845399307448291576437149535666242787714789239906342934704941405030076525765872992789956732781267370054218830961331386937938227", + "1560874275157996115690798614896583152874299071332485575429578479812685869409882810060153051531745985579913465561313554715590246120137719243860309" + ], + "source_word_sha256": "c46ccd7d3eedabc6a7efbb1a91ec13f2928bad8ee0bac57c6e593a092d13d96c", + "status": "PASS_NEW_CUT_SCREEN", + "tests": 180 +} diff --git a/research/bit-copy-path/construction/PROOF.md b/research/bit-copy-path/construction/PROOF.md new file mode 100644 index 0000000000..082b804b08 --- /dev/null +++ b/research/bit-copy-path/construction/PROOF.md @@ -0,0 +1,314 @@ +# The actual R438 centre cannot be rerouted cheaply through its postscatter suffix + +This is a source-bound obstruction, not a changed supplier. For each of the +nine actual retained totals, replacing its COPY7 by any finite family of +independent snapshots of that **same unchanged total**, delivered during the +postscatter suffix through closed nested private-frame walks, costs at least +seven query ranks, with no reduction of the corresponding unweighted paid +rank moment. This remains true with free public retiming, scalar ADDs and +final source/helper restoration. The proposed private +COPY1 / promote1 / demote1 episode is a valid local operator, but cannot +deliver the whole required centre contribution. + +The result uses exact rational frame geometry and the literal Boolean word. +It does not exclude mixed new query totals, a changed upstream factory, new +public downward queries, or another public source/target interface. It does +not improve the fixed allocation ceiling or establish a new kappa. + +## Actual word and independent inputs + +The input word has compressed SHA256 +`c46ccd7d3eedabc6a7efbb1a91ec13f2928bad8ee0bac57c6e593a092d13d96c`. +Its geometry has SHA256 +`a353999bceac4ac957f540adfd3e5f8768297138b27132eeb83332e2ea4b1022`. +They are the copied inputs named in `../PREDECESSOR.json`, whose predecessor +commit is `b4ec15cbdb017776c04b0bf369a807f2278113cd`. Code and contextual +source pins are in `SOURCES.json`. + +Order the 660 persistent decoded inputs as 111 sources, 111 arbitrary +targets, and 438 arbitrary dirty helpers. There is one private zero field. +Over F2 the original boundary is `(x,y,h) -> (x,y+x,h)`. +`acquisition_screen.py` replays all 660 independent columns, all nine actual +snapshots, and each actual ERASE equality. No helper is assumed zero. +`walk_certificate.py` additionally checks all original frame events, all +36,005 ADDs, and the complete histogram + +``` +{1:1322, 2:1342, 3:79, 4:39, 5:102, 6:11, 7:18}. +``` + +Its paid rank mass is 5,101. The nine COPY/ERASE intervals run from record +23524 through record 24068, inclusive. At those cuts the actual totals are + +``` +q_j = R_j x + h_(471+j), j=0,...,8. +``` + +Each contains exactly its own independent helper input after the zero-frame +preconditioning. The nine-by-111 matrix R, the 111-by-nine scatter matrix K, +and the nine complete snapshot rows each have binary rank nine. The complete +literal rows are retained in `WALK-CERTIFICATE.json`. + +Let `k_j` denote column j of K. Deleting the j-th original scatter changes +the decoded target by `k_j q_j`. Because `q_j` is an arbitrary input +functional, a replacement that uses only that same total must supply the +coefficient vector `k_j`, not merely match one initialized state. + +The source of `q_j` is register `471+j` at its actual hyperplane +`F_j = frame(510+j)`. The postscatter target-writing suffix is records +`[24069,25768)`. In that interval: + +- Every public MOVE is upward between nested rational frames. +- Every ADD has equal actual source and destination frames. +- There are no COPYs. +- Every retained centre's value and frame are unchanged. + +The checker also verifies that no actual target register is ever an ADD +source, so the literal scatter column `k_j` survives unchanged to the output. + +The final target frame at port p is exactly `K_p = u_p^perp`, where `u_p` +is the actual rational port vector of squared norm four. All 636 catalogue +frames are independently reconstructed over Q, and their supplied live-root +lists are checked to equal their full rational root closures. + +For every j, the roots with `k_j[p]=1` span all of Q^8. Moreover, their +orthogonal projections onto `F_j` span that seven-dimensional frame. +The certificate contains exact rank-eight pivot ports and independent +rational projected-rank checks for all nine centres. These are actual +geometry checks; no abstract rank-nine scatter argument substitutes for +them. + +## The copied-field contract and tariff + +Fix one j and write `F=F_j`, `q=q_j`. A branch copies q from unchanged F +into a private frame `G subset F`, then follows a finite sequence of nested +private MOVE edges, reads into public receivers at equal frames, and returns +to G before its guarded ERASE against the same source q. + +Write + +``` +r = 7 - dim(G), m = sum of private MOVE ranks. +``` + +The charged branch tariff is `r+m`. Every private MOVE in both directions +is included. A rank-zero same-frame copy is granted free for this lower +bound, even if a chosen concrete API does not provide it. Scalar reads, +workspace, source preparation, all public retiming and final cleanup are +also granted free. Charging any of those costs only strengthens the result. + +COPY/ERASE is the inherited exact copied-total pair. The private decoded +value is q throughout its reads, and ERASE sees its original q at G. +`../composition/` supplies the separate exact address-operator derivation +for private promotion. The old fixed private ABI's prohibition of MOVE +inside a snapshot lifetime is not used as a mathematical obstruction here. + +Different branches may copy the same q independently. The theorem does not +permit a branch to become a privately mixed new query total, borrow an +uncounted downward native query, omit its return, or erase against a +different snapshot. + +## A single closed private walk + +Let L be the intersection of every private frame visited by a branch, +including its entrance, every read frame, and its returned frame. An upward +MOVE cannot shrink the running intersection. A downward MOVE of rank d +shrinks it by at most d. Because the private walk is closed, its total +upward and downward ranks are both `m/2`. Therefore + +``` +L subset G subset F, +dim(L) >= dim(G) - m/2 = 7-r-m/2. (1) +``` + +At an ADD from the private field, the public receiver's frame contains L. +The contribution subsequently propagates only through upward public MOVEs +and equal-frame ADDs. Consequently a nonzero contribution from this branch +can reach target p only if + +``` +L subset K_p, equivalently u_p is orthogonal to L. (2) +``` + +This is a support restriction. Cancellations over F2 can remove paths or +output coefficients; they cannot create a coefficient at a target to which +no permitted path exists. Reads at G before the first MOVE or after the +return satisfy the same condition. + +If this one branch supplies all of `k_j`, its recipient roots span Q^8. +Equation (2) forces L=0, and (1) yields + +``` +r + m/2 >= 7, +r + m >= 14-r. (3) +``` + +Thus a strict reduction of the COPY rank to `r<7` requires total branch +tariff at least eight. At r=1 the complete path needs tariff at least +thirteen to supply the full `k_j`, although a rank-three local excursion +can serve a smaller set of targets. + +## Any finite number of independent copies of the same total + +For branch i, let `L_i`, `r_i`, and `m_i` be as above. A recipient root +projects onto F into + +``` +F intersect L_i^perp, +``` + +whose dimension is at most `r_i+m_i/2` by (1). The support of the total +delivered coefficient is contained in the union of the branch supports. +The span of the projections of that union therefore has dimension at +most the sum of these bounds. + +The required `k_j` recipient roots project onto a rank-seven subspace, +namely all of F. Hence + +``` +7 <= sum_i (r_i+m_i/2) <= sum_i (r_i+m_i). (4) +``` + +Direct public reads of q at unchanged F add only roots in `F^perp`; their +projections onto F are zero and do not improve (4). This proves the stated +seven-rank floor even for arbitrarily many independent same-query branches. + +The proof is over rational or real frame geometry. The decoded coefficient +calculation is over F2. No uniform assertion about every odd-characteristic +specialization of intersection dimensions is made. + +## Rank-moment consequence + +Keep the same rank-to-child conversion and the same stock and ownership +premises. For `0 <= b < 1`, define + +``` +Phi_b(r) = 64^b r^(1-b), Phi_b(0) = 0. +``` + +Let `t_l` be the positive integer paid ranks of all replacement COPYs and +private MOVEs. Equation (4) gives `sum_l t_l >= 7`. For `a=1-b` in `(0,1]`, +the function `t^a` is subadditive on the nonnegative reals: writing +`s=x+y>0`, the facts `(x/s)^a >= x/s` and `(y/s)^a >= y/s` give +`x^a+y^a >= s^a`. Repeated application and monotonicity yield + +``` +sum_l Phi_b(t_l) >= Phi_b(sum_l t_l) >= Phi_b(7). +``` + +Thus dividing the same-total path among independent branches cannot save +this paid rank moment either. All extra public retiming costs were granted +free, so pricing them cannot weaken the inequality. This does not compare +different recursive owner allocations, changed helper stock, or different +rank-to-child conversions; those remain separate obligations. + +## A concrete control: the valid rank-three episode is insufficient + +Use actual source frame510, and the final cap of port3. Their orthogonal +normals make their intersection G rank six. COPY to G has rank one; +promotion to the cap and its inverse cost one each. The episode can add q +to port3 and return exactly for ERASE. + +Port2 also belongs to `support(k_0)`, but its root has nonzero projection +onto G. Its cap does not contain G, so that same episode cannot supply the +required port2 contribution through the monotone suffix. The exact checker +verifies both facts using the actual vectors. Omitting the inverse MOVE +leaves the private field in the wrong frame for the original ERASE. + +These controls separate valid local private transport from delivery of the +complete target pattern. They do not reject the local transport identity. + +## Earlier-cut and acquisition screens + +The written postscatter theorem has no downstream COPY. Two separate finite +screens examine changes outside that premise. + +`acquisition_screen.py` deletes each of the nine original COPY episodes in +turn and permits every source-to-helper ADD at an already visited common +frame throughout literal A. Sources remain their original x throughout A; +the helper-helper network and B are fixed. An inserted acquisition +`h += x_i` contributes a computable suffix target vector. For each of the +523 required `(deleted centre, source column)` repairs, the desired missing +column lies outside the span of all such influences. The emitted dual +functional annihilates every candidate influence and detects the missing +column. Failure persists when all partial-helper restoration is granted +free. This is a complete screen of that acquisition class, not a theorem +about changed helper-helper networks or new source frame paths. + +`late_factory_screen.py` starts when each retained centre first becomes its +complete q, through the first COPY cut. The nine cut counts are +`56,49,42,35,28,21,14,8,1`, totalling 254. The eight remaining downstream +COPYs are included in each suffix influence calculation. The screen allows +a new COPY of rank at most four, reads into arbitrary public registers, +and early upward retiming within an existing MOVE after the last old-frame +use, counting COPY and ERASE source reads as uses. It grants all scalar +work, MOVE splitting and inverse cleanup free. +It rejects all 90 catalogue-frame cases. A further nine relaxations allow +a different arbitrary real frame for every receiver and cut, while +retaining the source-hyperplane and existing-next-MOVE constraints; all +nine also fail. Each rejection has an exact binary separator. + +`newcut_screen.py` records the analogous 1,700-cut B screen and its simple +separators. This finite calculation is secondary to (4). It also tests +all 301 rank-six, rank-seven and full catalogue reader frames for a fixed +promoted private reader frame: all 2,709 cases fail. Taking their completely +free union has rank 111 and permits every desired pattern. This is a useful +scope control: changing private frames can enlarge the reachable target +support. Equations (1)--(4), including the required return and every MOVE, +provide the tariff obstruction. Initial COPY rank alone does not price +that arbitrary promoted walk. + +None of the failed screens produces a candidate word. In particular no new +integer lift, inverse cleanup, positive envelope, owner invoice or physical +ABI is silently inferred from a solved scalar equation. + +## Prior work, provenance and reproducibility + +The P111 label-decoder floor 63, including arbitrary numbers of terms, is +prior work of Russell Seligmann, PR757 +`e2ec7d7e719031e8f8b8012a4ab5b20faed074ce`, and Cody Harris, PR761 +`287758702129e6cbca187356b92b311e8a1c843d`, in the PR446 Theorem C pricing +model. `research/bit-orthogonal-pair-decoders/README.md` records those +premises and credit. `direct_scatter.py` is only a separate actual-R438 +binding/corroboration: 512 unit-query subsets plus an exact forced 3x3 +minor. It does not discover a new 63 floor, and its ten-term floor is +weaker than the older P111-specific bound. + +Credit the source word's Ali Romman R438 construction, Sagar Thakrar +transport/rehosting, and DreamingOfClouds/eumemic/PR589/PR555 lineage, retained +in the parent package. The finite-branch projection argument was suggested +by the independent review agent; the root agent supplied the compact +3x3 minor for the secondary direct-scatter corroboration. The caller agent +independently derived the valid private transport law and reviewed (3). +Substantial OpenAI Codex assistance; Apache-2.0. + +From the worktree root, primary replay: + +```sh +python3 -I -B research/bit-copy-path/construction/walk_certificate.py --out /tmp/r438-walk.json +``` + +Scoped finite supplements: + +```sh +python3 -I -B research/bit-copy-path/construction/acquisition_screen.py --all --allow-partial --out /tmp/r438-acquisition.json +python3 -I -B research/bit-copy-path/construction/late_factory_screen.py --out /tmp/r438-late-factory.json +python3 -I -B research/bit-copy-path/construction/newcut_screen.py --out /tmp/r438-newcut.json +python3 -I -B research/bit-copy-path/construction/direct_scatter.py --out /tmp/r438-direct-scatter.json +``` + +Sibling modules execute only their authenticated source bytes using +`compile(..., dont_inherit=True, optimize=0)`. A cached `.pyc` is never used +for these imports. This is finite exact computation plus the written +all-branch argument above, not a formal-kernel proof. + +## Open next questions + +A construction must change a premise to evade this result: acquire a +different useful total before the fixed completed-centre window, change the +helper-helper network or source frame path, introduce and price new public +downward queries, or compose several genuinely different query totals. +The known coded-source construction changes the source endpoint and has a +separate caller/decoder obligation. No source-preserving cheaper actual +word, full compiler admission, precision improvement, or meaningful kappa +improvement is established here. diff --git a/research/bit-copy-path/construction/SOURCES.json b/research/bit-copy-path/construction/SOURCES.json new file mode 100644 index 0000000000..da0e14f55d --- /dev/null +++ b/research/bit-copy-path/construction/SOURCES.json @@ -0,0 +1,31 @@ +{ + "code": { + "acquisition_screen.py": "7c2786fb369724df55cc4b7920d6f3da6f394398d104e05dac69f7a0595e4a19", + "direct_scatter.py": "ba48fa08864c0ddd3914d0bb934ffdab9f20dc5d4af36f8436824c702f0eb49b", + "late_factory_screen.py": "8297b1be1c240e4c1635803a7253dfbd3e21a29b9dd2045970e22414a408c55d", + "newcut_screen.py": "6d986ee44950d1dfccbe6146061e04cb052d9fc6b6185e7782262d103e67acea", + "walk_certificate.py": "9d7360b301641e9fcd9a400a5ad0b401f85038db5f1ea82ee7a9b240dcf7af4d" + }, + "context_checkout_commit": "12fa8c5c844be21bfefb38a19503fbd867a11244", + "documents": { + "PROOF.md": "0993a370d83b7a3cae2fe98b8b36b274719f94c66ac189b2b87be118863762bd" + }, + "prior_art": { + "credited_commits": { + "PR757 Russell Seligmann": "e2ec7d7e719031e8f8b8012a4ab5b20faed074ce", + "PR761 Cody Harris": "287758702129e6cbca187356b92b311e8a1c843d" + }, + "credited_floor_PRs": [ + 757, + 761 + ], + "meaning": "The P111 direct label-decoder cost63 floor is prior work; direct_scatter.py is source-bound corroboration.", + "path": "research/bit-orthogonal-pair-decoders/README.md", + "sha256": "7c833024359891e15766be310c3d1a4579dd0231fe7ffdbf981fd0814bac2dfb" + }, + "scientific_inputs": { + "../inputs/GEOMETRY.json.gz": "a353999bceac4ac957f540adfd3e5f8768297138b27132eeb83332e2ea4b1022", + "../inputs/WORD.json.gz": "c46ccd7d3eedabc6a7efbb1a91ec13f2928bad8ee0bac57c6e593a092d13d96c" + }, + "scientific_predecessor_commit": "b4ec15cbdb017776c04b0bf369a807f2278113cd" +} diff --git a/research/bit-copy-path/construction/WALK-CERTIFICATE.json 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104, + 107, + 108 + ] + } + ], + "complete_original_F2_map": "(x,y,h)->(x,y+x,h)", + "controls": { + "deficient_rank_witness_rejected": true, + "missing_private_return_rejected": true, + "valid_private_excursion": { + "copy_rank": 1, + "private_move_ranks": [ + 1, + 1 + ], + "reader_frame": 528, + "served_port": 3, + "source_frame": 510, + "total": 3, + "unserviceable_required_port": 2 + } + }, + "exact_frame_count": 636, + "geometry_sha256": "a353999bceac4ac957f540adfd3e5f8768297138b27132eeb83332e2ea4b1022", + "independent_same_query_branch_total_floor": 7, + "no_new_kappa": true, + "original_frame_audit": { + "operation_counts": { + "ADD": 36005, + "COPY": 9, + "ERASE": 9 + }, + "paid_profile": { + "1": 1322, + "2": 1342, + "3": 79, + "4": 39, + "5": 102, + "6": 11, + "7": 18 + }, + "paid_rank_mass": 5101 + }, + "rank_moment_corollary": { + "conclusion": "sum Phi_b(paid COPY and private MOVE ranks)>=Phi_b(7)", + "definition": "Phi_b(r)=64^b*r^(1-b); Phi_b(0)=0", + "exponent_domain": "0<=b<1", + "scope": "same rank-to-child conversion and ownership/stock premises; unweighted paid rank moment" + }, + "scope": "rational frame geometry; unchanged actual centre total; independent exact private snapshots with closed nested MOVE walks; monotone public postscatter influence; no new downstream COPYs", + "single_branch_floor_by_copy_rank": { + "0": 14, + "1": 13, + "2": 12, + "3": 11, + "4": 10, + "5": 9, + "6": 8, + "7": 7 + }, + "source_sha256": "9d7360b301641e9fcd9a400a5ad0b401f85038db5f1ea82ee7a9b240dcf7af4d", + "status": "PASS_ACTUAL_R438_SAME_QUERY_WALK_FLOOR_7_NO_NEW_KAPPA", + "word_sha256": "c46ccd7d3eedabc6a7efbb1a91ec13f2928bad8ee0bac57c6e593a092d13d96c" +} diff --git a/research/bit-copy-path/construction/acquisition_screen.py b/research/bit-copy-path/construction/acquisition_screen.py new file mode 100644 index 0000000000..a9807bbefd --- /dev/null +++ b/research/bit-copy-path/construction/acquisition_screen.py @@ -0,0 +1,231 @@ +#!/usr/bin/env python3 +"""Bounded exact GF(2) screen for new source acquisitions in the literal A. + +No source endpoint code and no new MOVE is allowed in this screen. An option +is one literal h += x_i at a cut where both registers already have the same +frame. It is a screening witness, not a supplier admission. +""" +from collections import Counter +from hashlib import sha256 +from pathlib import Path +import argparse +import gzip +import json + +HERE = Path(__file__).resolve().parent +INPUT = HERE.parent / "inputs" +N, V = 660, 111 +WORD_SHA = "c46ccd7d3eedabc6a7efbb1a91ec13f2928bad8ee0bac57c6e593a092d13d96c" +GEOMETRY_SHA = "a353999bceac4ac957f540adfd3e5f8768297138b27132eeb83332e2ea4b1022" + + +def need(ok, message): + if not ok: + raise ValueError(message) + + +def load(): + wb = (INPUT / "WORD.json.gz").read_bytes() + gb = (INPUT / "GEOMETRY.json.gz").read_bytes() + need(sha256(wb).hexdigest() == WORD_SHA, "literal word pin") + need(sha256(gb).hexdigest() == GEOMETRY_SHA, "geometry pin") + return json.loads(gzip.decompress(wb)), json.loads(gzip.decompress(gb)) + + +def basis_add(basis, row, witness): + while row: + p = row.bit_length() - 1 + if p in basis: + old, old_witness = basis[p] + row ^= old + witness ^= old_witness + else: + basis[p] = row, witness + return True + return False + + +def solve(basis, row): + witness = 0 + while row: + p = row.bit_length() - 1 + if p not in basis: + return None + old, old_witness = basis[p] + row ^= old + witness ^= old_witness + return witness + + +def separator(rows, target): + """Return l with l.rows=0 and l.target=1, or reject if none exists.""" + equations = {} + for row, rhs in [(r, 0) for r in rows] + [(target, 1)]: + while row: + p = row.bit_length() - 1 + if p in equations: + old, old_rhs = equations[p] + row ^= old + rhs ^= old_rhs + else: + equations[p] = row, rhs + break + else: + need(rhs == 0, "no dual separator") + result = 0 + for p, (row, rhs) in sorted(equations.items()): + if ((row & result).bit_count() & 1) != rhs: + result ^= 1 << p + need(all((row & result).bit_count() % 2 == 0 for row in rows), + "dual annihilates every legal acquisition") + need((target & result).bit_count() % 2 == 1, "dual detects missing centre") + return result + + +def bits(row): + while row: + b = row & -row + yield b.bit_length() - 1 + row ^= b + + +def audit(word): + records = word["records"] + copies = [(i, r) for i, r in enumerate(records) if r[0] == 2] + erases = [(i, r) for i, r in enumerate(records) if r[0] == 3] + need(len(copies) == len(erases) == 9, "nine original copy episodes") + values = [1 << j for j in range(N)] + [0] + snapshots = [] + K = [0] * V + for index, (op, d, s, c, f, phase) in enumerate(records): + if op == 1: + values[d] ^= values[s] + if s == N: + need(V <= d < 2 * V, "scatter target") + K[d - V] ^= 1 << (len(snapshots) - 1) + elif op == 2: + need(values[N] == 0, "fresh private") + values[N] = values[d] + snapshots.append(values[d]) + elif op == 3: + need(values[N] == values[d], "snapshot restoration at actual ERASE") + values[N] = 0 + desired = [1 << j for j in range(N)] + [0] + for i in range(V): + desired[V + i] ^= 1 << i + need(values == desired, "complete original F2 boundary") + need(all(s >> (2 * V) == 1 << (471 + j - 2 * V) + for j, s in enumerate(snapshots)), "one independent centre dirt") + R = [s & ((1 << V) - 1) for s in snapshots] + return copies, erases, snapshots, R, K + + +def screen(word, removed, safe_partial=True): + """Screen all existing-frame A insertion cuts for one deleted episode.""" + records = word["records"] + copies, erases, snapshots, R, K = audit(word) + lo, hi = copies[removed][0], erases[removed][0] + A = [i for i, r in enumerate(records) if r[0] == 1 and r[-1] == 2] + a_start, a_stop = A[0], A[-1] + 1 + partial = [j for j in range(2 * V, N) if word["final"][j] != 1] + need(partial == [462, 463, 464, 467, 470], "actual partial terminals") + influence = [0] * (N + 1) + for j in range(V): + influence[V + j] = 1 << j + # Exact integer cotangents for partial endpoints. Individual zero options + # cannot add any error to those early-retired arbitrary-input fields. + terminal = [[int(j == p) for p in partial] for j in range(N + 1)] + cuts = {} + for index in range(len(records) - 1, a_start - 1, -1): + op, d, s, c, f, phase = records[index] + if not lo <= index <= hi: + if op == 1: + influence[s] ^= influence[d] + terminal[s] = [x + c * y for x, y in zip(terminal[s], terminal[d])] + elif op == 3: + influence[N] = 0 + terminal[N] = [0] * len(partial) + elif op == 2: + influence[d] ^= influence[N] + terminal[d] = [x + y for x, y in zip(terminal[d], terminal[N])] + influence[N] = 0 + terminal[N] = [0] * len(partial) + if a_start <= index <= a_stop: + cuts[index] = (influence[:], [any(x) for x in terminal]) + current = word["start"][:] + source = [1 << j for j in range(V)] + options = [[] for _ in range(V)] + seen = [set() for _ in range(V)] + for index, (op, d, s, c, f, phase) in enumerate(records): + if index in cuts: + inf, unsafe = cuts[index] + by_frame = {} + for h in range(2 * V, N): + if inf[h] and (not safe_partial or not unsafe[h]): + by_frame.setdefault(current[h], []).append(h) + for i in range(V): + for h in by_frame.get(current[i], []): + key = inf[h] + if key not in seen[i]: + seen[i].add(key) + options[i].append((index, h, current[i], key)) + if op == 0: + current[d] = c + elif op == 1 and d < V: + need(index >= a_stop, "source not clean through A") + if index >= a_stop: + break + target = sum(((row >> removed) & 1) << i for i, row in enumerate(K)) + results, insertions = [], [] + for i in bits(R[removed]): + basis = {} + for t, option in enumerate(options[i]): + basis_add(basis, option[3], 1 << t) + result = solve(basis, target) + chosen = [] if result is None else [options[i][t] for t in bits(result)] + insertions += [[cut, h, i, frame] for cut, h, frame, _ in chosen] + results.append(dict(source=i, options=len(options[i]), rank=len(basis), + solvable=result is not None, ADDs=len(chosen), + dual_separator=(hex(separator([t[3] for t in options[i]], target)) + if result is None else None), + options_sha256=sha256(json.dumps(options[i], separators=(",", ":")) + .encode()).hexdigest())) + return dict(removed=removed, source_register=471 + removed, + missing_source_weight=R[removed].bit_count(), + missing_target_weight=target.bit_count(), + safe_partial=safe_partial, all_solvable=all(r["solvable"] for r in results), + solvable=sum(r["solvable"] for r in results), + total_ADDs=len(insertions), sources=results, insertions=insertions) + + +def main(): + parser = argparse.ArgumentParser() + parser.add_argument("--out", type=Path) + parser.add_argument("--all", action="store_true") + parser.add_argument("--allow-partial", action="store_true") + args = parser.parse_args() + word, geometry = load() + copies, erases, snapshots, R, K = audit(word) + ranks = {} + for label, rows in (("R", R), ("K", K), ("snapshot", snapshots)): + b = {} + for i, row in enumerate(rows): + basis_add(b, row, 1 << i) + ranks[label] = len(b) + results = [] + for j in range(9) if args.all else [0]: + result = screen(word, j, not args.allow_partial) + results.append(result) + print(json.dumps({k: v for k, v in result.items() + if k not in ("sources", "insertions")}), flush=True) + report = dict(source_word_sha256=WORD_SHA, geometry_sha256=GEOMETRY_SHA, + persistent_inputs=N, arbitrary_helpers=438, cut_ranks=ranks, + R=[str(r) for r in R], K=K, + snapshot_rows=[str(r) for r in snapshots], results=results, + claim="fixed-existing-frame source-acquisition GF2 screen only") + if args.out: + args.out.write_text(json.dumps(report, indent=2, sort_keys=True) + "\n") + + +if __name__ == "__main__": + main() diff --git a/research/bit-copy-path/construction/direct_scatter.py b/research/bit-copy-path/construction/direct_scatter.py new file mode 100644 index 0000000000..1289cd8734 --- /dev/null +++ b/research/bit-copy-path/construction/direct_scatter.py @@ -0,0 +1,305 @@ +#!/usr/bin/env python3 +"""Exact finite floor for the R438 direct clean-query scatter class. + +The 512 cases are unit-query subsets, not an enumeration of GL(9,2). +No physical compiler, positive supplier or changed outer exponent is claimed. +""" +from collections import Counter +from fractions import Fraction +from hashlib import sha256 +from pathlib import Path +import argparse +import json +import types + +HERE = Path(__file__).resolve().parent + + +def load_source(filename): + path = HERE / filename + source = path.read_bytes() + pins = json.loads((HERE / "SOURCES.json").read_bytes())["code"] + if sha256(source).hexdigest() != pins[filename]: + raise ValueError("authenticated sibling source: " + filename) + module = types.ModuleType(filename[:-3]) + module.__file__ = str(path) + exec(compile(source, str(path), "exec", dont_inherit=True, optimize=0), module.__dict__) + return module + + +acquisition = load_source("acquisition_screen.py") + + +def need(ok, message): + if not ok: + raise ValueError(message) + + +def bits(row): + return acquisition.bits(row) + + +def rank(rows): + basis = {} + for row in rows: + acquisition.basis_add(basis, row, 0) + return len(basis) + + +def product(left, right): + result = [] + for row in left: + value = 0 + for j in bits(row): + value ^= right[j] + result.append(value) + return result + + +def inverse(rows): + n = len(rows) + augmented = [row | (1 << (n + j)) for j, row in enumerate(rows)] + for j in range(n): + p = next((i for i in range(j, n) if augmented[i] >> j & 1), None) + need(p is not None, "invertible centre pairing") + augmented[j], augmented[p] = augmented[p], augmented[j] + for i in range(n): + if i != j and augmented[i] >> j & 1: + augmented[i] ^= augmented[j] + need([r & ((1 << n) - 1) for r in augmented] == [1 << j for j in range(n)], + "inverse left block") + result = [r >> n for r in augmented] + need(product(rows, result) == product(result, rows) == [1 << j for j in range(n)], + "both exact binary inverse identities") + return result + + +def modular_rank(lines, support, modulus=101): + """Field rank supplies a certified lower bound on the rational rank.""" + basis, pivots = {}, [] + for i in support: + row = [x % modulus for x in lines[i]] + for j, old in sorted(basis.items()): + c = row[j] + row = [(x - c * y) % modulus for x, y in zip(row, old)] + j = next((j for j, x in enumerate(row) if x), None) + if j is not None: + c = pow(row[j], -1, modulus) + basis[j] = [(c * x) % modulus for x in row] + pivots.append(i) + if len(basis) == 8: + break + return len(basis), pivots + + +def rational_normal(lines, support): + basis = {} + for i in support: + row = list(map(Fraction, lines[i])) + for j, old in sorted(basis.items()): + c = row[j] + row = [x - c * y for x, y in zip(row, old)] + j = next((j for j, x in enumerate(row) if x), None) + if j is not None: + c = row[j] + basis[j] = [x / c for x in row] + need(len(basis) == 7, "actual rational hyperplane") + free = next(j for j in range(8) if j not in basis) + normal = [Fraction(0)] * 8 + normal[free] = Fraction(1) + for j, row in sorted(basis.items(), reverse=True): + normal[j] = -sum(a * b for a, b in zip(row, normal)) + need(all(sum(a * b for a, b in zip(lines[i], normal)) == 0 for i in support), + "normal annihilates every actual port line in support") + return normal + + +def frame_geometry(geometry): + """Check every catalogue span and its complete live-root closure over Q.""" + lines = geometry["lines"] + masks, dims = [], [] + for support in geometry["frames"]: + basis = {} + for i in support: + row = list(map(Fraction, lines[i])) + for j, old in sorted(basis.items()): + c = row[j] + row = [x - c * y for x, y in zip(row, old)] + j = next((j for j, x in enumerate(row) if x), None) + if j is not None: + c = row[j] + basis[j] = [x / c for x in row] + actual = [] + if len(basis) == 8: + actual = list(range(len(lines))) + else: + for p, line in enumerate(lines): + row = list(map(Fraction, line)) + for j, old in sorted(basis.items()): + c = row[j] + row = [x - c * y for x, y in zip(row, old)] + if not any(row): + actual.append(p) + need(actual == support, "catalogue frame is the exact rational live-root closure") + need(modular_rank(lines, support)[0] == len(basis), + "modular dimensions agree with exact rational frame dimensions") + masks.append(sum(1 << p for p in support)) + dims.append(len(basis)) + return masks, dims + + +def encode(vector): + return [[x.numerator, x.denominator] for x in vector] + + +def decode(vector): + return [Fraction(a, b) for a, b in vector] + + +def geometry_audit(R, K, M, geometry): + lines = geometry["lines"] + source, target = [], [] + source_seven, target_seven = {}, {} + for coefficient in range(1, 512): + query = product([coefficient], R)[0] + source_support = list(bits(query)) + output_support = [p for p, row in enumerate(K) + if (row & coefficient).bit_count() & 1] + srank, spivots = modular_rank(lines, source_support) + trank, tpivots = modular_rank(lines, output_support) + need(srank >= 7 and trank >= 7, "all nonzero support ranks at least seven") + if srank == 7: + source_seven[coefficient] = rational_normal(lines, source_support) + if trank == 7: + target_seven[coefficient] = rational_normal(lines, output_support) + source.append(dict(coefficient=coefficient, modular_rank=srank, pivot_ports=spivots)) + target.append(dict(coefficient=coefficient, modular_rank=trank, pivot_ports=tpivots)) + need(set(source_seven) == {1 << j for j in range(9)}, + "exactly the nine original source directions have rank seven") + need(set(target_seven) == set(M), + "exactly the nine pairing rows are target hyperplane supports") + need(all(source_seven[1 << j] == target_seven[M[j]] for j in range(9)), + "source and target special normals coincide in actual coordinates") + incidence = [] + for j in range(9): + for k in range(9): + dot = sum(a * b for a, b in zip(source_seven[1 << j], target_seven[M[k]])) + actual = dot == 0 + expected = j != k and (M[k] >> j) & 1 == 1 + need(actual == expected, "exact rank-six bridge incidence") + if actual: + incidence.append([k, j]) + return dict(source_rank_histogram=dict(sorted(Counter(x["modular_rank"] for x in source).items())), + target_rank_histogram=dict(sorted(Counter(x["modular_rank"] for x in target).items())), + source_directions=source, target_directions=target, + source_seven_normals={str(a): encode(n) for a, n in source_seven.items()}, + target_seven_normals={str(v): encode(n) for v, n in target_seven.items()}, + rank_six_cells=incidence) + + +def nine_term_audit(M): + cases = [] + for J in range(512): + zero_directions = [row for row in M if row & J == 0] + r = rank(zero_directions) + d = 9 - J.bit_count() + benefits = [j for j in bits(J) + if any((v & J) == 1 << j and k != j for k, v in enumerate(M))] + lower = 63 - len(benefits) + d - r + need(lower >= 63, "nine-query paid lower bound") + cases.append(dict(unit_query_subset=J, remaining_dimension=d, + special_reader_rank=r, rank_six_possible=benefits, lower_bound=lower)) + need([c["unit_query_subset"] for c in cases if c["lower_bound"] == 63] == [0, 511], + "equality subsets") + return dict(cases=cases, lower_bound_histogram=dict(sorted(Counter(c["lower_bound"] + for c in cases).items())), minimum=63, equality_unit_query_subsets=[0, 511]) + + +def all_count_audit(M, Minv, geometry_result): + minor = [1, 4, 8] + block = [sum(((Minv[row] >> col) & 1) << j for j, col in enumerate(minor)) + for row in minor] + need(block == [1, 4, 2] and rank(block) == 3, "forced rank-three exact minor") + low_cells = {tuple(cell) for cell in geometry_result["rank_six_cells"]} + need(all((row, col) not in low_cells for row in minor for col in minor), + "every rank-six contribution is zero on forced minor") + return dict(minor_rows=minor, minor_columns=minor, minor_binary_rows=block, + minor_rank=3, minimum_higher_rank_terms=3, + cost_floor_for_ten_or_more="6*N+3", overall_cost_floor=63) + + +def controls(word, R, K, M, Minv, geometric): + outcomes = [] + for name, mutate in ( + ("deleted_first_copy_fails_complete_boundary", + lambda records: [r for i, r in enumerate(records) if not 23524 <= i <= 23588]), + ("changed_live_snapshot_fails_actual_erase", + lambda records: records[:23525] + [[1, 471, 472, 1, 510, 99]] + records[23525:]), + ): + bad = dict(word) + bad["records"] = mutate(word["records"]) + try: + acquisition.audit(bad) + except ValueError: + outcomes.append(name) + else: + raise ValueError("semantic control accepted: " + name) + # The deleted-copy control above also fails the nine-episode contract. + # Separately replay the removed episode's public difference explicitly. + column = next(bits(R[0])) + target = next(p for p, row in enumerate(K) if row & 1) + need((R[0] >> column) & (K[target] & 1) == 1, + "single independent source input witnesses omitted scatter") + outcomes.append(dict(control="one_hot_source_detects_deleted_scatter", + source=column, target=111 + target)) + normal = decode(geometric["source_seven_normals"]["1"]) + need(sum(x*x for x in normal) != 0 and (M[0] & 1) == 1, + "diagonal M entry does not imply geometric containment") + outcomes.append("falsely_allowing_diagonal_rank_six_is_rejected") + need(rank([1, 2, 4]) == 3 and rank([1, 4, 0]) == 2, + "minor rank-sensitive control") + broken = Minv[:] + broken[8] ^= 1 << 4 + need(product(M, broken) != [1 << j for j in range(9)], "wrong inverse rejected") + outcomes.append("one_bit_inverse_defect_breaks_exact_identity") + dual = acquisition.separator([1, 2], 4) + need((dual & 4).bit_count() % 2 == 1, "dual witness positive control") + try: + acquisition.separator([1, 2, 4], 4) + except ValueError: + outcomes.append("adding_missing_influence_invalidates_acquisition_separator") + else: + raise ValueError("false acquisition obstruction control accepted") + return outcomes + + +def main(): + parser = argparse.ArgumentParser() + parser.add_argument("--out", type=Path, required=True) + args = parser.parse_args() + word, geometry = acquisition.load() + _, _, snapshots, R, K = acquisition.audit(word) + need(rank(R) == rank(K) == rank(snapshots) == 9, "literal rank-nine transfer") + M = product(R, K) + need(M == [507, 135, 238, 349, 185, 437, 205, 503, 425], "actual centre pairing") + need(all((M[i] >> j & 1) == (M[j] >> i & 1) for i in range(9) for j in range(9)), + "symmetric actual pairing") + Minv = inverse(M) + geometric = geometry_audit(R, K, M, geometry) + nine = nine_term_audit(M) + all_count = all_count_audit(M, Minv, geometric) + report = dict(status="PASS_DIRECT_QUERY_SCATTER_FLOOR_63_NO_NEW_KAPPA", + no_new_kappa=True, source_word_sha256=acquisition.WORD_SHA, + geometry_sha256=acquisition.GEOMETRY_SHA, + persistent_columns=660, R=[str(r) for r in R], K=K, + M=M, M_inverse=Minv, geometry=geometric, + nine_term_bound=nine, all_count_bound=all_count, + controls=controls(word, R, K, M, Minv, geometric), + scope="query rows in span(R), reader columns in image(K), direct immutable snapshots; arbitrary count", + script_sha256=sha256(Path(__file__).read_bytes()).hexdigest()) + args.out.write_text(json.dumps(report, indent=2, sort_keys=True) + "\n") + print(report["status"], "512 unit-query subsets; 511 source and 511 target supports; exact 3x3 minor") + + +if __name__ == "__main__": + main() diff --git a/research/bit-copy-path/construction/late_factory_screen.py b/research/bit-copy-path/construction/late_factory_screen.py new file mode 100644 index 0000000000..88e944a3cb --- /dev/null +++ b/research/bit-copy-path/construction/late_factory_screen.py @@ -0,0 +1,140 @@ +#!/usr/bin/env python3 +"""Retimed lower-rank centre snapshots before the other centre COPYs.""" +from hashlib import sha256 +from pathlib import Path +import argparse +import json +import types + +HERE = Path(__file__).resolve().parent + + +def load_source(filename): + path = HERE / filename + source = path.read_bytes() + pins = json.loads((HERE / "SOURCES.json").read_bytes())["code"] + if sha256(source).hexdigest() != pins[filename]: + raise ValueError("authenticated sibling source: " + filename) + module = types.ModuleType(filename[:-3]) + module.__file__ = str(path) + exec(compile(source, str(path), "exec", dont_inherit=True, optimize=0), module.__dict__) + return module + + +acquisition = load_source("acquisition_screen.py") +direct = load_source("direct_scatter.py") + + +def screen(word, geometry, centre, masks, dims): + records = word["records"] + copies, erases, snapshots, R, K = acquisition.audit(word) + lo, hi = copies[centre][0], erases[centre][0] + begin = 1 + max(i for i, r in enumerate(records) + if r[0] == 1 and r[1] == 471 + centre and r[-1] == 2) + end = copies[0][0] + current = word["start"][:] + for op, d, s, c, f, phase in records[:begin]: + if op == 0: + current[d] = c + source_frame = current[471+centre] + acquisition.need(source_frame == 510+centre, "completed source centre frame") + acquisition.need(not any(r[0] == 1 and r[1] == 471+centre for r in records[begin:end]), + "completed source remains immutable") + influence = [0] * 661 + for p in range(111): + influence[111+p] = 1 << p + suffix = {} + for index in range(len(records)-1, begin-1, -1): + op, d, s, c, f, phase = records[index] + if not lo <= index <= hi: + if op == 1: + influence[s] ^= influence[d] + elif op == 3: + influence[660] = 0 + elif op == 2: + influence[d] ^= influence[660] + influence[660] = 0 + if index <= end: + suffix[index] = influence[:660] + touches = [[] for _ in range(660)] + for index, (op, d, s, c, f, phase) in enumerate(records): + if index < begin or lo <= index <= hi: + continue + if op == 0: + touches[d].append((index, 0, c)) + elif op == 1: + touches[d].append((index, 1, f)) + if s < 660: + touches[s].append((index, 1, f)) + elif op in (2, 3): + # Both operations read their persistent source at frame c. + # An early public MOVE must not cross that unchanged read. + touches[d].append((index, op, c)) + pointers = [0] * 660 + frames = [f for f in range(len(masks)) + if 3 <= dims[f] < 7 and masks[f] & ~masks[source_frame] == 0] + options = {f: {} for f in frames} + relaxed = {} + for index in range(begin, end+1): + for d, vector in enumerate(suffix[index]): + if not vector or d == 471+centre: + continue + while pointers[d] < len(touches[d]) and touches[d][pointers[d]][0] < index: + pointers[d] += 1 + nxt = touches[d][pointers[d]] if pointers[d] < len(touches[d]) else None + old = current[d] + new = nxt[2] if nxt and nxt[1] == 0 else old + option = [index, d, old, new] + for f in frames: + if masks[old] & ~masks[f] == 0 and masks[f] & ~masks[new] == 0: + options[f].setdefault(vector, option) + intersection = dims[new] - int(bool(masks[new] & ~masks[source_frame])) + if (dims[old] < 7 and masks[old] & ~masks[source_frame] == 0 + and intersection >= max(3, dims[old])): + relaxed.setdefault(vector, option) + if index < end: + op, d, s, c, f, phase = records[index] + if op == 0: + current[d] = c + target = sum(((row >> centre) & 1) << p for p, row in enumerate(K)) + results = [] + for f, pool in list(options.items()) + [(None, relaxed)]: + basis = {} + vectors = list(pool) + for t, vector in enumerate(vectors): + acquisition.basis_add(basis, vector, 1 << t) + solution = acquisition.solve(basis, target) + results.append(dict(frame=f, distinct_influences=len(pool), influence_rank=len(basis), + solvable=solution is not None, + insertions=([pool[vectors[t]] for t in acquisition.bits(solution)] + if solution is not None else None), + separator=(hex(acquisition.separator(vectors, target)) + if solution is None else None), + option_set_sha256=sha256(json.dumps(list(pool.items()), separators=(",", ":")) + .encode()).hexdigest())) + return dict(centre=centre, source=471+centre, completed_cut=begin, final_cut=end, + cuts=end-begin+1, results=results) + + +def main(): + parser = argparse.ArgumentParser() + parser.add_argument("--out", type=Path, required=True) + args = parser.parse_args() + word, geometry = acquisition.load() + masks, dims = direct.frame_geometry(geometry) + results = [screen(word, geometry, j, masks, dims) for j in range(9)] + positives = [[r["centre"], q] for r in results for q in r["results"] if q["solvable"]] + report = dict(status="PASS_LATE_FACTORY_CENTRE_SCREEN" if not positives else "LATE_FACTORY_CANDIDATES", + source_word_sha256=acquisition.WORD_SHA, geometry_sha256=acquisition.GEOMETRY_SHA, + results=results, positives=positives, + scope="completed retained centre before first remaining COPY; unchanged other scalar gates; legal early MOVE splitting; lower COPY rank at most four", + relaxation="frame None permits a different arbitrary real frame for each receiver and cut", + rational_frame_audit=dict(frames=len(masks), all_exact_root_closures=True, + all_exact_rational_dimensions=True), + grants=["free restoration", "free added scalar gates", "free MOVE splitting"]) + args.out.write_text(json.dumps(report, indent=2, sort_keys=True) + "\n") + print(report["status"], sum(r["cuts"] for r in results), "cuts;", len(positives), "positives") + + +if __name__ == "__main__": + main() diff --git a/research/bit-copy-path/construction/newcut_screen.py b/research/bit-copy-path/construction/newcut_screen.py new file mode 100644 index 0000000000..a6e0b43a11 --- /dev/null +++ b/research/bit-copy-path/construction/newcut_screen.py @@ -0,0 +1,209 @@ +#!/usr/bin/env python3 +"""One lower-frame centre snapshot carried through all of literal phase B. + +Allows arbitrary target/source/helper receivers and free upward retiming within +an existing frame transition, after the register's last use at its old frame. +The existing phase-B scalar gates are fixed. Inverse cleanup and owner costs +are granted free by the negative screen. +""" +from collections import defaultdict +from hashlib import sha256 +from pathlib import Path +import argparse +import json +import types + +HERE = Path(__file__).resolve().parent + + +def load_source(filename): + path = HERE / filename + source = path.read_bytes() + pins = json.loads((HERE / "SOURCES.json").read_bytes())["code"] + if sha256(source).hexdigest() != pins[filename]: + raise ValueError("authenticated sibling source: " + filename) + module = types.ModuleType(filename[:-3]) + module.__file__ = str(path) + exec(compile(source, str(path), "exec", dont_inherit=True, optimize=0), module.__dict__) + return module + + +acquisition = load_source("acquisition_screen.py") +direct = load_source("direct_scatter.py") + +N = 660 + + +def main(): + parser = argparse.ArgumentParser() + parser.add_argument("--out", type=Path, required=True) + args = parser.parse_args() + word, geometry = acquisition.load() + copies, erases, snapshots, R, K = acquisition.audit(word) + records = word["records"] + begin = erases[-1][0] + 1 + end = max(i for i, r in enumerate(records) if r[0] == 1 and 111 <= r[1] < 222) + 1 + acquisition.need((begin, end) == (24069, 25768), "actual B interval") + masks, dims = direct.frame_geometry(geometry) + current = word["start"][:] + for op, d, s, c, f, ph in records[:begin]: + if op == 0: + current[d] = c + acquisition.need(all(current[471+j] == 510+j for j in range(9)), + "original retained centre frames") + acquisition.need(not any(r[0] == 1 and r[1] in range(471, 480) + for r in records[begin:end]), "snapshot sources immutable throughout B") + # At a cut, a register can be raised early only if its next use is the + # original MOVE. This permits splitting that MOVE with no rank-mass gain + # claimed. Every old scalar use remains at its original frame. + touches = [[] for _ in range(N)] + for index, (op, d, s, c, f, ph) in enumerate(records): + if index < begin: + continue + if op == 0: + touches[d].append((index, 0, c)) + elif op == 1: + touches[d].append((index, 1, f)) + if s < N: + touches[s].append((index, 1, f)) + elif op in (2, 3): + touches[d].append((index, op, c)) + pointers = [0] * N + influence = [0] * (N + 1) + for p in range(111): + influence[111+p] = 1 << p + suffix = {} + for index in range(len(records)-1, begin-1, -1): + op, d, s, c, f, ph = records[index] + if op == 1: + influence[s] ^= influence[d] + elif op == 3: + influence[N] = 0 + elif op == 2: + influence[d] ^= influence[N] + influence[N] = 0 + if index <= end: + suffix[index] = influence[:N] + frame_sets = [[f for f in range(len(masks)) + if 3 <= dims[f] < 7 and masks[f] & ~masks[510+j] == 0] + for j in range(9)] + acquisition.need(all(len(fs) == 10 for fs in frame_sets), "ten catalogue lower frames per centre") + high_frames = [f for f in range(len(masks)) if 6 <= dims[f] <= 8] + all_frames = sorted({f for fs in frame_sets for f in fs} | set(high_frames)) + options = {f: {} for f in all_frames} + actual_options = {f: {} for f in all_frames} + arbitrary_frame_options = [{} for _ in range(9)] + compatibility = {} + for index in range(begin, end+1): + inf = suffix[index] + for d, vector in enumerate(inf): + if not vector: + continue + while pointers[d] < len(touches[d]) and touches[d][pointers[d]][0] < index: + pointers[d] += 1 + nxt = touches[d][pointers[d]] if pointers[d] < len(touches[d]) else None + old = current[d] + new = nxt[2] if nxt and nxt[1] == 0 else old + pair = old, new + if pair not in compatibility: + compatibility[pair] = [f for f in all_frames + if masks[old] & ~masks[f] == 0 + and masks[f] & ~masks[new] == 0] + for f in compatibility[pair]: + option = [index, d, old, new, vector] + options[f].setdefault(vector, option) + if old == f: + actual_options[f].setdefault(vector, option) + # Stronger necessary screen: allow a different arbitrary real G + # for every receiver, at every cut. There is no common-frame or + # dyadic-projector demand in this relaxation. Since F_source is + # a hyperplane, dim(F_source intersect F_next) is rank(F_next) + # or rank(F_next)-1 according to containment. + for j in range(9): + fs = 510+j + intersection_rank = dims[new] - int(bool(masks[new] & ~masks[fs])) + if (dims[old] < 7 and masks[old] & ~masks[fs] == 0 + and intersection_rank >= max(3, dims[old])): + arbitrary_frame_options[j].setdefault(vector, [index, d, old, new, vector]) + if index == end: + break + op, d, s, c, f, ph = records[index] + if op == 0: + current[d] = c + results = [] + for j in range(9): + desired = sum(((row >> j) & 1) << p for p, row in enumerate(K)) + for f in frame_sets[j]: + for retiming, pool in ((False, actual_options[f]), (True, options[f])): + option_list = list(pool.values()) + basis = {} + for t, option in enumerate(option_list): + acquisition.basis_add(basis, option[-1], 1 << t) + solution = acquisition.solve(basis, desired) + chosen = None if solution is None else [option_list[t] for t in acquisition.bits(solution)] + row = dict(centre=j, source=471+j, copy_frame=f, copy_rank=7-dims[f], + retiming=retiming, distinct_influences=len(pool), influence_rank=len(basis), + solvable=solution is not None, insertion_options=chosen, + separator=(hex(acquisition.separator(pool, desired)) + if solution is None else None), + option_set_sha256=sha256(json.dumps(option_list, separators=(",", ":")) + .encode()).hexdigest()) + results.append(row) + positives = [r for r in results if r["solvable"]] + relaxed = [] + for j, pool in enumerate(arbitrary_frame_options): + desired = sum(((row >> j) & 1) << p for p, row in enumerate(K)) + basis = {} + option_list = list(pool.values()) + for t, option in enumerate(option_list): + acquisition.basis_add(basis, option[-1], 1 << t) + solution = acquisition.solve(basis, desired) + relaxed.append(dict(centre=j, distinct_influences=len(pool), rank=len(basis), + solvable=solution is not None, + separator=hex(acquisition.separator(pool, desired)) if solution is None else None, + insertion_options=([option_list[t] for t in acquisition.bits(solution)] + if solution is not None else None))) + promoted = [] + union_options = {} + for f in high_frames: + for vector, option in options[f].items(): + union_options.setdefault(vector, [f, option]) + union_basis = {} + for t, vector in enumerate(union_options): + acquisition.basis_add(union_basis, vector, 1 << t) + union_targets = [] + for j in range(9): + desired = sum(((row >> j) & 1) << p for p, row in enumerate(K)) + union_targets.append(acquisition.solve(union_basis, desired) is not None) + for f in high_frames: + pool = options[f] + basis = {} + option_list = list(pool.values()) + for t, option in enumerate(option_list): + acquisition.basis_add(basis, option[-1], 1 << t) + solution = acquisition.solve(basis, desired) + if solution is not None: + promoted.append(dict(centre=j, reader_frame=f, reader_rank=dims[f], + insertions=[option_list[t] for t in acquisition.bits(solution)])) + report = dict(status="PASS_NEW_CUT_SCREEN" if not positives else "NEW_CUT_CANDIDATES_REQUIRE_FULL_REPLAY", + source_word_sha256=acquisition.WORD_SHA, geometry_sha256=acquisition.GEOMETRY_SHA, + B_cuts_inclusive=[begin, end], cuts=end-begin+1, + persistent_inputs=N, source_snapshot_rows=[str(r) for r in snapshots], + frame_sets=frame_sets, tests=len(results), positives=positives, results=results, + arbitrary_frame_per_receiver_relaxation=relaxed, + high_frame_private_promotion_screen=dict( + reader_frames=high_frames, tests=9*len(high_frames), positives=promoted, + grants="arbitrary snapshot delivery to H, free private promotion/restoration; fixed H during all B reads", + all_H_union_rank=len(union_basis), all_H_union_targets_solvable=union_targets, + all_H_union_scope="diagnostic relaxation only: free arbitrary changes of H, no closed private walk or tariff asserted"), + rational_frame_audit=dict(frames=len(masks), all_exact_root_closures=True, + all_exact_rational_dimensions=True), + scope="one unchanged retained total; one catalogue lower frame for entire B; original B gates; legal early MOVE splitting", + grants=["inverse cleanup free", "all new scalar ADDs free", "all MOVE splitting free", + "arbitrary targets and helpers", "all chronological cut influences may be combined"]) + args.out.write_text(json.dumps(report, indent=2, sort_keys=True) + "\n") + print(report["status"], report["tests"], "screens;", len(positives), "positive candidates") + + +if __name__ == "__main__": + main() diff --git a/research/bit-copy-path/construction/walk_certificate.py b/research/bit-copy-path/construction/walk_certificate.py new file mode 100644 index 0000000000..3d925e014d --- /dev/null +++ b/research/bit-copy-path/construction/walk_certificate.py @@ -0,0 +1,182 @@ +#!/usr/bin/env python3 +"""Source-bound hypotheses and exact witnesses for the postscatter walk bound.""" +from collections import Counter +from fractions import Fraction +from hashlib import sha256 +from pathlib import Path +import argparse +import json +import types + +HERE = Path(__file__).resolve().parent + + +def need(ok, message): + if not ok: + raise ValueError(message) + + +def load_source(filename): + path = HERE / filename + source = path.read_bytes() + pins = json.loads((HERE / "SOURCES.json").read_bytes())["code"] + need(sha256(source).hexdigest() == pins[filename], "authenticated sibling source: " + filename) + module = types.ModuleType(filename[:-3]) + module.__file__ = str(path) + exec(compile(source, str(path), "exec", dont_inherit=True, optimize=0), module.__dict__) + return module + + +acquisition = load_source("acquisition_screen.py") +direct = load_source("direct_scatter.py") + + +def rational_rank(rows): + basis = {} + for row in rows: + row = list(map(Fraction, row)) + for j, old in sorted(basis.items()): + c = row[j] + row = [x-c*y for x, y in zip(row, old)] + j = next((j for j, x in enumerate(row) if x), None) + if j is not None: + c = row[j] + basis[j] = [x/c for x in row] + return len(basis) + + +def dot(a, b): + return sum(x*y for x, y in zip(a, b)) + + +def project_onto_hyperplane(row, normal): + c = Fraction(dot(row, normal), dot(normal, normal)) + return [x-c*y for x, y in zip(row, normal)] + + +def literal_frames(word, masks, dims): + current = word["start"][:] + [None] + live = None + counts = Counter() + profile = Counter() + for index, (op, d, s, c, f, phase) in enumerate(word["records"]): + need(0 <= d < 660 and 0 <= s <= 660, "actual persistent/private operands") + if op == 0: + need(current[d] == s, "actual MOVE entrance") + need(masks[s] & ~masks[c] == 0, "actual nested public MOVE") + need(dims[c]-dims[s] == f and f > 0, "actual positive MOVE tariff") + current[d] = c + profile[f] += 1 + elif op == 1: + need(current[d] == current[s] == f, "actual ADD common frame") + need(d != live, "literal copied source is immutable") + need(c in (-1, 1), "signed unit ADD") + counts["ADD"] += 1 + elif op == 2: + need(live is None and s == 660 and current[660] is None, "fresh private COPY") + need(current[d] == c and masks[f] & ~masks[c] == 0, "nested copied-total frames") + need(dims[c] > dims[f], "original paid COPY") + current[660], live = f, d + profile[dims[c]-dims[f]] += 1 + counts["COPY"] += 1 + else: + need(op == 3 and s == 660 and live == d, "actual ERASE source identity") + need(current[d] == c and current[660] == f, "actual ERASE frame identity") + current[660], live = None, None + counts["ERASE"] += 1 + need(current[:-1] == word["final"] and current[-1] is None and live is None, "all actual endpoint frames") + need(counts == {"ADD": 36005, "COPY": 9, "ERASE": 9}, "literal operation counts") + need(dict(profile) == {1:1322, 2:1342, 3:79, 4:39, 5:102, 6:11, 7:18}, + "entire original paid rank histogram") + return dict(operation_counts=dict(counts), paid_profile=dict(sorted(profile.items())), + paid_rank_mass=sum(r*n for r, n in profile.items())) + + +def main(): + parser = argparse.ArgumentParser() + parser.add_argument("--out", type=Path, required=True) + args = parser.parse_args() + word, geometry = acquisition.load() + copies, erases, snapshots, R, K = acquisition.audit(word) + masks, dims = direct.frame_geometry(geometry) + frame_receipt = literal_frames(word, masks, dims) + begin = erases[-1][0] + 1 + end = 1 + max(i for i, r in enumerate(word["records"]) + if r[0] == 1 and 111 <= r[1] < 222) + need((begin, end) == (24069, 25768), "actual postscatter target-writing interval") + B = word["records"][begin:end] + need(all(r[0] in (0, 1) for r in B), "no downstream COPY in target-writing suffix") + need(not any(r[0] == 1 and r[1] in range(471, 480) for r in B), + "all actual snapshot sources remain unchanged during B") + need(not any(r[0] == 0 and r[1] in range(471, 480) for r in B), + "all actual source hyperplanes remain fixed during B") + need(not any(r[0] == 1 and 111 <= r[2] < 222 for r in word["records"]), + "actual target registers never feed another field; scatter K survives unchanged") + lines = geometry["lines"] + for p in range(111): + cap = word["final"][111+p] + need(dims[cap] == 7 and dot(lines[p], lines[p]) == 4, "actual port norm and cap rank") + need(all(dot(lines[p], lines[q]) == 0 for q in geometry["frames"][cap]), + "actual final target frame is port-orthogonal cap") + centres = [] + for j in range(9): + F = 510+j + need(dims[F] == 7 and copies[j][1][3] == F, "actual rank-seven centre source frame") + support = [p for p, row in enumerate(K) if row >> j & 1] + full_rank, pivot_ports = direct.modular_rank(lines, support) + need(full_rank == rational_rank([lines[p] for p in support]) == 8, + "every original target pattern spans all eight rational coordinates") + normal = direct.rational_normal(lines, geometry["frames"][F]) + projected = [project_onto_hyperplane(lines[p], normal) for p in support] + need(rational_rank(projected) == 7, "recipient roots projected onto actual source frame span seven") + centres.append(dict(centre=j, source_register=471+j, source_frame=F, + source_normal=direct.encode(normal), target_support=support, + support_rank=8, exact_projected_rank=7, modular_pivot_ports=pivot_ports, + snapshot_row=str(snapshots[j]), source_row=str(R[j]))) + # A valid cheap private excursion serves some targets, but cannot supply + # the required whole centre pattern. Use actual F510 and cap of port3. + normal = direct.decode(centres[0]["source_normal"]) + reader_normal = list(map(Fraction, lines[3])) + need(dot(normal, reader_normal) == 0 and K[3] & 1, "actual orthogonal excursion pair") + def project_G(row): + first = project_onto_hyperplane(row, normal) + return project_onto_hyperplane(first, reader_normal) + G = [project_G([int(i == j) for i in range(8)]) for j in range(8)] + need(rational_rank(G) == 6, "actual excursion intersection dimension") + need(all(x == 0 for x in project_G(reader_normal)), "chosen reader cap contains G") + need(K[2] & 1 and any(project_G(lines[2])), "another required target cap excludes G") + closed_path_ranks = [6, 7, 6] + up = sum(max(b-a, 0) for a, b in zip(closed_path_ranks, closed_path_ranks[1:])) + down = sum(max(a-b, 0) for a, b in zip(closed_path_ranks, closed_path_ranks[1:])) + need(up == down == 1 and 1+up+down == 3, "complete cheap excursion tariff") + need([6, 7][0] != [6, 7][-1], "omitting private return fails original ERASE frame") + # Destroy the geometric witness deliberately: seven roots cannot certify + # full rank eight. The primary check must rely on actual coverage. + damaged = centres[0]["modular_pivot_ports"][:7] + need(rational_rank([lines[p] for p in damaged]) <= 7, "rank witness defect is detected") + result = dict(status="PASS_ACTUAL_R438_SAME_QUERY_WALK_FLOOR_7_NO_NEW_KAPPA", + no_new_kappa=True, word_sha256=acquisition.WORD_SHA, + geometry_sha256=acquisition.GEOMETRY_SHA, + arbitrary_inputs=660, arbitrary_dirty_helpers=438, actual_ERASE_checks=9, + complete_original_F2_map="(x,y,h)->(x,y+x,h)", + exact_frame_count=len(masks), actual_B_interval=[begin, end], + original_frame_audit=frame_receipt, centres=centres, + single_branch_floor_by_copy_rank={str(r):14-r for r in range(8)}, + independent_same_query_branch_total_floor=7, + rank_moment_corollary=dict( + definition="Phi_b(r)=64^b*r^(1-b); Phi_b(0)=0", + exponent_domain="0<=b<1", + conclusion="sum Phi_b(paid COPY and private MOVE ranks)>=Phi_b(7)", + scope="same rank-to-child conversion and ownership/stock premises; unweighted paid rank moment"), + controls=dict(valid_private_excursion=dict(source_frame=510, reader_frame=word["final"][114], + copy_rank=1, private_move_ranks=[1, 1], total=3, + served_port=3, unserviceable_required_port=2), + missing_private_return_rejected=True, deficient_rank_witness_rejected=True), + source_sha256=sha256(Path(__file__).read_bytes()).hexdigest(), + scope="rational frame geometry; unchanged actual centre total; independent exact private snapshots with closed nested MOVE walks; monotone public postscatter influence; no new downstream COPYs") + args.out.write_text(json.dumps(result, indent=2, sort_keys=True) + "\n") + print(result["status"], "9 full-support witnesses; 9 projected-rank witnesses; 636 exact frames") + + +if __name__ == "__main__": + main() diff --git a/research/bit-copy-path/independent/REVIEW.md b/research/bit-copy-path/independent/REVIEW.md new file mode 100644 index 0000000000..2a83324bca --- /dev/null +++ b/research/bit-copy-path/independent/REVIEW.md @@ -0,0 +1,319 @@ +# Independent review of the actual COPY path + +The postscatter obstruction is valid in its stated model. It is stronger than +the finite cut search: any finite collection of independent, exact copies of +one unchanged actual retained total, with closed nested private MOVE walks and +monotone public propagation, has total query rank at least seven. The original +COPY7 already has that rank. The coded R429 word changes the endpoint; direct +substitution in the nine-stage native caller has an exact public TAIL defect. +Neither result excludes a changed factory, mixed new query totals, or a caller +with additional paid native operations. + +This review separates three kinds of evidence: + +- `review_check.py` independently checks finite source facts, exact witnesses, + supplied dual separators, and several decisive moment inequalities. +- The dimension argument for arbitrarily many branches, the all-count + histogram minimization, and the native stage identities are written proofs. +- No new physical compiler, private-MOVE interface, complete changed owner + schedule, precision bound, or joint multiplication theorem is admitted. + +The historical 63-unit direct label-decoder floor is prior work of Russell +Seligmann (PR757) and Cody Harris (PR761), under their stated decoder model. +The present reconstruction binds that fact to the actual retained totals and +prices its complete caller consequence. It is not a new 63-unit floor. + +## Independent source reconstruction + +The checker executes no producer module and reads no producer receipt as an +oracle. Its inputs are fixed compressed word/geometry bytes and the selected +assembly record, individually authenticated by SHA256. It represents binary +rows by sets and symmetric difference; rational ranks and containment use +primitive integer elimination. The B and late-factory audits propagate only +the specified scalar dual observables, rather than reconstructing the +producer's entire influence span. + +It covers all 660 ordinary input columns of the retained word and all 651 +columns of the coded word, in both orientations. Helpers are independent +arbitrary inputs. Every private lifetime compares its complete snapshot +against the same source at ERASE. Frame replay separately checks every +monotone MOVE, every common-frame ADD, COPY source immutability, and all +physical endpoint labels. + +The actual snapshots are `q_j = R_j x + h_(471+j)`. R, K, and the nine +snapshots have rank nine. Removing the private scatter reads and replaying the +complete word produces exactly `K q` on the targets and zero change on every +other persistent row. Thus the required replacement vector is the actual +column `k_j`; the argument does not infer it from a clean-helper trial. + +All 636 supplied frame root sets equal their exact rational closures. The +nine centre frames have dimension seven. For every j, roots of ports in +`support(k_j)` have rank eight, and their projections onto the actual centre +hyperplane have rank seven. These finite facts supply the geometric +hypotheses of the written obstruction. + +The coded endpoint is independently recovered as + +``` +F_A(x,y,h) = (Ax, y+Ax, h), +F_A^-1(x,y,h) = (A^-1 x, y+x, h). +``` + +A has order 28, and `rank(A+A^-1)=8`. The actual factorization +`A-I=W'R`, with the first three rows of K suppressed in W', is verified +entry by entry. The checker also reconstructs all three support censuses: +511 source directions, 511 direct-reader directions, and 511 W' directions. +The first two have nine rank-seven supports and 502 rank-eight supports. +W' has twelve rank-seven supports and 499 rank-eight supports. + +## Why the finite-branch theorem is complete in its model + +Fix a centre hyperplane F of dimension seven. Branch i copies the same +unchanged q into `G_i subset F` at rank cost +`r_i=7-dim(G_i)`. It carries q through a closed sequence of nested private +frames, reads at common frames, and returns to its original G_i for exact +ERASE. Let m_i be the sum of every upward and downward MOVE rank, and L_i +the intersection of all visited frames. + +A downward edge of rank d removes at most d dimensions from the running +intersection. An upward edge removes none. Closure makes the total downward +rank `m_i/2`, so + +``` +L_i subset F, dim(L_i) >= 7-r_i-m_i/2. +``` + +After a private read, the affected public field has a frame containing L_i. +Every subsequent upward MOVE and common-frame ADD preserves this support +condition along each coefficient path. A final target p can therefore receive +a nonzero contribution from branch i only when its cap `u_p^perp` contains +L_i. Cancellation can remove such contributions, but cannot create an output +outside this permitted support. + +Projecting those recipient roots onto F confines their span to +`F intersect L_i^perp`, of dimension at most `r_i+m_i/2`. The support of the +sum of all branches is contained in the union of their supports. The actual +required roots project onto all seven dimensions of F. Consequently + +``` +7 <= sum_i (r_i+m_i/2) <= sum_i (r_i+m_i). +``` + +Direct reads of the original q at F contribute only roots in `F^perp`, whose +projection onto F is zero. They do not evade the argument. For one branch, +the unprojected rank-eight support forces `L=0`, yielding the stronger +`r+m >= 14-r`. + +The proof covers arbitrary real private subspaces and any finite number of +independent branches; it does not depend on the original 636-frame catalogue +or forbid a private promotion merely because the old ABI lacks it. It does +require the fixed logical-q chart, nested MOVE tariff, unchanged copied +source, closed return, common-frame scalar reads, and the absence of new +downstream public COPY/downward-query events. Mixing privately different +query totals, changing the factory, or introducing new paid native exchanges +requires a new argument. The rational/real statement is not a uniform claim +about rank under every odd-characteristic specialization. + +For `0=9`, child ranks +`{6,7,8}`, mass at least 63, and `0=11` costs at least `11 Phi_b(6)`, greater than the +nine-term profile. Its complete caller still rejects the target, including +the free-bad discount. + +The selected complex certificate remains an additional joint-transfer +obligation. A successful BIT planning inequality alone is not a 10% κ +improvement. + +## Reproduction and status + +From any working directory: + +```sh +python3 -I -B /absolute/path/to/independent/review_check.py --output /tmp/review.json +python3 -I -B -O /absolute/path/to/independent/review_check.py --output /tmp/review-opt.json +``` + +The normal and optimized receipts must agree byte for byte. The checker has +no repository-root import or Git dependency. Controls remove an actual +private scatter read, grant a missing influence to invalidate a separator, +and remove the native code edge. Each control must fail its named check; +unexpected acceptance is raised outside the caught-exception path. Reversing +all ADD signs is accepted as an equivalent binary presentation, without +claiming equivalence over the integers. + +The retained receipt passed on Python 3.14 in ordinary and optimized modes. +A minimal Git-free copy containing only the checker and its six authenticated +inputs passed on Python 3.12.12 in both modes with identical bytes. +In disposable copies, disabling the endpoint guard and the native-edge guard +made their corresponding negative controls fail by unexpected acceptance; +those controls do not certify themselves by catching that failure. +`VALIDATION.json` records the source and result hashes for this check. + +The remaining work is constructive: find a changed useful total, factory, or +coupled caller whose complete physical operation and paid invoice satisfy +the demonstrated budgets. The current package establishes scoped +obstructions and planning thresholds, not a positive improved supplier. diff --git a/research/bit-copy-path/independent/VALIDATION.json b/research/bit-copy-path/independent/VALIDATION.json new file mode 100644 index 0000000000..846c91cf32 --- /dev/null +++ b/research/bit-copy-path/independent/VALIDATION.json @@ -0,0 +1,37 @@ +{ + "status": "PASS_INDEPENDENT_COPY_PATH_NORMAL_OPTIMIZED_GIT_FREE_AND_CONTROL_SENSITIVITY", + "checker": "review_check.py", + "checker_sha256": "66f19765e975bc98e2b62ed2ce1a91a951c86f6a96682e8fc4f036353855aaba", + "receipt": "review-result.json", + "receipt_sha256": "00ca7891a5758e5c82b11149fb0f2c145ce814c70c8424d686395aa0dcfca171", + "runs": [ + { + "environment": "owned checkout", + "python": "3.14", + "modes": ["-I -B", "-I -B -O"], + "byte_identical": true + }, + { + "environment": "temporary Git-free package containing only checker and its six pinned inputs", + "python": "3.12.12", + "modes": ["-I -B", "-I -B -O"], + "byte_identical_to_checkout": true + } + ], + "disabled_guard_controls": [ + { + "disabled_expression": "need(rows == expected, \"complete original binary endpoint\")", + "result": "unexpected acceptance: omitted actual private read", + "nonzero_exit_required": true + }, + { + "disabled_expression": "need(0 in matrix[3], \"native witness code edge\")", + "result": "unexpected acceptance: delete native witness edge", + "nonzero_exit_required": true + } + ], + "producer_checkers_executed_by_independent_checker": false, + "formal_kernel_proof": false, + "physical_supplier_admission": false, + "new_kappa": false +} diff --git a/research/bit-copy-path/independent/review-result.json b/research/bit-copy-path/independent/review-result.json new file mode 100644 index 0000000000..ef1468131b --- /dev/null +++ b/research/bit-copy-path/independent/review-result.json @@ -0,0 +1,2424 @@ +{ + "caller_budget": { + "arithmetic": "24-term atanh after binary range reduction; 16-term exp; rational entire 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b/research/bit-copy-path/independent/review_check.py new file mode 100644 index 0000000000..24d604149e --- /dev/null +++ b/research/bit-copy-path/independent/review_check.py @@ -0,0 +1,742 @@ +#!/usr/bin/env python3 +# SPDX-License-Identifier: Apache-2.0 +"""Independent exact audit of the frozen COPY-path question. + +Set-valued binary rows, primitive integer elimination, scalar dual propagation, +and a short rational analytic bound. No producer module or receipt is executed. +Substantial OpenAI Codex assistance. +""" +import argparse +from collections import Counter +from fractions import Fraction as Q +from functools import lru_cache +import gzip +from hashlib import sha256 +import json +from math import gcd +from pathlib import Path + +HERE = Path(__file__).resolve().parent +ROOT = HERE.parent +PINS = { + "inputs/WORD.json.gz": + "c46ccd7d3eedabc6a7efbb1a91ec13f2928bad8ee0bac57c6e593a092d13d96c", + "inputs/GEOMETRY.json.gz": + "a353999bceac4ac957f540adfd3e5f8768297138b27132eeb83332e2ea4b1022", + "inputs/global_k.py.txt": + "6368196a393ac25bb2918880e4c4bd31ed733b762c27b6873ab391cbc0c5c3c8", + "composition/inputs/CODED-WORD.json.gz": + "1a6841fccba51fa922993c77b4b296e659c2c538252e5e06113d0438c5edb870", + "composition/inputs/FULL-E8-GEOMETRY.json.gz": + "1fb726ea3818fe0837cceb0dd22a71a0814f064e5e781628104610d6468e0e55", + "bounds/inputs/SELECTED-ASSEMBLY.json": + "1122cb3a575f8c788345fc1bf4c108ac7c3250fb81a5988671f5eb04b3ab3045", +} +SEPARATORS = [ + [104, 107], [104, 107], [100, 108], [110], [98, 108], + [106, 108], [106, 108], [106, 108], [107], +] +LATE_SEPARATORS = [ + [108, 110], [106, 109], [102, 108, 110], [101, 102], [109], + [109], [110], [102], [108], +] + + +def need(condition, label): + if not condition: + raise ValueError(label) + + +def reject(label, operation, intended): + try: + operation() + except ValueError as error: + need(str(error) == intended, "wrong rejection: " + label) + return label + raise ValueError("unexpected acceptance: " + label) + + +def read(relative): + data = (ROOT / relative).read_bytes() + need(sha256(data).hexdigest() == PINS[relative], "immutable input " + relative) + if relative.endswith(".gz"): + return json.loads(gzip.decompress(data)) + if relative.endswith(".json"): + return json.loads(data) + return data.decode() + + +def digest(value): + return sha256(json.dumps(value, sort_keys=True, separators=(",", ":")).encode()).hexdigest() + + +def sxor(rows): + answer = set() + for row in rows: + answer.symmetric_difference_update(row) + return answer + + +def matmul(left, right): + return [sxor(right[j] for j in row) for row in left] + + +def rank2(rows): + pivots = {} + for original in rows: + row = set(original) + while row: + p = min(row) + if p in pivots: + row.symmetric_difference_update(pivots[p]) + else: + pivots[p] = row + break + return len(pivots) + + +def bits(row): + return sum(1 << j for j in row) + + +def primitive(row): + common = 0 + for x in row: + common = gcd(common, x) + if common: + first = next(x for x in row if x) + common *= 1 if first > 0 else -1 + row = [x // common for x in row] + return tuple(row) + + +def reduceq(row, basis): + """Fraction-free exact reduction; no finite-field rank proxy.""" + row = primitive(row) + for p, old in basis: + if row[p]: + row = primitive([old[p]*x - row[p]*y for x, y in zip(row, old)]) + return row + + +def basisq(rows): + basis = [] + for row in rows: + row = reduceq(row, basis) + if any(row): + pivot = next(j for j, x in enumerate(row) if x) + basis.append((pivot, row)) + basis.sort() + return basis + + +def normal(basis): + need(len(basis) == 7, "hyperplane normal dimension") + free = next(j for j in range(8) if all(j != p for p, _ in basis)) + vector = [Q(int(j == free)) for j in range(8)] + for p, row in reversed(basis): + vector[p] = -sum((row[j]*vector[j] for j in range(p+1, 8)), Q(0))/row[p] + den = 1 + for x in vector: + den = den*x.denominator // gcd(den, x.denominator) + return primitive([int(x*den) for x in vector]) + + +def dot(a, b): + return sum(x*y for x, y in zip(a, b)) + + +def frame_audit(geometry): + roots = geometry["lines"] + need(len(roots) == 111 and all(dot(x, x) == 4 for x in roots), "actual root lines") + frames = [set(f) for f in geometry["frames"]] + bases = [basisq(roots[p] for p in f) for f in frames] + for f, basis in zip(frames, bases): + closure = {p for p, root in enumerate(roots) if not any(reduceq(root, basis))} + need(f == closure, "exact rational frame closure") + dimensions = [len(b) for b in bases] + need(dimensions[0] == 0 and dimensions[1] == 8, "zero/full frame dimensions") + need(all(dimensions[510+j] == 7 for j in range(9)), "nine centre hyperplanes") + return frames, bases, dimensions + + +def frame_word(word, frames, dimensions): + n = len(word["start"]) + current = word["start"][:] + [None] + live = None + histogram = Counter() + nesting = 0 + for op, dst, src, coefficient, frame, phase in word["records"]: + if op == 0: + need(current[dst] == src and frames[src] <= frames[coefficient], + "literal monotone MOVE") + move_rank = dimensions[coefficient]-dimensions[src] + need(frame == move_rank, "literal MOVE child rank") + histogram[move_rank] += 1 + current[dst] = coefficient + nesting += 1 + elif op == 1: + need(coefficient in (-1, 1), "literal ADD coefficient") + need(dst != src and dst != live, "immutable live COPY source") + need(current[dst] == current[src] == frame, "literal common-frame ADD") + elif op == 2: + need(live is None and src == n and current[dst] == coefficient, + "literal COPY source") + need(frames[frame] <= frames[coefficient], "literal nested COPY") + copy_rank = dimensions[coefficient]-dimensions[frame] + if "copy_schema" in word: + need(phase == copy_rank, "literal coded COPY child rank") + histogram[copy_rank] += 1 + current[n] = frame + live = dst + elif op == 3: + need(live == dst and current[dst] == coefficient and current[n] == frame, + "literal ERASE frame return") + current[n] = None + live = None + else: + raise ValueError("unknown literal operation") + need(current[:-1] == word["final"] and live is None, "literal final frames") + need(0 not in histogram, "only positive literal children") + return dict(sorted(histogram.items())), nesting + + +def literal(word, inverse=False, omitted_scatter=(), corrupt_scatter=None): + n = len(word["start"]) + rows = [{j} for j in range(n)] + [set()] + snapshots, episodes = [], [] + live, episode = None, -1 + records = list(enumerate(word["records"])) + if inverse: + records.reverse() + for index, (op, dst, src, coefficient, _frame, _phase) in records: + if op == 1: + if src == n: + need(live is not None and 111 <= dst < 222, "private target-only scatter") + if not inverse: + episodes[-1]["targets"].symmetric_difference_update({dst-111}) + if episode in omitted_scatter or index == corrupt_scatter: + continue + rows[dst].symmetric_difference_update(rows[src]) + elif op == (3 if inverse else 2): + need(live is None and not rows[n], "empty private at COPY") + live = dst + episode += 1 + rows[n] = set(rows[dst]) + snapshots.append(set(rows[dst])) + if not inverse: + episodes.append(dict(begin=index, source=dst, targets=set())) + elif op == (2 if inverse else 3): + need(live == dst and rows[n] == rows[dst], "complete private snapshot return") + if not inverse: + episodes[-1]["end"] = index + rows[n] = set() + live = None + else: + need(op == 0, "known operation in scalar replay") + need(live is None and not rows[n], "empty final private") + return rows, snapshots, episodes + + +def require_original_endpoint(rows): + n = len(rows)-1 + expected = [{p} for p in range(n)] + [set()] + for p in range(111): + expected[111+p].add(p) + need(rows == expected, "complete original binary endpoint") + + +def source_audit(word, geometry, coded): + rows, snapshots, episodes = literal(word) + reverse, _, _ = literal(word, inverse=True) + require_original_endpoint(rows) + require_original_endpoint(reverse) + need(len(snapshots) == 9 and [e["source"] for e in episodes] == list(range(471, 480)), + "actual nine snapshot sources") + R = [] + for j, snapshot in enumerate(snapshots): + need(snapshot - set(range(111)) == {471+j}, "one independent snapshot dirt") + R.append(snapshot & set(range(111))) + K = [{j for j, e in enumerate(episodes) if p in e["targets"]} for p in range(111)] + need(K == [set(row) for row in geometry["scatter"]], "literal K equals actual scatter") + erased, _, _ = literal(word, omitted_scatter=range(9)) + for p in range(660): + expected = sxor(snapshots[j] for j in K[p-111]) if 111 <= p < 222 else set() + need(rows[p] ^ erased[p] == expected, "all-column omitted-scatter defect") + need(rank2(R) == rank2(K) == rank2(snapshots) == 9, "full transfer rank") + M = matmul(R, K) + need([bits(row) for row in M] == [507,135,238,349,185,437,205,503,425], + "actual retained pairing") + forward, _, _ = literal(coded) + backward, _, _ = literal(coded, inverse=True) + A, Ainv = forward[:111], backward[:111] + identity = [{p} for p in range(111)] + need(matmul(A, Ainv) == matmul(Ainv, A) == identity, "actual coded inverse") + for p in range(111): + need(forward[111+p] == {111+p} ^ A[p], "complete coded forward target") + need(backward[111+p] == {111+p, p}, "complete coded inverse target") + need(all(forward[p] == backward[p] == {p} for p in range(222, 651)), + "arbitrary coded helper return") + W = [set() if p < 3 else set(K[p]) for p in range(111)] + need(matmul(W, R) == [A[p] ^ {p} for p in range(111)], "actual A-I factorization") + power = identity + orders = [] + for exponent in range(1, 29): + power = matmul(A, power) + if power == identity: + orders.append(exponent) + need(orders == [28], "coded order twenty-eight") + defect = [A[p] ^ Ainv[p] for p in range(111)] + need(rank2(defect) == 8, "nonzero coded last-pair residual") + bad_index = next(i for i, r in enumerate(word["records"]) if r[0] == 1 and r[2] == 660) + controls = [reject("omitted actual private read", lambda: require_original_endpoint( + literal(word, corrupt_scatter=bad_index)[0]), "complete original binary endpoint")] + # Set symmetric difference also accepts an equivalent presentation of signs. + sign_changed = dict(word) + sign_changed["records"] = [ + [op, d, s, -c if op == 1 else c, f, ph] + for op, d, s, c, f, ph in word["records"]] + require_original_endpoint(literal(sign_changed)[0]) + controls.append("all ADD signs reversed is the same binary operator") + return R, K, M, A, Ainv, W, snapshots, episodes, { + "original_columns": 660, "coded_columns": 651, + "R": [sorted(row) for row in R], "K": [sorted(row) for row in K], + "A_sha256": digest([sorted(row) for row in A]), + "A_inverse_sha256": digest([sorted(row) for row in Ainv]), + "M": [bits(row) for row in M], + "coded_order": 28, "last_pair_scalar_residual_rank": 8, + "controls": controls, + } + + +def geometry_census(geometry, frames, bases, R, K, M, W): + roots = geometry["lines"] + source_seven, target_seven = {}, {} + hist = {name: Counter() for name in ("source", "target", "coded_defect")} + coded_seven = [] + for mask in range(1, 512): + coefficients = {j for j in range(9) if mask >> j & 1} + source = sxor(R[j] for j in coefficients) + target = {p for p, row in enumerate(K) if len(row & coefficients) % 2} + coded_target = {p for p, row in enumerate(W) if len(row & coefficients) % 2} + sb = basisq(roots[p] for p in sorted(source)) + tb = basisq(roots[p] for p in sorted(target)) + cb = basisq(roots[p] for p in sorted(coded_target)) + for name, basis in (("source", sb), ("target", tb), ("coded_defect", cb)): + hist[name][len(basis)] += 1 + if len(sb) == 7: + source_seven[mask] = normal(sb) + if len(tb) == 7: + target_seven[mask] = normal(tb) + if len(cb) == 7: + coded_seven.append(mask) + need(hist["source"] == hist["target"] == {7: 9, 8: 502}, "complete direct support census") + need(hist["coded_defect"] == {7: 12, 8: 499}, "complete coded support census") + need(set(source_seven) == {1 << j for j in range(9)}, "source special directions") + need(set(target_seven) == set(map(bits, M)), "target special directions") + low_cells = set() + for j in range(9): + need(source_seven[1 << j] == target_seven[bits(M[j])], "matching support normals") + for k in range(9): + orthogonal = dot(source_seven[1 << j], target_seven[bits(M[k])]) == 0 + need(orthogonal == (j != k and j in M[k]), "rank-six incidence") + if orthogonal: + low_cells.add((k, j)) + projections = [] + for j in range(9): + support = [p for p, row in enumerate(K) if j in row] + need(len(basisq(roots[p] for p in support)) == 8, "each K column spans all eight") + n = normal(bases[510+j]) + norm = dot(n, n) + projected = [[norm*x - dot(root, n)*y for x, y in zip(root, n)] + for root in (roots[p] for p in support)] + need(len(basisq(projected)) == 7, "projected K column spans centre hyperplane") + need(all(dot(v, n) == 0 for v in projected), "projected roots stay in F") + projections.append(dict(centre=j, target_count=len(support), root_rank=8, + projected_rank=7, source_normal=list(n))) + return low_cells, { + "support_histograms": {k: dict(sorted(v.items())) for k, v in hist.items()}, + "coded_rank7_directions": coded_seven, + "same_total_branch_theorem_finite_premises": projections, + } + + +def old_direct_floor(M, low_cells): + histogram = Counter() + equality = [] + for mask in range(512): + J = {j for j in range(9) if mask >> j & 1} + r = rank2(row for row in M if not row & J) + benefits = sum(any(row & J == {j} and k != j for k, row in enumerate(M)) for j in J) + lower = 63-benefits+9-len(J)-r + need(lower >= 63, "minimum-basis direct cost floor") + histogram[lower] += 1 + if lower == 63: + equality.append(mask) + need(equality == [0, 511], "direct equality subsets") + # Solve M X = I with sets, retaining a different representation than producer. + augmented = [set(row) | {9+p} for p, row in enumerate(M)] + for p in range(9): + q = next(j for j in range(p, 9) if p in augmented[j]) + augmented[p], augmented[q] = augmented[q], augmented[p] + for j in range(9): + if j != p and p in augmented[j]: + augmented[j].symmetric_difference_update(augmented[p]) + inverse = [{j-9 for j in row if j >= 9} for row in augmented] + need(matmul(M, inverse) == [{j} for j in range(9)], "direct pairing inverse") + selected = [1, 4, 8] + minor = [{k for k, col in enumerate(selected) if col in inverse[row]} for row in selected] + need(rank2(minor) == 3 and not any((r, c) in low_cells for r in selected for c in selected), + "all-count direct forced minor") + return {"prior_result": "63 direct scatter floor; not a new lower bound", + "unit_subset_cases": 512, "lower_bound_histogram": dict(sorted(histogram.items())), + "forced_minor_rows": [bits(row) for row in minor], + "ten_or_more_floor": "6*n+3"} + + +def newcut_duals(word, geometry, frames, dimensions, K, episodes): + records = word["records"] + begin = episodes[-1]["end"]+1 + end = max(i for i, r in enumerate(records) if r[0] == 1 and 111 <= r[1] < 222)+1 + need((begin, end) == (24069, 25768), "actual B cut range") + need(all(r[0] in (0, 1) for r in records[begin:]), "no public COPY after B starts") + need(not any(r[0] == 1 and 471 <= r[1] < 480 for r in records[begin:end]), + "immutable source centres during B") + need(not any(r[0] == 0 and 471 <= r[1] < 480 for r in records[begin:end]), + "unchanged source centre frames during B") + for p, root in enumerate(geometry["lines"]): + cap = frames[word["final"][111+p]] + need(cap == {q for q, line in enumerate(geometry["lines"]) if dot(root, line) == 0}, + "actual final target caps") + for j, separator in enumerate(SEPARATORS): + need(sum(j in K[p] for p in separator) % 2 == 1, "dual detects actual deleted column") + # Nine scalar terminal observables, packed into nine bits per register. + # This checks only supplied duals, not a reconstructed 111-vector span. + dual = [0]*660 + for j, separator in enumerate(SEPARATORS): + for p in separator: + dual[111+p] |= 1 << j + current = word["final"][:] + next_move = [None]*660 + compatibility = {} + eligible = [0]*9 + checked = 0 + for index in range(len(records)-1, begin-1, -1): + op, dst, src, coefficient, frame, phase = records[index] + if op == 0: + need(current[dst] == coefficient, "reverse frame state") + current[dst] = src + next_move[dst] = coefficient + else: + need(op == 1, "only MOVE/ADD in B suffix") + dual[src] ^= dual[dst] + next_move[dst] = next_move[src] = None + if index > end: + continue + for d in range(660): + old = current[d] + new = old if next_move[d] is None else next_move[d] + pair = old, new + if pair not in compatibility: + need(frames[old] <= frames[new], "legal early MOVE splitting") + allow = 0 + for j in range(9): + F = frames[510+j] + intersection = dimensions[new] - int(not frames[new] <= F) + if (dimensions[old] < 7 and frames[old] <= F and + intersection >= max(3, dimensions[old])): + allow |= 1 << j + compatibility[pair] = allow + allowed = compatibility[pair] + need(dual[d] & allowed == 0, "every permitted B insertion annihilated") + for j in range(9): + eligible[j] += (allowed >> j) & 1 + checked += 1 + need(checked == 1700*660, "every B cut and persistent receiver checked") + # If the desired missing vector itself is granted, the dual must reject. + def verify_added_vector(j, vector): + need(sum(p in vector for p in SEPARATORS[j]) % 2 == 0, + "every permitted B insertion annihilated") + controls = [reject("granting missing output invalidates dual", + lambda: verify_added_vector(0, {p for p, row in enumerate(K) if 0 in row}), + "every permitted B insertion annihilated")] + # A zero vector is a harmless equivalent null insertion. + verify_added_vector(0, set()) + return {"cuts": 1700, "cut_receiver_pairs": checked, + "per_centre_eligible_cut_receiver_counts_including_zero_influences": eligible, + "dual_target_ports": SEPARATORS, "controls": controls, + "scope": "arbitrary real G subset of original F, dim G >= 3, original B gates, early MOVE splitting"} + + +def late_factory_duals(word, frames, dimensions, K, episodes): + """Verify fixed duals without constructing or solving influence spans.""" + records = word["records"] + end = episodes[0]["begin"] + cases = [] + for j, separator in enumerate(LATE_SEPARATORS): + begin = 1+max(i for i, r in enumerate(records) + if r[0] == 1 and r[1] == 471+j and r[-1] == 2) + lo, hi = episodes[j]["begin"], episodes[j]["end"] + need(all(r[0] != 0 for r in records[lo:hi+1]), + "removed original COPY episode has no ordinary MOVE") + need(sum(j in K[p] for p in separator) % 2 == 1, "late dual detects missing column") + current = word["final"][:] + [None] + dual = [0]*661 + next_move = [None]*661 + for p in separator: + dual[111+p] = 1 + eligible = checked = 0 + compatibility = {} + for index in range(len(records)-1, begin-1, -1): + op, dst, src, coefficient, frame, phase = records[index] + if lo <= index <= hi: + continue_event = True + else: + continue_event = False + if op == 0: + need(current[dst] == coefficient, "late reverse frame state") + current[dst] = src + next_move[dst] = coefficient + elif op == 1: + dual[src] ^= dual[dst] + next_move[dst] = next_move[src] = None + elif op == 2: + dual[dst] ^= dual[660] + dual[660] = 0 + next_move[dst] = None + else: + need(op == 3, "late known operation") + dual[660] = 0 + next_move[dst] = None + if index > end: + continue + # The cut at a removed COPY still exists in the changed word. + need(not continue_event or index == end == lo, + "only the first removed COPY can meet the late cut range") + need(current[471+j] == 510+j, "completed late source has its actual frame") + for d in range(660): + old = current[d] + new = old if next_move[d] is None else next_move[d] + pair = old, new + if pair not in compatibility: + F = frames[510+j] + need(frames[old] <= frames[new], "late legal early MOVE splitting") + intersection = dimensions[new]-int(not frames[new] <= F) + compatibility[pair] = ( + dimensions[old] < 7 and frames[old] <= F + and intersection >= max(3, dimensions[old])) + allowed = compatibility[pair] and d != 471+j + need(not allowed or dual[d] == 0, "every late factory insertion annihilated") + eligible += int(allowed) + checked += 1 + need(checked == (end-begin+1)*660, "complete late cut-receiver census") + cases.append({"centre": j, "cuts": end-begin+1, "completed_cut": begin, + "dual_target_ports": separator, "cut_receiver_pairs": checked, + "eligible_pairs_including_zero_influences": eligible}) + need(sum(row["cuts"] for row in cases) == 254, "all 254 late cuts") + return {"cases": cases, "cuts": 254, + "scope": "completed unchanged centre, before first original COPY, dim G>=3 in F, other gates fixed"} + + +def native_witness(geometry, full, A, Ainv, R, K): + roots = geometry["lines"] + need(geometry["ports_geom"][0] == 0 and full["lines"][0] == roots[0], + "actual identity root-zero chart") + need(roots[0] == [2,0,0,0,0,0,0,0] and roots[3] == [0,0,0,2,0,0,0,0], + "actual root-zero and root-three axes") + need(0 in A[3], "native witness code edge") + head = [Q(int(j == 59)) for j in range(64)] + need(59 not in list(range(8))+list(range(50, 57)), "stage-seven fresh-block exclusion") + # Identity cap chart at port zero sends fresh coordinate 59 to core e3. + routed = [Q(int(j == 3)) for j in range(64)] + root = roots[3] + tail = [Q(root[i]*dot(root, routed[:8]), 4) if i < 8 else Q(0) for i in range(64)] + need(tail == routed, "destination rho fixes root and N produces TAIL") + need(head != tail and tail[3] == 1 and head[3] == 0, "uncancelled native impulse") + # The inverse stage-seven branch, whatever A^-1[3,0], remains at e59. + residual_addresses = {(tuple([Q(0)]*64), tuple(tail))} + if 0 in Ainv[3]: + residual_addresses.add((tuple(head), tuple([Q(0)]*64))) + need(any(any(t) for _, t in residual_addresses), "nonzero quotient by TAIL-zero support") + clean = [row ^ {p} for p, row in enumerate(matmul(K, R))] + edge_witnesses = [] + for p, row in enumerate(clean): + q = next((q for q in sorted(row) if q != p), None) + need(q is not None, "off-diagonal clean-map entry on every row") + entries = [(i, j) for i in range(8) for j in range(8) + if roots[p][i]*roots[p][j] != roots[q][i]*roots[q][j]] + need(entries, "distinct actual root projector") + edge_witnesses.append([p, q, *entries[0]]) + mutant = [set(row) for row in A] + mutant[3].remove(0) + def require_edge(matrix): + need(0 in matrix[3], "native witness code edge") + controls = [reject("delete native witness edge", lambda: require_edge(mutant), + "native witness code edge")] + return {"input_last_target_port": 0, "HEAD_axis": 59, "TAIL": 0, + "output_source_ports": 3, "output_TAIL_axis": 3, + "coefficient": 1, "all_four_source_banks": True, + "integral_witness_all_odd_ADDRESS_fields": True, + "restricted_row_repair_obstructions": edge_witnesses, "controls": controls, + "scope": "native two-stage identity is a written operator proof; this is its exact integral witness"} + + +@lru_cache(None) +def log_interval(x): + """atanh series after binary range reduction, with the entire tail.""" + x = Q(x) + need(x >= 1, "logarithm domain") + exponent = 0 + while x >= 2: + x /= 2 + exponent += 1 + def small(y): + z = (y-1)/(y+1) + s = sum((2*z**(2*n+1)/Q(2*n+1) for n in range(24)), Q(0)) + tail = 2*z**49/(49*(1-z*z)) + return s, s+tail + a, b = small(x) + c, d = small(Q(2)) + return a+exponent*c, b+exponent*d + + +@lru_cache(None) +def power_interval(r, b): + lo, hi = log_interval(Q(64, r)) + lo, hi = b*lo, b*hi + need(0 <= lo <= hi < 1, "exponential domain") + def polynomial(x): + term = Q(1) + total = term + for n in range(1, 17): + term *= x/n + total += term + return total, term + lower, _ = polynomial(lo) + upper, term = polynomial(hi) + tail = term*hi/17/(1-hi/18) + return lower, upper+tail + + +def bill(hist, b, surcharge=True, free_bad=False): + lo = hi = Q(0) + for r, count in hist.items(): + a, z = power_interval(r, b) + lo += r*count*a + hi += r*count*z + if free_bad: + lo *= 1-Q(1, 10**16) + hi *= 1-Q(1, 10**16) + if surcharge: + a, z = power_interval(1, b) + c = Q(32*64**2, 10**16)*sum(hist.values()) + lo += c*a + hi += c*z + return lo-64*28416, hi-64*28416 + + +def compact(interval): + grid = 10**18 + a, b = interval + lower = a.numerator*grid//a.denominator + upper = -((-b.numerator*grid)//b.denominator) + return [str(Q(lower, grid)), str(Q(upper, grid))] + + +def complete_hist(local, full, partial): + F, D = 288*full, 288*partial + t = Q(F-56832, 5) + need(0 <= D <= 14208 and 0 <= t <= 14208, "owner LP resource regime") + hist = {r: Q(288*n) for r, n in local.items()} + for r, n in ((7,21312), (6,14208), (4,14208-D), (2,t), (42,14208-t)): + hist[r] = hist.get(r, 0)+n + return hist + + +def budgets(original_hist, coded_hist, selected): + K = Q(11, 10)*Q(selected["kappa"]) + b = K/(1-K) + need(b == Q(9199631083923049, 4990800368916076951), "exact target primitive") + G = complete_hist(original_hist, 433, 5) + C = complete_hist(coded_hist, 429, 0) + need(sum(r*n for r, n in G.items()) == 1808352 and + sum(r*n for r, n in C.items()) == 1791072, "complete returned owner mass") + one = G.copy() + one[7] -= 288 + one_bill = bill(one, b) + allowance = -one_bill[1]/288, -one_bill[0]/288 + coded_bill = bill(C, b) + coded_allowance = -coded_bill[1]/288, -coded_bill[0]/288 + need(Q(49779,10000) < allowance[0] <= allowance[1] < 5, + "one-copy allowance below five") + need(Q(583826,10000) < coded_allowance[0] <= coded_allowance[1] < Q(583827,10000), + "coded allowance") + five = one.copy() + five[5] = five.get(5,0)+288 + five_excess = bill(five, b, surcharge=False, free_bad=True) + need(five_excess[0] > 0, "rank-five replacement fails raw LP with free bad packets") + repair = C.copy() + repair[7] = repair.get(7,0)+9*288 + repair_excess = bill(repair, b, surcharge=False, free_bad=True) + need(repair_excess[0] > 0, "nine common-cap repairs fail favorable raw LP") + direct = G.copy() + for r, n in ((6,4), (7,-8), (8,4)): + direct[r] = direct.get(r,0)+288*n + direct_excess = bill(direct, b, surcharge=False, free_bad=True) + need(direct_excess[0] > 0, "direct relaxed histogram target rejected") + return {"target": str(b), "one_COPY7_allowance_per_call": compact(allowance), + "coded429_allowance_per_call": compact(coded_allowance), + "free_bad_rank5_replacement_excess": compact(five_excess), + "free_bad_nine_rank7_repair_excess": compact(repair_excess), + "free_bad_direct_histogram_excess": compact(direct_excess), + "arithmetic": "24-term atanh after binary range reduction; 16-term exp; rational entire tails", + "scope": "fixed-stock raw owner LP premise; charged displayed-primal allowances"} + + +def main(): + parser = argparse.ArgumentParser() + parser.add_argument("--output", type=Path, default=HERE/"review-result.json") + args = parser.parse_args() + word = read("inputs/WORD.json.gz") + geometry = read("inputs/GEOMETRY.json.gz") + coded = read("composition/inputs/CODED-WORD.json.gz") + full = read("composition/inputs/FULL-E8-GEOMETRY.json.gz") + selected = read("bounds/inputs/SELECTED-ASSEMBLY.json") + read("inputs/global_k.py.txt") # Identity derivation reviewed against these exact bytes. + frames, bases, dimensions = frame_audit(geometry) + ordinary_hist, ordinary_moves = frame_word(word, frames, dimensions) + coded_hist, coded_moves = frame_word(coded, frames, dimensions) + R, K, M, A, Ainv, W, snapshots, episodes, source = source_audit(word, geometry, coded) + low_cells, census = geometry_census(geometry, frames, bases, R, K, M, W) + output = { + "status": "PASS_INDEPENDENT_COPY_PATH_REVIEW", + "no_new_kappa": True, + "input_pins": PINS, + "checker_sha256": sha256(Path(__file__).read_bytes()).hexdigest(), + "source_operators": source, + "exact_geometry": { + "rational_closed_frames": len(frames), + "frame_dimension_histogram": dict(sorted(Counter(dimensions).items())), + "ordinary_monotone_moves": ordinary_moves, "coded_monotone_moves": coded_moves, + **census, + }, + "ordinary_local_histogram": ordinary_hist, + "coded_local_histogram": coded_hist, + "credited_direct_floor": old_direct_floor(M, low_cells), + "newcut_separators": newcut_duals(word, geometry, frames, dimensions, K, episodes), + "late_factory_separators": late_factory_duals(word, frames, dimensions, K, episodes), + "native_counterexample": native_witness(geometry, full, A, Ainv, R, K), + "caller_budget": budgets(ordinary_hist, coded_hist, selected), + "limits": [ + "The finite-branch dimension theorem and native stage composition are written proofs.", + "The 63 direct factorization floor is credited prior work.", + "No new physical private-MOVE ABI, dirty-owner schedule, rounded precision contract, or supplier admission.", + "No general changed-total, changed-factory, changed-caller, or routed-decoder lower bound.", + ], + } + args.output.write_text(json.dumps(output, sort_keys=True, indent=2)+"\n") + print("PASS_INDEPENDENT_COPY_PATH_REVIEW no_new_kappa") + + +if __name__ == "__main__": + main() diff --git a/research/bit-copy-path/inputs/FULL-CALLER-PROOF.md b/research/bit-copy-path/inputs/FULL-CALLER-PROOF.md new file mode 100644 index 0000000000..f08bcbbf1a --- /dev/null +++ b/research/bit-copy-path/inputs/FULL-CALLER-PROOF.md @@ -0,0 +1,528 @@ +# Full BIT caller: the price of retaining a rank-four helper for COPY + +Apache-2.0. Substantial OpenAI Codex assistance. + +This note has two results. First, it prices every local call and idle +completion in the six constructed nine-stage callers. A protected old host +preserves the earlier strict COPY saving, but all three consumer callers have +a strictly larger recursive query tariff, including the inherited surcharge, +than the constructed ordinary caller that reuses those hosts. This comparison +holds for every power exponent `03`, strict concavity gives +`Phi(r+4)-Phi(r) < Phi(7)-Phi(3)`, so (2) saves query cost. +For `r=0`, it costs `Phi(4)+Phi(3)-Phi(7)>0` by strict subadditivity and +also adds one child to the surcharge. At `r=3` the power change is zero. + +The owner construction emits the following six complete invoices. Each +has all 288 local calls, all 126,144 helper roles, all 71,040 idle endpoint +rows including rank zero, 10,369,440 local ADDs, 63,936 bridge ADDs and +2,592 COPY/ERASE episodes. There is no additional helper bank. +`check_invoices.py` reconstructs each histogram directly from the literal +idle rows and the actual local H, rather than importing the owner checker. + +| Policy | Ordinary reference root bracket, including surcharge | Consumer root bracket, including surcharge | +|---|---|---| +| Reuse old hosts fully, `r=0` | `(0.001742200652684, 0.001742200652686)` | `(0.001742059196608, 0.001742059196610)` | +| Limit later old-host fill to 16, `r=8` | `(0.001741631982308, 0.001741631982311)` | `(0.001741716310006, 0.001741716310008)` | +| Freeze old hosts, `r=24` | `(0.001741582528043, 0.001741582528045)` | `(0.001741780146956, 0.001741780146958)` | + +The first reference has 901,625 children. The other five invoices have +901,725. Every row has the same stock and rank mass. Each lower endpoint +has a certified upper bill strictly below `64W`; each upper endpoint has +a certified lower bill strictly above it. Positive coefficients make the +bill strictly increasing, so these are root brackets, not failed +sufficient tests. + +The two protected-host policies preserve the matched consumer saving. +However, protection has an opportunity cost: the ordinary reference that +fully reuses the old hosts is cheaper than all three consumer versions. + +This last statement is uniform in `b`, not just a comparison of decimal +roots. For each consumer let +`D=H_consumer-H_reuse_reference`. The exact receipt verifies + +\[ +\sum_r rD(r)=0,\qquad +\sum_rD(r)\min(r,t)\ge0 +\] + +at every rank knot, with strict positivity somewhere. Between knots the +second expression is linear and outside them it is either zero or +constant. The integral representation of every strictly concave power +with exponent in `(0,1)` then gives +`sum D(r)Phi_b(r)>0`. Each consumer also has 100 additional children. +Thus the complete inherited tariff is strictly larger for all `00, \tag{6} +\] +\[ +5\Phi(28)+3\Phi(2)-5\Phi(4)-3\Phi(42)>0. \tag{7} +\] + +To prove both, cancel the common factor `64^b` and put `p=1-b`. +Each expression is a nonzero sum of four functions `r^p` with distinct +positive bases. Such a function has at most three real zeros counting +multiplicity: divide by its smallest exponential and differentiate, then +apply Rolle's theorem inductively to the remaining three exponentials. + +Both expressions vanish at `p=0` and `p=1`. Their derivative at zero is +strictly positive and their derivative at one is strictly negative. These +signs are exact integer comparisons: + +\[ +\frac{2\,14^5}{6^5\,42}>1,\qquad +\frac{2^2\,14^{70}}{6^{30}\,42^{42}}<1, +\] +\[ +\frac{2^3\,28^5}{4^5\,42^3}>1,\qquad +\frac{2^6\,28^{140}}{4^{20}\,42^{126}}<1. +\] + +They imply positivity immediately inside both endpoints. A failure of +strict positivity between them would contribute at least two further +zeros counting multiplicity, contradicting the three-zero bound. +Thus (6) and (7) hold throughout `00`. +This is an opportunity-cost theorem for the whole stated resource +regime, with the same bin shapes and whole-helper contracts. The checker +verifies the resource and donation identities coefficient by coefficient, +and verifies the actual nine-stage inventory as a specialization; it +does not infer this corollary from a finite sample of parameters. + +Consequently (4) is the optimal relaxed bill for all those exponents. +Its per-donor change is + +\[ +\Phi_b(4)+\Phi_b(3)-\Phi_b(7)>0. \tag{9} +\] + +An optimal allocation retains each rank-four helper to erase a rank-four +public remainder. The apparent COPY saving forfeits that more valuable +use. This is a global opportunity-cost statement within the stated +inventory, not a claim that every individual legal consumer placement +must worsen its matched reference. + +The optimistic roots, with zero surcharge, are: + +| Donated helpers k | Strict root bracket of H*(k) | +|---:|---| +| 0 | `(0.001743393260913,0.001743393260914)` | +| 100 | `(0.001743251611103,0.001743251611104)` | +| 900 | `(0.001742119241980,0.001742119241981)` | +| 1,440 | `(0.001741355725345,0.001741355725346)` | + +Equations (4) and (9) cover all intermediate donor counts without +1,441 separate searches. The zero-donor upper endpoint is a ceiling +for the entire mechanism, regardless of chronology or integrality. + +### Granting every bad packet free + +The LP theorem concerns the rank tariff. Dropping its positive fallback +surcharge does not by itself discount the good work by the exceptional +residue fraction. To strengthen the fixed-stock screen, assume the +inherited good/bad split guarantees at least a `1-epsilon` fraction of +each child's good work, where `epsilon=10^-16`. Grant every bad packet +zero cost and check the still more favorable bill + +\[ +\widetilde B_k(b)=(1-\epsilon)\sum_r H^*(k)(r)\Phi_b(r). +\] + +At every upper endpoint in the table and at the necessary 10% endpoint, +the checker proves `widetilde B_k(b)>64W` using exact lower bounds. +For `k=0` at its stated upper endpoint the raw excess is greater than +`2.6803*10^-9`, whereas `epsilon*64W=1.818624*10^-10`; the exact +discounted excess remains positive. Thus the same displayed decimal +ceiling survives this discount. Multiplying (9) by `1-epsilon` retains +strict positivity, so the zero-donor rejection covers every donor count. + +This statement fixes the public stock at `W=28416` and concerns the +native moment screen under the stated good-fraction premise. It does +not reinterpret the rank tariff as a lower bound on every possible +algorithm's actual running time. + +### Additional full-progress helper banks + +At every certified upper endpoint and target in `SCREENS.json`, exact +signs also give `lambda8<-8` and `lambda4<-4`. Suppose `A` full helpers +and `B` partial helpers move from public hosts to extra full-progress +stock `X`. Grant that stock an ideal rank-64 completion, even though +rank 64 is not a contracting recursive child. Its idle mass is +`64X-8A-4B>=0`; its cost minus its output-stock credit is at least +`-8A-4B`. Applying the public dual to the reduced helper inventories +raises the total lower bound by + +\[ +A(-\lambda_8-8)+B(-\lambda_4-4)\ge0. +\] + +Thus added fully paid full-progress banks cannot evade any certified +ceiling or target rejection. This extension is asserted at the certified +exponents, not for all `0\frac K{1-K} +=\frac{9199631083923049}{4990800368916076951} +>0.001843317785504024003736989724. \tag{10} +\] + +At (10), even `H*(0)` exceeds `64W` by more than +`590.603137635021650149691091093154` raw units. The 100-donor relaxation +exceeds it by more than `591.488492471965400762795010866872`. +Every further donor worsens the bill by (9). All omitted costs favor +acceptance, so the 10% target is excluded for the entire stated mechanism. +The exact discounted checks above also reject that target while granting +every bad packet free at fixed public stock; the optional extra-bank +extension retains its separate unit-mass completion premise. + +The best relaxed ceiling yields + +\[ +\kappa<\frac{0.001743393260914}{1+0.001743393260914}, +\] + +less than a `4.047382%` gain over the selected record, even with an ample +complex supplier. This is a ceiling, not an achieved gain. With the +selected complex supplier unchanged, the certified substitution gives +zero κ improvement. + +## 6. Verification, dependencies and remaining choices + +Run from the checkout root: + +```sh +python3 -B research/bit-full-caller/bounds/check_screens.py --output /tmp/bit-screens.json +python3 -O -B research/bit-full-caller/bounds/check_screens.py --output /tmp/bit-screens-opt.json +python3 -B research/bit-full-caller/bounds/check_invoices.py --output /tmp/bit-invoices.json +python3 -O -B research/bit-full-caller/bounds/check_invoices.py --output /tmp/bit-invoices-opt.json +``` + +Normal and optimized modes must produce identical receipts. The interval +engine uses only integers and exact fractions: a positive atanh series +after binary range reduction for logarithms, a positive Taylor series for +exponentials, explicit omitted-tail bounds, and outward rounding to a +256-bit dyadic grid. No floating-point result decides a comparison. + +Negative controls reject a missing helper in the primal resource rows, +inflation of a tight dual price, omission of public idle mass, retention +of the old stage-zero delta after full reuse, and the selected endpoint's +next grid point. Source extraction retains assertions as explicit checks +under `python -O`. Local arithmetic sources are loaded from their actual +read bytes with `compile(..., dont_inherit=True, optimize=0)`, bypassing +local `.pyc` caches. Semantic controls ignore a deliberately poisoned +module cache and preserve a deliberately failing source assertion even +when the parent interpreter runs with `-O`. + +`../independent/check_inventory.py` provides a separate arithmetic route: +integer nth-root bisection at `b=1/573`, with independently derived +source inventories and no imported moment engine. Its weaker endpoint +also rejects the 10% target. It does not independently certify the tight +decimal root brackets above. + +The schedule specialist owns physical-family disjointness, literal frame +charts, arbitrary-input restoration and every later source-window +obligation. The native specialist owns copied-total ABI, real adapter +costs, changed integer-prefix growth, precision, source access and tape. +Those obligations must hold before any positive query endpoint becomes +a supplier endpoint. + +The allocation theorem assumes unchanged nine-stage public window ranks, +whole rank-eight/rank-four helper progress that occupies disjoint +directions within each window, the fixed local H, and direct rank-four +completion of rank-seven COPY jobs. It does not exclude a changed local +word, genuine deletion/sharing of an entire COPY path, new public +chronology, nonorthogonal operator fusion, or independently useful +partial-output demand. Each changes the mathematical problem and needs +a new complete inventory and invoice. +This is a ceiling for the stipulated recursive-query tariff and transfer +criterion, not a lower bound against every possible multiplication +algorithm or a changed native tariff. + +Credit: the retained word and geometry are predecessor R438 work; +the public controller credits DreamingOfClouds, Zhihao Chen, eumemic and +chafreaky, with the hosting/owner lineage retained in the frozen package. +The selected assembly is Rohan Arun's PR1094 composition using the cited +PR589 supplier; the exact outer functions are Wronskode's retained +seed81 audit. This note contributes complete consumer pricing, the +whole-inventory mixed-width primal/dual optimum and its explicit ceiling. diff --git a/research/bit-copy-path/inputs/FULL-CALLER-README.md b/research/bit-copy-path/inputs/FULL-CALLER-README.md new file mode 100644 index 0000000000..3d9d55a889 --- /dev/null +++ b/research/bit-copy-path/inputs/FULL-CALLER-README.md @@ -0,0 +1,219 @@ +# Full recursive BIT caller: the opportunity cost of a cheaper COPY + +Apache-2.0. Substantial OpenAI Codex assistance. + +The saved 100-job COPY consumer integrates into the complete nine-stage +Boolean caller, but it does **not** improve the complete recursive query +bill over the best ordinary caller constructed here. A stronger result +explains why: in a relaxation permitting every allocation of the fixed +whole-helper inventory, donating a rank-four helper to these COPY queries +strictly worsens the optimum for every exponent `03`. Filling that host completely gives `e_4+e_3-e_7`, a strict +increase for every `0Phi_b(7)`, every donor strictly +increases this optimum. Keeping the helper to eliminate an entire +rank-four idle query is more valuable than reducing COPY7 to COPY3. + +The same dual also proves this opportunity-cost result throughout a +resource regime: for bin counts `C7,C14,C28,C42`, full-helper inventory +`F` between `C14+3*C28` and `C14+3*C28+5*C42`, and partial inventory +`0<=D<=C28`. It is therefore not an isolated numerical property of the +288-call witness. Available distinct COPY jobs and the same whole-helper +and bin contracts remain premises. + +The zero-donor relaxed exponent satisfies + +```text +0.001743393260913 < b_relaxed < 0.001743393260914. +``` + +The fixed-stock ceiling remains valid after discounting every good-path +query cost by `1-10^-16` and granting all bad packets zero cost. +This additional check avoids treating rare fallback packets as though +they necessarily incur the full good-path tariff. + +Exact point certificates extend the ceiling to additional full-progress +helper banks **under the explicit tariff premise that their completion +cost is at least the remaining rank**. No free-bad-packet discount is +applied to that bank premise. Independently useful partial outputs or +unconstrained physical bank implementations are outside this extension. + +[The full proof](bounds/PROOF.md) includes the resource equations, +all-parameter inequalities, exact interval certificates and negative +controls. A separate [integer-root checker](independent/check_inventory.py) +derives the source inventory independently and verifies a weaker ceiling +at `b=1/573`, without the logarithm/exponential engine. + +## Consequences for κ + +The best constructed ordinary query profile has a charged exponent in +`(0.001742200652684,0.001742200652686)`. The consumer with unrestricted +reuse is lower, in `(0.001742059196608,0.001742059196610)`. +The protected-host consumer profiles are lower still. + +The ordinary R438 profile improves the query screen over the selected +R439 BIT endpoint by about 0.506%. That comparison is conditional on +missing native implementation premises, and the new consumer is not +responsible for it. + +The pinned selected assembly has +`kappa=0.001672660197076918` and binds on the complex supplier. Substitution +of any of these six BIT screens leaves that selected κ unchanged; all +47 retained outer inequalities are checked. + +Even with an ample complex supplier, the necessary outer inequality +`kappa < b/(1+b)` and the optimistic ceiling above permit less than a +4.047382% gain over the pinned selected record. They exclude a 10% gain +for the **entire fixed mechanism**, before charging its missing +implementation costs. This is a ceiling, not an achieved improvement. + +## Native costs and remaining physical boundary + +The 100 rank-three complements have exact dyadic charts valid in every +odd prime-power ring. On its good branch, a positive rank-r child uses +the pinned generic adapter's `2[r(127-r)+4096] f` affine slots; a zero +remainder uses no child or adapter. Unrestricted host reuse therefore +adds **824,000 f** good-adapter slots as well as 100 recursive children. +The two protected +policies save 8,000 f and 33,600 f relative to their own references, but +their full query profiles already lose to ordinary reuse. + +All six schedules retain 28,416 public families, the inherited nine-family +private reserve, 10,369,440 local ADDs, 63,936 bridge ADDs and 2,592 +COPY/ERASE episodes per common cover class. The operation counts multiply +by the cover size; physical family stock does not. Complete packet +volume, spectators, held consumer packets, routing, descriptors, physical +head returns and parking remain charged. + +An exact whole-caller signed scalar calculation gives the bound +`2^18 * 1874 * 1302^4 * 1446^4 < 2^113` for unit-bounded inputs at +completed primitives. This does **not** admit an overall 128-bit rounded +recursive implementation. + +The existing physical Catalogue hardcodes the old word, private slot, +replica-symmetric owners and whole-call order. It has not been adapted +here. Its complete source/packet ABI, faster chart routing, rounded +precision and concrete tape/storage installation remain open. +The unfavorable necessary query ceiling does not require those +costs to be free in a real implementation: granting them for free only +makes the negative conclusion stronger. + +See [the native proof and ledger](native/PROOF.md). A positive supplier +claim would require closing those implementation obligations. + +## Reproduction and provenance + +The scientific base is `12fa8c5c844be21bfefb38a19503fbd867a11244`. +The entire predecessor export is preserved under `inputs/consumer`; +[PREDECESSOR.json](PREDECESSOR.json) binds its 118 files to commit +`b19253416b4a29002b25699ba37920f99c1670fc`. Historical receipts remain +historical evidence; they are not relabeled as fresh full-suite execution. + +Run the new package from any directory with Python 3.12 or later: + +```sh +python3 -I -B /path/to/bit-full-caller/verify.py --output /tmp/bit-full-caller.json +``` + +The verifier checks the immutable source manifest, executes every new +checker normally and with `-O`, compares the complete receipts, and +rechecks source bytes. It requires no Git checkout or network. +See [REPLAY.json](REPLAY.json) and [the independent review](independent/review.md). + +The new contribution is the complete delayed-idle integration, the +full-profile reversal and dominance result, and the mixed-width +whole-inventory optimum and its outer ceiling. It does not exclude a +changed local word, deletion or sharing of an entire COPY path, changed +public chronology, nonorthogonal residual fusion, or a genuinely useful +partial-output contract. Those are changes to the mathematical model, +not further packing searches within this one. + +## Open next questions + +* Can an actual local query path be removed or shared, changing `Hlocal`? + The first useful step is an exact arbitrary-dirty operator witness for + one concrete COPY/ADD interval, including its source reads and inverse + cleanup. The obstruction here applies while that complete local + histogram remains fixed. +* Can a required partial-progress output absorb the exported helper's + progress? Start with a caller that already demands that exact output + projector, and price its entrance, terminal and later lifetime. An + unused new output would not change the obstruction. +* Can the ordinary R438 query profile be installed as a native supplier + for an independent reason? Adapt the concrete Catalogue and routing + grammar first, then verify complete volume, rounded precision and + parking. Its modest BIT improvement still needs an improved complex + supplier before it can improve the selected κ. + +Attribution and the read-only overlap scope are in +[RELATED-WORK.md](RELATED-WORK.md). 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zD^uA+gRkgp*g~687wQThhJmU_tMyNsZFKV7W(o}SL^Wy8bI#4yU39jZV{*8kJd83- zl^Ep%ky7p#X-4;*-@%0>IEtpr6B!DzbY6x40oDHDL>*p<4kMf@D$8Y;ufgs2Ce%~Oi{Ib{2OmJsGzO@AD5pG^?h{adw%%4Aw2xzYf+++lC^xB=8*pp H`HlYziN5}> literal 0 HcmV?d00001 diff --git a/research/bit-copy-path/inputs/full-check-screens.py.txt b/research/bit-copy-path/inputs/full-check-screens.py.txt new file mode 100644 index 0000000000..319287343a --- /dev/null +++ b/research/bit-copy-path/inputs/full-check-screens.py.txt @@ -0,0 +1,502 @@ +#!/usr/bin/env python3 +# SPDX-License-Identifier: Apache-2.0 +"""Exact full-inventory BIT allocation ceiling, with primal and dual witnesses. + +Standard library only. Explicit checks remain active with python -O. +Substantial OpenAI Codex assistance. No new kappa or native admission. +""" +import argparse +import ast +from collections import Counter +from fractions import Fraction as Q +import gzip +import hashlib +import json +from math import prod +from pathlib import Path +import sys +import types + +HERE = Path(__file__).resolve().parent + + +def compile_source(data, filename): + return compile(data, str(filename), "exec", dont_inherit=True, optimize=0) + + +def source_module(name, path): + """Execute precisely these source bytes; never import local bytecode caches.""" + module = types.ModuleType(name) + module.__file__ = str(path) + sys.modules[name] = module + exec(compile_source(path.read_bytes(), path), module.__dict__) + return module + + +_interval = source_module("_bit_bounds_exact_interval", HERE / "exact_interval.py") +add, exact, linear_phi, mul, need, phi, report, sub = ( + getattr(_interval, name) for name in + ("add", "exact", "linear_phi", "mul", "need", "phi", "report", "sub")) + +PKG = HERE.parent +CONSUMER = PKG / "inputs/consumer" +M, W, CALLS = 64, 28416, 288 +EPSILON = Q(1, 10**16) +H = {1: 1322, 2: 1342, 3: 79, 4: 39, 5: 102, 6: 11, 7: 18} +BIN_COUNTS = {7: 21312, 14: 14208, 28: 14208, 42: 14208} +N8, N4 = 433 * CALLS, 5 * CALLS +PINS = { + "inputs/selected-assembly.json": + "1122cb3a575f8c788345fc1bf4c108ac7c3250fb81a5988671f5eb04b3ab3045", + "inputs/selected-bit-expected.json": + "b9f0e5ca038abe837b0007c8fe006f366a6f02009244ac57705a8a200e2301f4", +} +ROOT_BRACKETS = { + 0: ("0.001743393260913", "0.001743393260914"), + 100: ("0.001743251611103", "0.001743251611104"), + 900: ("0.001742119241980", "0.001742119241981"), + 1440: ("0.001741355725345", "0.001741355725346"), +} + + +def digest(path): + return hashlib.sha256(path.read_bytes()).hexdigest() + + +def combine(*terms): + out = Counter() + for c, coefficients in terms: + for r, n in coefficients.items(): + out[r] += c * n + return {r: Q(n) for r, n in sorted(out.items()) if n} + + +def local_histogram(k): + need(0 <= k <= N4, "rank-four donor inventory") + return combine((CALLS, H), (k, {3: 1, 7: -1})) + + +def relaxed_idle(k): + return {2: Q(67872, 5), 4: Q(12768 + k), 6: Q(14208), + 7: Q(21312), 42: Q(3168, 5)} + + +def relaxed_histogram(k): + return combine((1, local_histogram(k)), (1, relaxed_idle(k))) + + +def raw_excess(histogram, b): + return sub(linear_phi(histogram, b), exact(M * W)) + + +def discounted_excess(histogram, b): + """Grant every bad packet free and retain only the uniform good fraction.""" + return sub(mul(linear_phi(histogram, b), 1-EPSILON), exact(M*W)) + + +def validate_primal(p, k): + need(all(n >= 0 for n in p.values()), "nonnegative primal counts") + for R, count in BIN_COUNTS.items(): + need(sum(n for (r, _, _), n in p.items() if r == R) == count, + "all bins accounted") + need(sum(a * n for (_, a, _), n in p.items()) == N8, "all rank-eight helpers") + need(sum(b * n for (_, _, b), n in p.items()) == N4-k, "all retained rank-four helpers") + idle = Counter() + for (R, a, b), n in p.items(): + q = R - 8 * a - 4 * b + need(q >= 0, "pattern capacity") + if q: + idle[q] += n + need(dict(idle) == relaxed_idle(k), "primal idle histogram") + + +def primal(k): + """Fractional bin pattern counts (R, full-eight helpers, partial-fours).""" + n4 = N4 - k + p = { + (7, 0, 0): Q(21312), + (14, 1, 0): Q(14208), + (28, 3, 0): Q(14208 - n4), + (28, 3, 1): Q(n4), + (42, 5, 0): Q(67872, 5), + (42, 0, 0): Q(3168, 5), + } + validate_primal(p, k) + return [{"R": R, "full8": a, "partial4": b, "count": str(n)} + for (R, a, b), n in sorted(p.items())] + + +def dual_coefficients(R, a, b): + """Coefficients of Phi(q)-lambda8*a-lambda4*b-mu_R.""" + q = R - 8 * a - 4 * b + l8 = {2: Q(1, 5), 42: -Q(1, 5)} + l4 = {4: Q(-1)} + mus = {7: {7: Q(1)}, + 14: combine((1, {6: 1}), (-1, l8)), + 28: combine((1, {4: 1}), (-3, l8)), + 42: {42: Q(1)}} + return combine((1, {q: 1} if q else {}), (-a, l8), (-b, l4), (-1, mus[R])) + + +def certify_dual(b): + """Every feasible integer occupancy pattern is covered, including rank zero.""" + result, equality = [], [] + for R in BIN_COUNTS: + for a in range(R // 8 + 1): + for p in range((R - 8 * a) // 4 + 1): + coefficients = dual_coefficients(R, a, p) + if not coefficients: + equality.append([R, a, p]) + continue + interval = linear_phi(coefficients, b) + need(interval[0] > 0, f"strict dual feasibility R={R},a={a},p={p}") + result.append({"pattern": [R, a, p], "slack": report(interval)}) + # Every positive primal column is an exact equality column, for all k. + for k in (0, N4): + for row in primal(k): + if Q(row["count"]) > 0: + need(not dual_coefficients(row["R"], row["full8"], row["partial4"]), + "complementary slackness") + l8 = linear_phi({2: Q(1, 5), 42: -Q(1, 5)}, b) + l4 = mul(phi(4, b), -1) + bank8, bank4 = sub(mul(l8, -1), exact(8)), sub(mul(l4, -1), exact(4)) + need(bank8[0] > 0 and bank4[0] > 0, "full-progress banks cannot improve the lower bound") + discounted_bank8 = sub(mul(l8, -(1-EPSILON)), exact(8)) + discounted_bank4 = sub(mul(l4, -(1-EPSILON)), exact(4)) + need(discounted_bank8[0] > 0 and discounted_bank4[0] > 0, + "discounted public dual still pays for unit-priced bank completions") + return {"b": str(b), "patterns_checked": len(result) + len(equality), + "exact_equality_patterns": equality, "strict_slacks": result, + "extra_full_progress_bank_penalties": + {"per_full8": report(bank8), "per_partial4": report(bank4), + "discounted_public_per_full8": report(discounted_bank8), + "discounted_public_per_partial4": report(discounted_bank4), + "premise": "extra bank completions retain undiscounted unit-mass cost"}} + + +def all_parameter_dual_controls(): + """Integer-product endpoints for the two four-power inequalities in PROOF.md.""" + rows = { + "rank14_chord": [(2, 1), (6, -5), (14, 5), (42, -1)], + "rank28_chord": [(2, 3), (4, -5), (28, 5), (42, -3)], + } + result = {} + for name, row in rows.items(): + need(sum(c for _, c in row) == 0, "power polynomial vanishes at zero") + need(sum(r*c for r,c in row) == 0, "power polynomial vanishes at one") + def ratio(weights): + return Q(prod(r**c for r,c in weights if c > 0), + prod(r**(-c) for r,c in weights if c < 0)) + d0, d1 = ratio(row), ratio([(r, r*c) for r,c in row]) + need(d0 > 1 and d1 < 1, "strict derivative endpoint signs") + result[name] = {"coefficients": row, "exp_derivative_at_zero": str(d0), + "exp_derivative_at_one": str(d1), + "zero_count_bound": 3, + "written_consequence": "strictly positive for every 0 D-k and append the genuine COPY change k(e3-e7). + delta = {} + for r, count in idle.items(): + changed = combine((1, count), (-count.get("D", 0), donated)) + difference = combine((1, changed), (-1, count)) + if difference: + delta[r] = difference + delta[3] = combine((1, delta.get(3, {})), (1, donated)) + delta[7] = combine((1, delta.get(7, {})), (-1, donated)) + need(delta == {3: donated, 4: donated, 7: {"k": Q(-1)}}, + "parametric complete donation delta") + actual = {"C7": Q(21312), "C14": Q(14208), "C28": Q(14208), + "C42": Q(14208), "F": Q(N8), "D": Q(N4)} + need({r: sum(c*actual[v] for v,c in count.items()) for r,count in idle.items()} == + relaxed_idle(0), "actual inventory is a specialization") + encode = lambda form: {v: str(c) for v,c in sorted(form.items())} + return { + "domain": ["C7,C14,C28,C42>=0", "0<=D<=C28", + "C14+3*C28<=F<=C14+3*C28+5*C42", + "0<=k<=D; at least k genuine rank-seven COPY jobs"], + "t": encode(t), + "primal": [{"R": R, "full8": a, "partial4": d, "count": encode(count)} + for (R,a,d), count in sorted(rows.items())], + "idle_histogram": {str(r): encode(count) for r,count in sorted(idle.items())}, + "complete_donation_delta": {str(r): encode(count) for r,count in sorted(delta.items())}, + "identities": ["all bins", "F full helpers", "D partial helpers", "idle mass", + "all primal patterns dual-tight", "D-to-D-k substitution", + "actual nine-stage inventory specialization"], + "scope": "fractional counts of whole legal bin patterns; all 0"), {}) + except AssertionError: + assertion_kept = True + need(assertion_kept, "loader pins optimize=0 under every interpreter mode") + return {"poisoned_module_cache_ignored": True, "source_assertion_kept_under_optimized_python": True, + "local_bytecode_cache_read": False, + "compiled_source_sha256": {"check_screens.py": digest(HERE / "check_screens.py"), + "exact_interval.py": digest(HERE / "exact_interval.py")}} + + +def main(): + ap = argparse.ArgumentParser() + ap.add_argument("--output", type=Path) + args = ap.parse_args() + source = source_controls() + assembly = json.loads((HERE / "inputs/selected-assembly.json").read_bytes()) + selected_k = Q(assembly["kappa"]) + target_k = Q(11, 10) * selected_k + # Every retained outer construction has kappa < c/(1+c). + necessary_c = target_k / (1 - target_k) + dual_points = {necessary_c, Q(1, 500)} + roots = {} + for k, endpoints in ROOT_BRACKETS.items(): + lo, hi = map(Q, endpoints) + lv, hv = raw_excess(relaxed_histogram(k), lo), raw_excess(relaxed_histogram(k), hi) + need(lv[1] < 0 and hv[0] > 0, "strict relaxed root bracket") + discounted_hv = discounted_excess(relaxed_histogram(k), hi) + need(discounted_hv[0] > 0, "ceiling survives free bad packets at fixed public stock") + dual_points.add(hi) + roots[str(k)] = { + "lower": str(lo), "upper": str(hi), "lower_excess": report(lv), + "upper_excess": report(hv), + "upper_excess_after_good_fraction_discount": report(discounted_hv), + "primal": primal(k), + "histogram": {str(r): str(n) for r, n in relaxed_histogram(k).items()}, + "hypothetical_outer_upper": str(hi / (1 + hi)), + "hypothetical_gain_over_selected_upper_percent": + str(100 * ((hi / (1 + hi)) / selected_k - 1)), + } + duals = [certify_dual(b) for b in sorted(dual_points)] + target_rows, discounted_target_rows = {}, {} + for k in (0, 100, 900, 1440): + z = raw_excess(relaxed_histogram(k), necessary_c) + need(z[0] > 0, "10 percent target rejected") + target_rows[str(k)] = report(z) + discounted = discounted_excess(relaxed_histogram(k), necessary_c) + need(discounted[0] > 0, "10 percent target rejected after good-fraction discount") + discounted_target_rows[str(k)] = report(discounted) + delta = linear_phi({4: 1, 3: 1, 7: -1}, necessary_c) + need(delta[0] > 0, "relaxed exported rank-four helper worsens cost") + # The complete difference is affine in k, not a 1441-point search. + need(combine((1, relaxed_histogram(1440)), (-1, relaxed_histogram(0))) == + {3: Q(1440), 4: Q(1440), 7: Q(-1440)}, "entire donor interval") + # Controls deliberately break a resource row, a dual, and complete accounting. + controls = {} + mutated = {(r["R"], r["full8"], r["partial4"]): Q(r["count"]) for r in primal(0)} + mutated[(42, 5, 0)] -= 1 + mutated[(42, 4, 0)] = 1 + try: + validate_primal(mutated, 0) + except ValueError: + controls["one_missing_full_helper_rejected"] = True + need(controls.get("one_missing_full_helper_rejected"), "primal resource negative control") + z = sub(exact(0), exact(Q(1, 10**9))) + need(z[1] < 0, "inflating a tight dual price breaks its equality pattern") + controls["tight_dual_mu_inflation_rejected"] = True + incomplete = sum(r*n for r,n in local_histogram(100).items()) + need(incomplete != 1808352, "omitting public idle work cannot retain full mass") + controls["omit_idle_invoice_rejected"] = True + for r in (0, 8, 24): + z = linear_phi({r+4: 1, 3: 1, r: -1, 7: -1}, Q(1, 500)) + need(z[0] > 0 if r == 0 else z[1] < 0, "matched completed-host sign") + controls["old_host_remainder_zero_reverses_saving"] = True + result = { + "status": "PASS_EXACT_FULL_INVENTORY_BIT_CEILING", + "no_new_kappa": True, + "source": source, + "arithmetic": {"floating_point_decisions": False, + "log": "positive atanh series after binary range reduction, explicit tail", + "exp": "positive Taylor series, explicit geometric tail", + "outward_grid_bits": 256}, + "model": {"width": M, "public_stock": W, "full8_helpers": N8, + "partial4_helpers_before_donation": N4, + "bin_counts": BIN_COUNTS, "donations": "all integers 0<=k<=1440", + "relaxations": ["fractional numbers of whole legal bin patterns", + "no chronology or per-call disjointness restriction", + "zero fallback, native, precision, routing and controller cost", + "optional full-progress helper banks retain ideal unit-priced idle work"], + "each_pattern": "a,b nonnegative integers; 8a+4b<=R; one residual child of rank R-8a-4b"}, + "roots": roots, + "dual_certificates": duals, + "all_parameter_dual": all_parameter_dual_controls(), + "parametric_resource_corollary": parametric_resource_controls(), + "selected_kappa": str(selected_k), + "ten_percent_target_kappa": str(target_k), + "necessary_bit_endpoint": str(necessary_c), + "necessary_bit_endpoint_decimal": report(exact(necessary_c)), + "target_excess": target_rows, + "free_bad_packet_screen": { + "epsilon": str(EPSILON), "good_fraction": str(1-EPSILON), + "formula": "(1-epsilon)*sum H*[r]*Phi_b(r)-64W", + "fixed_public_stock": W, + "premise": "each child's good-work fraction is at least 1-epsilon", + "bad_packet_cost_granted": 0, + "discounted_target_excess": discounted_target_rows, + "all_k_argument": "the discounted optimal donor slope is " + "(1-epsilon)*(Phi4+Phi3-Phi7)>0", + "bank_scope": "optional extra bank completions are not granted a free-bad " + "discount; they must pay their full residual rank mass", + }, + "per_donor_delta_at_target": report(delta), + "all_donor_counts": "same dual works for all k; H*(k)=H*(0)+k(e4+e3-e7)", + "negative_controls": controls, + "execution_controls": execution_controls(), + "scope": "finite LP primal/dual certificate and written inventory reduction; " + "not an attained fractional caller, all-size native bound or formal-kernel proof", + } + data = json.dumps(result, sort_keys=True, indent=2) + "\n" + if args.output: + args.output.parent.mkdir(parents=True, exist_ok=True) + args.output.write_text(data) + else: + print(data, end="") + + +if __name__ == "__main__": + main() diff --git a/research/bit-copy-path/inputs/full-exact-interval.py.txt b/research/bit-copy-path/inputs/full-exact-interval.py.txt new file mode 100644 index 0000000000..d1409bf724 --- /dev/null +++ b/research/bit-copy-path/inputs/full-exact-interval.py.txt @@ -0,0 +1,117 @@ +#!/usr/bin/env python3 +# SPDX-License-Identifier: Apache-2.0 +"""Small exact rational log/exp intervals; no floating-point decisions. + +Substantial OpenAI Codex assistance. This is an arithmetic certificate +engine, not a formal-kernel proof. +""" +from fractions import Fraction as Q +from functools import lru_cache + +BITS = 256 +SCALE = 1 << BITS +LOG_TERMS = 100 +EXP_TERMS = 40 + + +def need(test, message): + if not test: + raise ValueError(message) + + +def exact(x): + x = Q(x) + return x, x + + +def add(x, y): + return x[0] + y[0], x[1] + y[1] + + +def mul(x, c): + c = Q(c) + return (x[0] * c, x[1] * c) if c >= 0 else (x[1] * c, x[0] * c) + + +def sub(x, y): + return add(x, mul(y, -1)) + + +def round_out(x): + lo = (x[0].numerator * SCALE) // x[0].denominator + hi = -((-x[1].numerator * SCALE) // x[1].denominator) + return Q(lo, SCALE), Q(hi, SCALE) + + +def _small_log(x): + """Enclose log(x), 1 <= x <= 2, by the positive atanh series.""" + need(1 <= x <= 2, "range-reduced log domain") + y = (x - 1) / (x + 1) + z, term, total = y * y, y, Q(0) + for j in range(LOG_TERMS): + total += term / (2 * j + 1) + term *= z + tail = 2 * term / ((2 * LOG_TERMS + 1) * (1 - z)) + return round_out((2 * total, 2 * total + tail)) + + +@lru_cache(None) +def log(x): + x = Q(x) + need(x >= 1, "log domain") + shift = 0 + while x > 2: + x /= 2 + shift += 1 + return round_out(add(_small_log(x), mul(_small_log(Q(2)), shift))) + + +def exp_point(x): + """Positive Taylor sum and geometric bound on its entire omitted tail.""" + need(0 <= x < 1, "exp domain") + term = total = Q(1) + for j in range(1, EXP_TERMS + 1): + term *= x / j + total += term + first = term * x / (EXP_TERMS + 1) + tail = first / (1 - x / (EXP_TERMS + 2)) + return round_out((total, total + tail)) + + +@lru_cache(None) +def power(base, exponent): + exponent = Q(exponent) + need(exponent >= 0, "nonnegative exponent") + y = mul(log(Q(base)), exponent) + return exp_point(y[0])[0], exp_point(y[1])[1] + + +@lru_cache(None) +def phi(rank, exponent, width=64): + if rank == 0: + return exact(0) + need(0 < rank <= width, "rank domain") + return mul(power(Q(width, rank), Q(exponent)), rank) + + +def linear_phi(coefficients, exponent, width=64): + total = exact(0) + for rank, coefficient in sorted(coefficients.items()): + total = add(total, mul(phi(rank, exponent, width), coefficient)) + return total + + +def report(interval, digits=30): + # For a negative number ceil(x*S)=-floor(-x*S), so this is sign-aware. + lo, hi = interval + scale = 10 ** digits + nlo = (lo.numerator * scale) // lo.denominator + nhi = -((-hi.numerator * scale) // hi.denominator) + def show(n): + sign = "-" if n < 0 else "" + n = abs(n) + return f"{sign}{n // scale}.{n % scale:0{digits}d}" + return { + "lower": str(lo), "upper": str(hi), + "decimal_lower": show(nlo), "decimal_upper": show(nhi), + } diff --git a/research/bit-copy-path/inputs/full-invoices.json b/research/bit-copy-path/inputs/full-invoices.json new file mode 100644 index 0000000000..a4ced28342 --- /dev/null +++ b/research/bit-copy-path/inputs/full-invoices.json @@ -0,0 +1,369 @@ +{ + "all_consumer_variants_dominated_by_constructed_ordinary_caller": { + "cap16": { + "consequence": "strictly dearer than reuse/reference for every 0=5 and a.stages%2==1; L=a.stages; K=(L-1)//2; T=a.replicas; M=1+7*L; NOBANK=a.no_banks + if L!=9: assert NOBANK, 'bank tiling is only emitted for the nine-stage layout; use --no-banks and host every residual' + DEFICIT=T*(2*K*120-9*H*L) + wp=a.word; w=json.loads(gzip.decompress(wp.read_bytes()) if wp.suffix=='.gz' else wp.read_bytes()) + V,R=w['v'],w['R']; LOCAL=2*V+R; DATA=2*K*V*T; LIVE=DATA+L*R + assert (w['h'],V,w['form'])==(8,120,'/2') + geo=json.loads((out/'GEOMETRY.json').read_text()) + assert geo['status']=='PASS_COMPRESSED_%s_BIT_GEOMETRY_AND_BANK_BLOCKS'%NAMES[L] and geo['stages']==L and geo['m']==M + # the port routes depend only on the labels: same 120 label vectors, in the same order + assert len(geo['ports'])==V and all(p['port']==t and p['q']==w['lines'][t] for t,p in enumerate(geo['ports'])) + rp=a.local_check/'records/249-records.bin'; raw=rp.read_bytes(); rec=list(struct.iter_unpack('<6i',raw)) + state=json.loads((a.local_check/'records/249-states.json').read_text()) + fs=json.loads((a.local_check/'records/frames.json').read_text())['frames'] + dims={int(f):z['dim'] for f,z in fs.items()} + lg=json.loads((a.local_check/'cout/legality.json').read_text()) + cc=json.loads((a.local_check/'cout/columns.json').read_text()) + assert lg['status'].startswith('PASS') and cc['status'].startswith('PASS') + assert (state['v'],state['n'])==(V,LOCAL) and lg['copies']==lg['erasures']==9 + ADDS=lg['common_frame_ADDs'] + assert cc['formal_columns_checked']==4*V+R and cc['local_scalar_additions']==ADDS + initial=[state['initial'][str(i)] for i in range(LOCAL)] + final=[state['final'][str(i)] for i in range(LOCAL)] + assert all(dims[final[i]]==H for i in range(2*V,LOCAL)) + widths=[H-dims[initial[i]] for i in range(2*V,LOCAL)] + ENT=sum(H-z for z in widths) + if NOBANK: place,NB=None,None; LIVE=DATA + else: + place,NB=placement(widths,T,M) + assert T%8==0 and T*(H*R-ENT)%M==0 and NB==T*(H*R-ENT)//M + LIVE=DATA+L*NB + isometries=json.loads(a.isometries.read_text()) + assert validate_isometries(isometries,w['frames'],w['lines'],dims)==set(initial[2*V:]) + streams={}; checks={}; localH=None + for reverse in (False,True): + cur=(final if reverse else initial)[:]+[-1]; val=[1<=0 + cur[x]=newf + if f: hist[f]+=1 + events.append([0,x,oldf,newf,f,int(reverse)]) + elif op==1: + assert cur[x]==cur[y]==f and x!=y and x Date: Wed, 21 Oct 2026 00:39:10 +1100 Subject: [PATCH 2/2] Clarify BIT COPY floor scope and prepare focused publication --- research/bit-copy-path/README.md | 27 +++++++++-- research/bit-copy-path/RELATED-WORK.md | 43 +++++++++++++++-- research/bit-copy-path/SOURCES.json | 21 +++++---- research/bit-copy-path/TASK.json | 2 +- research/bit-copy-path/VALIDATION.md | 49 ++++++++++++++++++++ research/bit-copy-path/independent/REVIEW.md | 8 ++-- 6 files changed, 130 insertions(+), 20 deletions(-) create mode 100644 research/bit-copy-path/VALIDATION.md diff --git a/research/bit-copy-path/README.md b/research/bit-copy-path/README.md index 38a5d29bae..08940db493 100644 --- a/research/bit-copy-path/README.md +++ b/research/bit-copy-path/README.md @@ -1,4 +1,4 @@ -# Actual BIT COPY paths and the price of repairing the coded caller +# The BIT COPY floor for closed snapshot paths **Scoped obstruction and complete planning budgets; no new supplier or κ.** @@ -30,7 +30,7 @@ The written geometric argument proves \sum_i\left(r_i+\frac{m_i}{2}\right)\geq 7. \] -Consequently the replacement cannot reduce either the COPY rank mass or +Consequently the replacement cannot reduce either the total paid query rank or its recursive moment \(\sum_\ell 64^\beta r_\ell^{\,1-\beta}\), for \(0\leq\beta<1\). For a single copy of rank \(r<7\), the complete rank cost is at least @@ -56,8 +56,9 @@ Two finite screens test ways around the late-continuation premise. - Source acquisitions at all existing common-frame cuts in the actual factory cannot replace any deleted centre scatter. Even relaxing the five partial-helper terminal conditions leaves exact binary separators. -- Copying each completed centre earlier, before the remaining centre - copies, fails across all **254** eligible cuts. The screen includes the +- Copying each completed centre earlier at **COPY rank at most four**, + before the remaining centre copies, fails across all **254** eligible + cuts. The screen includes the unchanged downstream COPYs and allows an arbitrary real intermediate frame separately for each receiver. It does not cover a changed incomplete total or a changed factory graph. @@ -135,8 +136,10 @@ in ordinary and optimized Python modes, compares their exact receipts, and writes all logs and generated evidence outside the package. See [REPLAY.json](REPLAY.json) for the command inventory. It does not reexecute the predecessor's complete historical suite. +The [validation record](VALIDATION.md) separates the retained mathematical +replays, publication review and hosted CI. -## Scope and next mathematical boundary +## Open next questions A successful construction must change something excluded by the proved class: the queried totals, their factory or public query events, the @@ -146,6 +149,20 @@ chronological owners, rounding, storage, and outer costs. Merely moving the same completed snapshot later, or treating a source code as a free terminal decoder, does not meet those requirements. +Two concrete questions remain: + +- **Can different queried totals share a paid path?** This changes the + unchanged-total premise of the theorem. Start with one complete changed + factory/query word, check every dirty-input column and restoration, and + price its full child histogram and returned helper owners. The blocker is + the absence of a favorable literal candidate. +- **Can a shared routed correction repair the coded caller cheaply?** This + lies outside the direct-reader and fixed-order wrapper bounds. Start with + one complete nine-call operator correction, charging every native query + and source observation against the stated repair allowance. The blocker + is the absence of a complete legal correction; scalar endpoint agreement + alone does not establish native caller compatibility. + The source and attribution record is in [RELATED-WORK.md](RELATED-WORK.md), [PREDECESSOR.json](PREDECESSOR.json), and [SOURCES.json](SOURCES.json). The changed word is not installed as a physical supplier. Selected diff --git a/research/bit-copy-path/RELATED-WORK.md b/research/bit-copy-path/RELATED-WORK.md index 06d2a692fe..dc125ae0b5 100644 --- a/research/bit-copy-path/RELATED-WORK.md +++ b/research/bit-copy-path/RELATED-WORK.md @@ -13,8 +13,11 @@ not claimed as freshly rerun here. BIT word and COPY pricing model. The R438 transport, five partial rank-four terminals, and their independent geometry checks are retained source constructions, not discoveries here. Their word and geometry bytes are - copied without alteration. -- Ross Richardson's `research/bit-r438-source-code` supplies the existing + copied without alteration. Credit Ali Romman's R438/PR1043 construction + and boundary-persistence work, Sagar Thakrar's transport/rehosting work, + and the DreamingOfClouds/eumemic/PR589/PR555 lineage retained by those sources. +- Ross Richardson's [PR3645](https://github.com/CrocSwap/integer-mult-bounds/pull/3645), + `research/bit-r438-source-code`, supplies the existing three-pivot repair and 429-helper source-code word. Its scientific source pin is `20ecf2891d80b5acf18d980b2164fb44ea7faa90`. The local 132-unit saving and 72-unit owner return were already established there. @@ -32,6 +35,32 @@ The present caller contribution is the actual nine-stage last-pair impulse, the all-column endpoint audit, and the explicitly bounded paid repair screens. The general warning about physical noncommutation is prior work. +## Closest later interface and budget results + +The publication overlap check also inspected Ross Richardson's +[PR4033](https://github.com/CrocSwap/integer-mult-bounds/pull/4033), +[PR4039](https://github.com/CrocSwap/integer-mult-bounds/pull/4039), +[PR4051](https://github.com/CrocSwap/integer-mult-bounds/pull/4051), and +[PR4054](https://github.com/CrocSwap/integer-mult-bounds/pull/4054). They cover, +respectively, binary cold-factor/separate-reader floors, a FULL-target +interface and its caller invoice, the older C511 mutable-source interface, +and kernel encoding at actual native cuts. Their premises and results are +retained. The present main theorem instead fixes one actual completed total +and allows arbitrarily many independent snapshots with paid closed private +frame walks through the monotone continuation. None of those earlier interface +results is claimed as new here. + +Sagar Thakrar's +[PR3948](https://github.com/CrocSwap/integer-mult-bounds/pull/3948) prices +target-scale centre-sharing and BIT levers against the selected PR589 +invoice. The present planning comparisons use the explicitly reconstructed +R438/five-partial-terminal inventory and the predecessor's fractional +owner relaxation, or the separately stated full-helper coded inventory. +Their numerical thresholds belong to those different inventories and +relaxations; they do not correct or replace PR3948's fixed-invoice result. +Both analyses leave the selected complex supplier binding and establish +no new κ. + ## Prior direct-decoder floor Russell Seligmann's [PR757](https://github.com/CrocSwap/integer-mult-bounds/pull/757) @@ -68,7 +97,15 @@ The entry overlap refresh read all 308 open PRs, 2703 discussion records, and 5234 local records, with no pagination warnings, as well as discussion hubs 1163, 1166 and 1167. This is an overlap search, not a novelty proof: it is not a census of closed remote PRs or every code diff. No public -research claim, discussion post, or PR was made by this run. +research claim, discussion post, or PR was made during that original +local-only investigation. + +Before publication, a fresh snapshot on 2026-10-20 covered 314 open PRs, +3540 closed/merged PRs, 2713 discussion records and 5234 local records. +All three remote corpora reported complete pagination and no warnings. +The search included titles, bodies, discussion replies and local package +metadata; it did not inspect every code diff or establish global novelty. +The canonical discussion and submission record are linked from the PR. ## Evidence boundaries diff --git a/research/bit-copy-path/SOURCES.json b/research/bit-copy-path/SOURCES.json index 36270c4931..a2d2f09f7c 100644 --- a/research/bit-copy-path/SOURCES.json +++ b/research/bit-copy-path/SOURCES.json @@ -12,14 +12,14 @@ "sha256": "844d8ac9cdcaaccb531c1019fe90a9c0afcdbdb39f15946643b951249fe9b3f3" }, { - "bytes": 7680, + "bytes": 8605, "path": "README.md", - "sha256": "c4a068fdaa3f20662d2b4f1058c50e4555bdadcb6d99c380e70688fdb50be6f1" + "sha256": "1ec764e5c5a2352fa2e6a370f1669351af1a111b0b0ef6b2bae28ecb0921286e" }, { - "bytes": 4232, + "bytes": 6466, "path": "RELATED-WORK.md", - "sha256": "1874787e93161ad1a01a9fad87b26999512eb66dfb8a1e54bb4139efb9250e13" + "sha256": "6cbc0c016897e1a2b65b16b23c8ce69f8d250b7ef612211a410027a5b18b9d5d" }, { "bytes": 2518, @@ -27,9 +27,14 @@ "sha256": "6844a8d4936f728f60f68b7f7f09266764a4eda421dedc59d101ad0b9ab48431" }, { - "bytes": 5656, + "bytes": 5658, "path": "TASK.json", - "sha256": "cd460deaa32b93f4d4753e6434fce132c7924a04b2f20a10887d275010b57448" + "sha256": "725989ca5eca1b5c327aae867cc56f13669314a9f0a81e5a0e99c31ebeb3f8ea" + }, + { + "bytes": 2607, + "path": "VALIDATION.md", + "sha256": "b80c4b8abd444a12955462a1a1682acb0ff1ba045b3637ed9a5acdd0886b342b" }, { "bytes": 433672, @@ -262,9 +267,9 @@ "sha256": "9d7360b301641e9fcd9a400a5ad0b401f85038db5f1ea82ee7a9b240dcf7af4d" }, { - "bytes": 16231, + "bytes": 16298, "path": "independent/REVIEW.md", - "sha256": "e266c3b6a0451660003896f6e774ac028429c29be154e2360c11c3e765b974c2" + "sha256": "a02c11d01331b89e38beed47776832a520043ba20e988ea4ca9283b570350b36" }, { "bytes": 1322, diff --git a/research/bit-copy-path/TASK.json b/research/bit-copy-path/TASK.json index 8bc84237eb..7fc5ee84ec 100644 --- a/research/bit-copy-path/TASK.json +++ b/research/bit-copy-path/TASK.json @@ -64,7 +64,7 @@ }, { "question": "Can an ADDRESS-dependent shared repair beat the coded caller allowance?", - "why_it_matters": "The actual native counterexample excludes a free terminal decoder, while the63 and111 floors cover specified paid interfaces only.", + "why_it_matters": "The actual native counterexample excludes a free terminal decoder, while the 63 and 111 floors cover specified paid interfaces only.", "first_step": "Derive a full nine-call operator correction with every native query and source observation priced; do not commute a source code through its own port exchange.", "blocked_by": "No complete legal shared routed repair has been supplied." } diff --git a/research/bit-copy-path/VALIDATION.md b/research/bit-copy-path/VALIDATION.md new file mode 100644 index 0000000000..073c344f7e --- /dev/null +++ b/research/bit-copy-path/VALIDATION.md @@ -0,0 +1,49 @@ +# Validation and publication review + +The mathematical package was checked at +`d82a679985888b703f689833c2b12780a8243382`, based on canonical +`12fa8c5c844be21bfefb38a19503fbd867a11244`. + +The standalone runner executed all ten checks in [REPLAY.json](REPLAY.json) +in normal and optimized Python, first in the owned checkout and then in a +Git-free export of the committed package: 40 fresh check invocations in +total. All mathematical receipt bytes agreed. The final committed export +included a formatting-only correction made after the checkout replay. +The original export's source-manifest SHA256 was +`8bcedf654ca08fadabe1cd56c54684b813f4d89e83d4b8883983173f2b416dac`. +Run logs and process-cleanup receipts were retained locally; the mathematical +receipts and checkers are included here for independent reproduction. + +The publication review rederived the finite-branch argument and narrowed two +summary phrases: the earlier-cut search has COPY rank at most four, and the +main floor concerns total COPY-plus-MOVE rank. It also added explicit next +questions and refreshed overlap/credit references. These publication changes +leave all mathematical proof files, executable checkers, scientific inputs +and fixed mathematical receipts byte-identical to the verified package. +The source inventory is updated to authenticate the revised documentation. +This reuses the matching mathematical replay evidence; it does not report +those executions as newly repeated during publication. + +[The separate review](independent/REVIEW.md) distinguishes independent finite +checks from written all-parameter arguments. Its +[validation record](independent/VALIDATION.json) includes normal/optimized +Python 3.14 and a minimal Git-free Python 3.12.12 check, plus disabled-guard +controls. The independent checker does not execute producer modules. +This is independent **agent** review, not independent human or formal-kernel +certification. + +From the repository root, reproduce with CPython 3.12 or later: + +```sh +python3.12 -I -B research/bit-copy-path/verify.py --output /tmp/bit-copy-path-check.json +``` + +Use a fresh output path. The expected final marker is +`PASS_ACTUAL_BIT_COPY_PATH_SCOPED_OBSTRUCTION_NO_NEW_KAPPA`. +This package uses its own explicit replay format and standalone verifier. +The predecessor's complete historical suite, selected supplier cold replay, +physical compiler admission and full multiplication theorem are not covered. +Hosted CI status is reported separately on the PR. κ and the selected +certificates remain unchanged. + +Apache-2.0. Prepared for Ross Richardson with substantial OpenAI Codex assistance. diff --git a/research/bit-copy-path/independent/REVIEW.md b/research/bit-copy-path/independent/REVIEW.md index 2a83324bca..e4e5f4b138 100644 --- a/research/bit-copy-path/independent/REVIEW.md +++ b/research/bit-copy-path/independent/REVIEW.md @@ -143,11 +143,13 @@ K column. This certifies the necessary span obstruction without assuming that independent insertion choices could coexist legally. It grants the extra choices, cleanup and scalar work, which only weakens the obstruction. -The nine stronger late-factory duals are also independently checked on all -254 completed-centre cuts before the first COPY, including the eight +The nine stronger late-factory duals are also independently checked for +replacement COPY rank at most four on all 254 completed-centre cuts before +the first COPY, including the eight remaining COPYs in each changed suffix. The oracle treats COPY and ERASE source reads as uses when deciding whether a MOVE may be retimed. These -checks cover arbitrary real lower frames in the actual centre hyperplane; +checks cover arbitrary real lower frames of dimension at least three in +the actual centre hyperplane; the fixed catalogue cases are subcases. They do not enumerate arbitrary earlier factory modifications or private frame walks across later COPYs.

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zsh4)Wav@ZP6Mj*$2hGw5c|T-}tTFDSBW18{MD=|ukDNa>@S5fo?B$NpWDCD=y4&mT z%19I+wioEK;Ttjhs69yX4IG`X%Dra1ppjr-Cg14!)2U2eC_}ohcg7rz5Z@pYSBTq$ zkYl3GEx47laxnnu? z#^uxjTBBcP(TazR^DD3Gf}gYc{kLg=|e)i@_EBW{< z=IW4&{;=EOZKOo7ujFjUe!Jt+y!YI!t|_dw`Q5MnUrRU=nU zF5knyb2xFj_ODPU=085%UG2ahSK;u-?~iEDWDA$g$}mv=52tYpy#N3J literal 0 HcmV?d00001 diff --git a/research/bit-copy-path/composition/PRIVATE-MOVE-CHECK.py b/research/bit-copy-path/composition/PRIVATE-MOVE-CHECK.py new file mode 100644 index 0000000000..26dcc5d870 --- /dev/null +++ b/research/bit-copy-path/composition/PRIVATE-MOVE-CHECK.py @@ -0,0 +1,360 @@ +#!/usr/bin/env python3 +"""Exact private-frame transport and restricted coded-wrapper controls. + +Only the standard library is used. Default operation compares a saved receipt; +--output explicitly writes a receipt. No compiler or supplier admission. +""" +from fractions import Fraction as Q +from hashlib import sha256 +from pathlib import Path +from types import SimpleNamespace +import argparse +import ast +import gzip +import json + +HERE = Path(__file__).resolve().parent +ROOT = HERE.parent +PINS = { + "inputs/GEOMETRY.json.gz": + "a353999bceac4ac957f540adfd3e5f8768297138b27132eeb83332e2ea4b1022", + "inputs/global_k.py.txt": + "6368196a393ac25bb2918880e4c4bd31ed733b762c27b6873ab391cbc0c5c3c8", + "composition/inputs/CODED-WORD.json.gz": + "1a6841fccba51fa922993c77b4b296e659c2c538252e5e06113d0438c5edb870", + "composition/PRIVATE-MOVE-control.py.txt": + "d3a93e54bb7d333b73917586407d11674d38b5c3fda4fbdc436fb5293ae7786a", + "composition/PRIVATE-MOVE-episode.py.txt": + "38f79cb2a370548471063952d5f7c74bd795a2a8496fdcf3ce0996d79e3adaff", + "composition/PRIVATE-MOVE-volume.md": + "19683df4394de299cbd7e1cce39bf9434ad260b1a37006587cf9aed84c9c0a03", + "composition/PRIVATE-MOVE-native-closure.md": + "74cc8de20f21acb5123fe2d20996fe0925ea6e51b3dec633bd5221fb719b3344", +} + + +def need(ok, message): + if not ok: + raise ValueError(message) + + +def read(relative): + raw = (ROOT / relative).read_bytes() + need(sha256(raw).hexdigest() == PINS[relative], "Source pin: " + relative) + return json.loads(gzip.decompress(raw)) if relative.endswith(".gz") else raw.decode() + + +def eye(n): + return [[Q(i == j) for j in range(n)] for i in range(n)] + + +def add(a, b, coefficient=1): + return [[x + coefficient*y for x, y in zip(r, s)] for r, s in zip(a, b)] + + +def mm(a, b): + return [[sum(x*y for x, y in zip(r, c)) for c in zip(*b)] for r in a] + + +def rref(rows): + basis = {} + for row in rows: + row = list(map(Q, row)) + for p, old in sorted(basis.items()): + c = row[p] + row = [x-c*y for x, y in zip(row, old)] + p = next((p for p, x in enumerate(row) if x), None) + if p is not None: + c = row[p] + basis[p] = [x/c for x in row] + return basis + + +def normal(lines): + basis = rref(lines) + if len(basis) != 7: + return None + p = next(p for p in range(8) if p not in basis) + result = [Q(0)]*8 + result[p] = Q(1) + for p, row in sorted(basis.items(), reverse=True): + result[p] = -sum(x*y for x, y in zip(row, result)) + need(all(sum(x*y for x, y in zip(row, result)) == 0 for row in lines), + "Exact hyperplane normal") + return result + + +def outer(x, y): + return [[a*b for b in y] for a in x] + + +def dot(x, y): + return sum(a*b for a, b in zip(x, y)) + + +def scale(a, c): + return [[x*c for x in row] for row in a] + + +def cap(n): + return add(eye(8), scale(outer(n, n), -1/dot(n, n))) + + +def intersection(n, m): + nn, nm, mm_ = dot(n, n), dot(n, m), dot(m, m) + determinant = nn*mm_-nm*nm + need(determinant > 0, "Distinct rational hyperplanes") + complement = add(scale(outer(n, n), mm_), scale(outer(m, m), nn)) + complement = add(complement, scale(add(outer(n, m), outer(m, n)), -nm)) + return add(eye(8), scale(complement, -1/determinant)) + + +def swap(p): + rest = add(eye(len(p)), p, -1) + return [a+b for a, b in zip(rest, p)] + [a+b for a, b in zip(p, rest)] + + +def serial(a): + return [[str(x) for x in row] for row in a] + + +def frame_checks(geometry): + normals = {} + for i, support in enumerate(geometry["frames"]): + n = normal([geometry["lines"][p] for p in support]) + if n is not None: + normals[i] = n + need(len(normals) == 120, "Actual 120 rank-seven frame closures") + examples = [] + for other in (511, 512): + n, m = normals[510], normals[other] + f, h, g = cap(n), cap(m), intersection(n, m) + r, s = add(f, g, -1), add(h, g, -1) + for name, p, dimension in (("F", f, 7), ("H", h, 7), ("G", g, 6), + ("COPY", r, 1), ("MOVE", s, 1)): + need(mm(p, p) == p and p == list(map(list, zip(*p))), + name + " exact orthogonal idempotent") + need(sum(p[i][i] for i in range(8)) == dimension, + name + " exact rational rank") + df, dg, dh, dr, ds = map(swap, (f, g, h, r, s)) + need(mm(dr, df) == dg, "COPY transports the whole arbitrary source") + need(mm(ds, dg) == dh, "Private promotion preserves logical payload") + need(mm(ds, dh) == dg, "Private inverse restores the exact copied frame") + need(mm(ds, mm(ds, mm(dr, df))) == dg, "Complete private return") + need(dh != dg, "Omitting return is detected on an address basis column") + examples.append({ + "centre_frame": 510, "receiver_frame": other, + "centre_normal": [str(x) for x in n], + "receiver_normal": [str(x) for x in m], + "normal_gram_determinant": str(dot(n, n)*dot(m, m)-dot(n, m)**2), + "intersection_projector": serial(g), + "copy_projector": serial(r), "move_projector": serial(s), + "denominators": sorted({x.denominator for p in (g, r, s) + for row in p for x in row}), + "rank_invoice": [1, 1, 1], + "checked_address_columns_per_identity": 16, + }) + need(3 not in examples[0]["denominators"] and + any(d % 3 == 0 for d in examples[1]["denominators"]), + "Actual non-dyadic intersection control") + f = cap(normals[510]) + full = eye(8) + s = add(full, f, -1) + need(mm(swap(s), swap(f)) == swap(full), "Zero-rank COPY then FULL promotion") + need(mm(swap(s), swap(full)) == swap(f), "Paid FULL return") + return {"rank_seven_catalogue_frames": len(normals), "examples": examples, + "zero_copy_full_visit_invoice": [0, 1, 1]} + + +def original_controller_rejection(source): + tree = ast.parse(source) + classes = [n for n in tree.body if isinstance(n, ast.ClassDef)] + need(len(classes) == 1 and classes[0].name == "LocalState", "Frozen controller class") + namespace = {"need": need} + exec(compile(ast.Module(body=classes, type_ignores=[]), "", "exec", + dont_inherit=True, optimize=0), + namespace) + call = SimpleNamespace(owners=(0, 1), start=(510, 512), reverse=False, + events=((2, 0, 2, 510, 636, 1, 0), + (0, 2, 636, 512, 1, 0)), + requests=(object(), object())) + state = namespace["LocalState"](call) + state.accept(0) + try: + state.accept(1) + except ValueError as error: + need(str(error) == "live local destination/source role", + "Existing ABI fails at the private destination gate") + return str(error) + raise ValueError("Old ABI unexpectedly admitted a private MOVE") + + +# A group-ring control over F2. N is involutive; C and c are mutual inverses. +# No commutation of N with C is introduced. F is factored off the active pair. +def reduce_word(word): + out = [] + inverse = {"N": "N", "C": "c", "c": "C"} + for letter in word: + if out and inverse[letter] == out[-1]: + out.pop() + else: + out.append(letter) + return tuple(out) + + +def poly(word=""): + return {reduce_word(word)} + + +ZERO = set() +ONE = poly() +NN = poly("N") +CC = poly("C") +CI = poly("c") + + +def pmul(a, b): + result = set() + for x in a: + for y in b: + result.symmetric_difference_update({reduce_word(x+y)}) + return result + + +def matmul(a, b): + result = [] + for row in a: + line = [] + for col in zip(*b): + entry = set() + for x, y in zip(row, col): + entry.symmetric_difference_update(pmul(x, y)) + line.append(entry) + result.append(line) + return result + + +def diag(a, b): + return [[a, ZERO], [ZERO, b]] + + +def wrapper_checks(word, recipe_source): + rows = [1 << i for i in range(word["public_registers"])] + [0] + for op, dst, src, coefficient, _frame, _phase in word["records"]: + if op == 1 and coefficient % 2: + rows[dst] ^= rows[src] + elif op == 2: + rows[-1] = rows[dst] + elif op == 3: + need(rows[-1] == rows[dst], "Actual copied code source retained") + rows[-1] = 0 + need(rows[3] & 1, "Actual A[3,0] coefficient for noncommutation") + # Geometry pins port0=2e0, port3=2e3 separately in the main checker. + # On a port0 HEAD=e0 impulse, (NA)[3] stays HEAD=e0 while + # (AN)[3] is TAIL=e0; the two induced VALUE permutation entries differ. + b = [[poly("CN"), ZERO], [poly("NCN"), NN]] + reverse = [[NN, ONE], [ZERO, poly("Nc")]] + clean_b = [[NN, ZERO], [ONE, NN]] + clean_reverse = [[NN, ONE], [ZERO, NN]] + need(matmul(b, diag(poly("NcN"), ONE)) == clean_b, + "Exact forward paid source wrapper") + need(matmul(diag(ONE, poly("NCN")), reverse) == clean_reverse, + "Exact inverse paid source wrapper") + source_restored = [[NN, ZERO], [poly("NCN"), NN]] + wrong = matmul(diag(ONE, CI), matmul(source_restored, diag(ONE, CC))) + need(wrong != clean_b and wrong[1][1] == poly("cNC"), + "False current-cap commutation rejected") + z = NN ^ ONE + allowed_forward, allowed_reverse = [], [] + for mask in range(16): + p = [[(mask >> (2*i+j)) & 1 for j in range(2)] for i in range(2)] + q = [[((ONE if i == j else ZERO) ^ (z if p[i][j] else ZERO)) + for j in range(2)] for i in range(2)] + if matmul(q, b) == matmul(b, q): + allowed_forward.append(mask) + need(p[0][0] == 0, "No active forward-source sweep commutes through the call") + if matmul(q, reverse) == matmul(reverse, q): + allowed_reverse.append(mask) + need(p[1][1] == 0, "No active inverse-source sweep commutes through the call") + tree = ast.parse(recipe_source) + nodes = [n for n in tree.body if isinstance(n, ast.FunctionDef) and n.name == "recipe"] + need(len(nodes) == 1, "Frozen recipe") + namespace = {} + exec(compile(ast.Module(body=nodes, type_ignores=[]), "", "exec", + dont_inherit=True, optimize=0), namespace) + events = namespace["recipe"](4) + tight = [{"stage": e["stage"], "pair": e["pair"], + "bank": 2*e["pair"]+int(e["reverse"]), + "interval": "after inverse call, before C" if e["reverse"] + else "after inverse C, before forward call"} + for e in events if e["kind"] == "local"] + need(len(tight) == 9, "Nine disjoint coded-wrapper intervals") + return { + "literal_code_A_3_0": 1, + "wrong_free_wrapper_bottom_right": "C^-1 N C, not N (F factored off)", + "paid_forward_wrapper": "diag(N C_s^-1 N,I) before B_s", + "paid_inverse_wrapper": "diag(I,N C_s N) after R_s", + "two_by_two_commuting_bank_masks_forward": allowed_forward, + "two_by_two_commuting_bank_masks_inverse": allowed_reverse, + "actual_recipe_tight_intervals": tight, + "restricted_bank_sweep_lower_bound": 9, + "rank_one_requests_per_whole_port_sweep": 111, + "restricted_rank_mass_lower_bound": 999, + "scope": "Fixed ordered barriers: retain the chronological order of both " + "coded calls and individual C_s or inverse-C_s factors; move, " + "combine and cancel only N sweeps. Scalar bank changes transport " + "P and the boundary functional. No reordering code factors or " + "commuting across composite barrier products. Not a minimum for " + "changed calls, partial-port queries, or general routed repairs.", + } + + +def main(): + parser = argparse.ArgumentParser() + parser.add_argument("--output", type=Path) + args = parser.parse_args() + sources = {name: read(name) for name in PINS} + geometry = sources["inputs/GEOMETRY.json.gz"] + need(geometry["lines"][0] == [2, 0, 0, 0, 0, 0, 0, 0] and + geometry["lines"][3] == [0, 0, 0, 2, 0, 0, 0, 0], + "Actual noncommuting port roots") + a = lambda r: 0 if r == 0 else 2*(r*(127-r)+4096) + output = { + "status": "PASS_PRIVATE_FRAME_TRANSPORT_AND_RESTRICTED_WRAPPER_CONTROLS", + "no_new_kappa": True, + "source_pins": PINS, + "private_operator_checks": frame_checks(geometry), + "existing_controller_rejection": original_controller_rejection( + sources["composition/PRIVATE-MOVE-control.py.txt"]), + "rank_three_invoice": {"rank_histogram": {"1": 3}, "positive_requests": 3, + "good_adapter_slots_per_f": 3*a(1), + "reference_rank7_adapter_slots_per_f": a(7), + "additional_adapter_slots_per_f": 3*a(1)-a(7), + "additional_bad_allowance": "2 * 10^-16 * 131072"}, + "zero_copy_full_invoice": {"rank_histogram": {"1": 2}, "positive_requests": 2, + "good_adapter_slots_per_f": 2*a(1), + "additional_adapter_slots_per_f": 2*a(1)-a(7)}, + "wrapper_controls": wrapper_checks( + sources["composition/inputs/CODED-WORD.json.gz"], + sources["inputs/global_k.py.txt"]), + "limits": [ + "Fresh private workspace; arbitrary ordinary source/helper payloads.", + "No emitted extended controller, request provider or physical supplier.", + "Non-dyadic G=F intersect H can exclude q=3; fixed q>2^80 is sufficient " + "for the displayed rational examples.", + "The same-query receiver floor is proved in construction/PROOF.md.", + "No rounded precision, whole-storage or outer-recurrence admission.", + ], + } + receipt = json.dumps(output, indent=2, sort_keys=True)+"\n" + if args.output: + args.output.write_text(receipt) + else: + need((HERE / "PRIVATE-MOVE-RESULT.json").read_text() == receipt, + "Saved private-MOVE receipt differs from exact recomputation") + print(json.dumps({"status": output["status"], "no_new_kappa": True, + "restricted_wrapper_rank_mass": 999}, sort_keys=True)) + + +if __name__ == "__main__": + main() diff --git a/research/bit-copy-path/composition/PRIVATE-MOVE-PROOF.md b/research/bit-copy-path/composition/PRIVATE-MOVE-PROOF.md new file mode 100644 index 0000000000..67f75264af --- /dev/null +++ b/research/bit-copy-path/composition/PRIVATE-MOVE-PROOF.md @@ -0,0 +1,306 @@ +# Private copied-total transport and paid coded-stage wrappers + +This appendix answers two interface questions. A copied-total private packet +can leave its original centre frame and return without changing its logical +payload. That algebra requires a new controller contract and does not by +itself deliver the desired target correction. The proposed free coded-stage +decoder instead fails an actual current-frame commutation test. Its explicit +paid wrapper has a restricted cancellation lower bound above the available +budget. + +These are exact operator statements and finite source checks. No new physical +supplier, precision theorem, owner schedule or kappa is admitted. The +same-query receiver obstruction is proved in +[`../construction/PROOF.md`](../construction/PROOF.md), with its finite +chronological checks in `walk_certificate.py`; that is the primary result +for the receiver question. + +## Private payload, frame and ownership contract + +Write the ADDRESS involution for an orthogonal projector P as + + D_P(h,t) = ((I-P)h+Pt, Ph+(I-P)t). + +The same notation below denotes its induced permutation of complete VALUE +arrays. Composition identities for ADDRESS permutations hold for arbitrary +VALUE payloads, including correlated dirty payloads and complete spectators. +No linearity of the input payload as a function of ADDRESS is assumed. + +The ordinary source and helper packets are arbitrary. The private packet is +fresh owned workspace, empty before COPY and empty after ERASE. This is the +actual copied-total interface, not permission to overwrite or discard an +arbitrary preexisting dirty private packet. + +Let the immutable centre be stored as D_F q. Choose G contained in F and +copy into a new private packet by applying the residual D_G D_F. The result +is D_G q. A private MOVE from current frame U to frame V applies + + D_V D_U. + +For nested U,V this is the single projector interchange on their orthogonal +difference, in either direction, with rank |dim V-dim U|. If the current +private packet is D_U q, the new packet is exactly D_V q. Each equal-frame +ADD from the private packet into a receiver at V therefore adds logical q. +All other receiver data remain arbitrary. + +For any sequence G=U0,U1,...,Uk=G of nested adjacent frames, + + (D_Uk D_U(k-1)) ... (D_U1 D_U0) D_G q = D_G q. + +This telescopes without commuting nonnested frames. Readers do not modify +the private packet. The final packet is therefore exactly the original +transported snapshot, and the original source remains D_F q. The original +transported-source ERASE equality holds. Returning merely the frame label +without applying the residual would not establish that equality. + +In particular, for distinct seven-dimensional F,H and G=F intersect H of +dimension six, the sequence is + + COPY F -> G; MOVE private G -> H; read at H; + MOVE private H -> G; ERASE at G. + +Its complete residual-rank list is **(1,1,1)**. Neither MOVE is free. +There is no extra logical query kernel or compensation term caused solely +by the private excursion. An arbitrary preexisting private dirt would +invalidate the fresh-copy premise and is outside this statement. + +The zero-rank COPY case is also legitimate algebraically: COPY F to F, +promote to FULL, return to F, ERASE. Its residual-rank list is **(0,1,1)**. +The rank-zero COPY still clones and later clears a complete packet; zero +native child rank does not mean zero copying or storage cost. + +## What must change in the retained implementation + +The exact consumed source files and hashes are in `PRIVATE-MOVE-SOURCES.json`. +They are frozen alongside this appendix. + +`PRIVATE-MOVE-control.py.txt` rejects a private MOVE at two distinct gates: +the destination must be an ordinary role, and every MOVE requires no open +COPY interval. Its ERASE also requires the stored private physical version +to be unchanged. The finite check executes the frozen `LocalState` class +and verifies rejection at the private-destination gate. + +`PRIVATE-MOVE-episode.py.txt` accepts a COPY, a sequence of target ADDs from +the snapshot, and its ERASE. It clones the complete source, obtains a +distinct owned residual result, reads that private packet, then clears +the private result and clone. The existing episode grammar does not +include intervening residual calls on the private packet. + +A valid extension must distinguish a logical snapshot generation from +physical frame/version changes. It must: + +* Keep the original source packet and its logical version unchanged. +* Admit only checked private transport operations and read-only target ADDs + during this interval; a private ADD destination is not a transport. +* Associate every transport with its actual source-bound residual request, + current frame, owner chart, complete packet and current incoming weights. +* Maintain the invariant `private = D_current q`; require the initial + private frame again at ERASE and the unchanged source generation. +* Clear each superseded owned residual output and all residual work, while + protecting the borrowed current private packet until its replacement is + available. Restore physical heads and retain full spectators. + +It is unsound simply to reset the old version counter after moving the +private packet. Conversely, a checked closed frame path need not be a +literal stack of reversed edges: the displayed potential identity already +proves closure for any admitted closed path. Each edge remains paid. + +No such extended controller or provider has been emitted here. The generic +arbitrary-input residual theorem supports the operator being requested, +but it does not make a new private destination or request label accepted +by the old executable controller. + +## Exact frame domain and native invoice + +The checker reconstructs the actual 120 rank-seven catalogue hyperplanes. +Two examples use centre F=frame510 and receiver H=frame511 or frame512. +Their normals are + + n510=(0,1,1,0,-1,0,0,1), + n511=(0,-1,1,0,1,0,0,1), + n512=(0,-1,-1,0,0,1,1,0). + +Each squared norm is four. The first pair is orthogonal; its intersection +and residual projectors are dyadic. The second has inner product -2 and +normal Gram determinant 12. Its intersection and residual entries have +denominators among 1,3,6,12. The exact sixteen-column checks prove the +COPY, promotion and return identities and reject omission of the return. + +Thus arbitrary intersections are not automatically members of the old +636-frame dyadic catalogue. The second example does not define the +displayed projector modulo 3. It is valid over every Z/q^w with odd +prime q other than 3, and in particular in the inherited fixed-prime +regime q>2^80. A finite extended catalogue needs its own authenticated +projector and split-basis labels. There is no claim here of admission over +every odd prime-power ring. + +For incoming unit weights A and tail reversal C, the retained residual is + + v=P(CD-A^-1 H), + T_A(P)(H,D)=(H+A v,D-C v). + +It is a conjugate of D_P. At a fixed outer cut the copied source's weighted +view, raw header and spectators must be retained; one must not apply the +producer's chart a second time. Source-bound preparation of a split +projector uses its full raw header, including high digits. In the fixed +large-prime regime each displayed rational rank-one projector is split, +so the generic good/fallback construction applies once the new request +provider is implemented. Every complete residual child includes its +own cleanup and inverse adapters. + +For n=64f+delta with f>=1, each positive rank r residual uses, on a good +packet, one complete width-rf child and + + A(r) f ordered affine ADD slots, + A(r)=2[r(127-r)+4096]. + +Both adapter halves and all zero coefficient slots are included. Set +A(0)=0 and emit no native child at rank zero. Therefore replacing a +rank-seven COPY by the rank-three private excursion gives + +| Invoice item | Original COPY7 | COPY1, MOVE1, MOVE1 | +| --- | ---: | ---: | +| Positive residual requests | 1 | 3 | +| Child rank mass | 7 | 3 | +| Good affine slots divided by f | 9,872 | 25,332 | +| Extra affine slots divided by f | — | 15,460 | + +For a power moment Phi_b(r)=r(64/r)^b, the child change is +`3 Phi_b(1)-Phi_b(7)`. This is distinct from the increased adapter count. +For zero-COPY/FULL/return the positive-request count is two and the +affine count is 16,888f, an increase of 7,016f over COPY7. + +The inherited bad-packet allowance is +`epsilon*131072` per positive request, epsilon=10^-16. It increases by +`2*epsilon*131072` for the rank-three excursion, and by +`epsilon*131072` for zero-COPY/FULL/return. This is a conservative per-edge +allowance, not equality of bad classes for different projectors. + +The source clone, private reads, erasure, coefficient preparation, routes, +rewinds and complete packet scans remain charged. If K distinct staged +packets are live, including an existing private packet, the frozen +residual lifetime result bounds MOVE by `(K+2)v+3R` and initial COPY +by `(K+3)v+3R`. In the 660-ordinary-role R438 word, both the initial COPY +and a private MOVE fit the scoped bound `663v+3R`: COPY has K=660; +private MOVE has K=661. This counts staged roles and the active residual, +not whole families, protected input, directories, routing, scatter, +peel/merge, control tapes or the longer private lifetime. + +The exact VALUE transform preserves the logical q, so private ADDRESS +transport alone introduces no new signed VALUE coefficient growth. +Nevertheless new residual calls, numerical coefficient work, receiving +ADDs and a longer live snapshot need their own complete rounded/error +and storage analysis before supplier admission. + +## The coded-stage wrapper and its actual forbidden free operation + +Use the notation of `PROOF.md`: C_s is the actual routed source code A_s, +N the diagonal port-root interchange, F_s=N G_s, and F_s commutes with C_s. +The code commutes with G_t for t different from s. It generally does +**not** commute with the current G_s: + + [C_s,G_s]=F_s[C_s,N] != 0. + +The actual A[3,0]=1 supplies a witness. Ports 0 and 3 have roots 2e0 and +2e3. On a port-zero impulse HEAD=e0, TAIL=0, `(NA)[3]` has HEAD=e0, +whereas `(AN)[3]` has TAIL=e0. Conjugation by the stage routes preserves +noncommutation because each route fixes and commutes with its port root. +The same conclusion holds for C_s^-1. + +For the source-restored B'_s, the purported free target wrapper +`diag(I,C_s^-1) B'_s diag(I,C_s)` has target-only block +`C_s^-1 G_s C_s`, not G_s. This fails even when its source is zero. +The required target postdecoder is + + G_s C_s^-1 G_s = N C_s^-1 N, + +which mixes HEAD and TAIL. It is not an equal-frame scalar ADD circuit +at the capped target cut. + +There is a simpler exact paid wrapper for the original coded word, without +first restoring its source: + + B_s diag(N C_s^-1 N,I) = B_clean,s, + diag(I,N C_s N) R_s = R_clean,s. (W) + +The checker verifies both identities over a noncommutative group algebra, +factoring off the central F_s on the two active banks. Every source code +and its inverse retain their stage index; no equality C_s=C_t is assumed. +The literal inverse code has 1,499 signed scalar ADDs. Each N sweep on +one complete bank consists of 111 positive rank-one port requests. +The direct implementation of (W) uses two such sweeps per local stage, +18 across the actual nine-stage recipe, or 1,998 rank-one requests. +Its full good-adapter, fallback and data-access invoice cannot improve +upon that child count. + +## A restricted cancellation bound for those wrappers + +This subsection covers a fixed ordered-barrier rewriting model. Retain +the chronological order of both the nine coded-call factors and every +individual C_s or C_s^-1 factor in (W). Move, combine and cancel only +the N sweeps. Ordinary scalar bank basis changes and the actual bridge +ADDs transport sweep matrices and the corresponding boundary functionals. +Reordering code factors or commuting a sweep across a composite product +of barriers is not allowed by this model. It does not cover modifying a +coded call, replacing an N sweep by a partial-port or different routed +query, or synthesizing a different coupled caller. + +Over binary VALUE arrays put z=N+I, so z^2=0. A scalar bank-conjugate of +an N sweep has form `I+zP`, where P is a rank-one bank projector. +Products of such sweeps between fixed noncommuting operations add the +bank matrices P. Allowing arbitrary P and charging at least rank(P) is +a relaxation, so a lower bound in that model also applies to sweep +words. Scalar bank changes conjugate P and preserve rank. + +At a code factor acting on bank i, any commuting sweep matrix must have +P_ii=0: its (i,i) commutator is `P_ii [N,C_s]`. At a forward coded call, +any commuting sweep similarly has P_ii=0 on the active source x. +For the role-swapped inverse, P_ii=0 on its active source y. + +For completeness, write the active 2-by-2 bank block of P as +`[[a,b],[c,d]]` and factor off F_s. The forward code has matrix +`[[C_s N,0],[N C_s N,N]]`. Its upper-left commutator equation reduces to + + (a+b) z C_s = a C_s z. + +If a=1, this either says C_s z=0, impossible since C_s is invertible, +or says C_s commutes with N, contrary to the actual witness. Hence a=0. +For the inverse `[[N,I],[0,N C_s^-1]]`, its lower-right equation with +d=1 would imply `[z,C_s^-1]=c z`. The diagonal port entries of C_s^-1 +are scalar identities, so the left-hand side has zero diagonal port +blocks while c z does not unless c=0. The nonzero off-diagonal +commutator then rejects d=1. Blocks outside the active pair cannot +alter these active diagonal equations. + +In each forward wrapper the last N_x is therefore trapped between +C_s^-1 and its forward coded call; in each inverse wrapper the first +N_y is trapped between the coded inverse and C_s. The actual +`recipe(4)` has nine such disjoint chronological intervals. Within +each interval the active diagonal bank coefficient is one. A sweep +commuting out through either boundary has coefficient zero there, +and scalar basis changes preserve the equivalent rank-one functional. +Thus each interval retains a nonzero bank matrix and costs at least +one bank sweep. Other commuting sweeps, even after bridge transport, +cannot remove this requirement. + +The resulting restricted lower bound is **9 sweeps, or 999 rank-one +requests**. It already exceeds the parent allowance of less than +525.45 rank-tariff units over all nine stages. The true positive-power +cost is at least `999 Phi_b(1)`, before adapters, fallback and access. +This stops this precise commutation-only wrapper optimization; it is +not a universal paid-decoder lower bound. + +## Reproduction + +Run with standard-library Python 3.12: + + python3.12 -I -B -Werror composition/PRIVATE-MOVE-CHECK.py + +The default recomputes and compares `PRIVATE-MOVE-RESULT.json` without +writing. `--output /absolute/new/path.json` explicitly emits a receipt. +The check authenticates every consumed source, reconstructs exact +rational projectors, executes the frozen old-controller rejection, +checks the exact group-algebra wrapper identities and lists the actual +nine chronological intervals. The all-path private transport proof, +receiver lower bound and restricted sweep argument are written +arguments; finite checks do not replace their stated hypotheses. diff --git a/research/bit-copy-path/composition/PRIVATE-MOVE-RESULT.json b/research/bit-copy-path/composition/PRIVATE-MOVE-RESULT.json new file mode 100644 index 0000000000..833b3f49d3 --- /dev/null +++ b/research/bit-copy-path/composition/PRIVATE-MOVE-RESULT.json @@ -0,0 +1,686 @@ +{ + "existing_controller_rejection": "live local destination/source role", + "limits": [ + "Fresh private workspace; arbitrary ordinary source/helper payloads.", + "No emitted extended controller, request provider or physical supplier.", + "Non-dyadic G=F intersect H can exclude q=3; fixed q>2^80 is sufficient for the displayed rational examples.", + "The same-query receiver floor is proved in construction/PROOF.md.", + "No rounded precision, whole-storage or outer-recurrence admission." + ], + "no_new_kappa": true, + 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chronological order of both coded calls and individual C_s or inverse-C_s factors; move, combine and cancel only N sweeps. Scalar bank changes transport P and the boundary functional. No reordering code factors or commuting across composite barrier products. Not a minimum for changed calls, partial-port queries, or general routed repairs.", + "two_by_two_commuting_bank_masks_forward": [ + 0 + ], + "two_by_two_commuting_bank_masks_inverse": [ + 0 + ], + "wrong_free_wrapper_bottom_right": "C^-1 N C, not N (F factored off)" + }, + "zero_copy_full_invoice": { + "additional_adapter_slots_per_f": 7016, + "good_adapter_slots_per_f": 16888, + "positive_requests": 2, + "rank_histogram": { + "1": 2 + } + } +} diff --git a/research/bit-copy-path/composition/PRIVATE-MOVE-SOURCES.json b/research/bit-copy-path/composition/PRIVATE-MOVE-SOURCES.json new file mode 100644 index 0000000000..1eadcb6257 --- /dev/null +++ b/research/bit-copy-path/composition/PRIVATE-MOVE-SOURCES.json @@ -0,0 +1,26 @@ +[ + { + "bytes": 2559, + "file": "PRIVATE-MOVE-control.py.txt", + "sha256": "d3a93e54bb7d333b73917586407d11674d38b5c3fda4fbdc436fb5293ae7786a", + "source": "/Users/ross/git/math/integer-mult-bounds/research/selected-bit-recursive-caller/predecessor/control.py" + }, + { + "bytes": 5712, + "file": "PRIVATE-MOVE-episode.py.txt", + "sha256": "38f79cb2a370548471063952d5f7c74bd795a2a8496fdcf3ce0996d79e3adaff", + "source": "/Users/ross/git/math/integer-mult-bounds/research/selected-bit-recursive-caller/predecessor/vendor/episode.py" + }, + { + "bytes": 9909, + "file": "PRIVATE-MOVE-volume.md", + "sha256": "19683df4394de299cbd7e1cce39bf9434ad260b1a37006587cf9aed84c9c0a03", + "source": "/Users/ross/git/math/integer-mult-bounds/research/selected-bit-recursive-caller/evidence/pr2731-README.md" + }, + { + "bytes": 23425, + "file": "PRIVATE-MOVE-native-closure.md", + "sha256": "74cc8de20f21acb5123fe2d20996fe0925ea6e51b3dec633bd5221fb719b3344", + "source": "/Users/ross/git/math/.research-automation/runs/bit-full-caller-20261020T1028Z/release/bit-full-caller/native/sources/RECURSIVE-CLOSURE.md" + } +] diff --git a/research/bit-copy-path/composition/PRIVATE-MOVE-VALIDATION.json b/research/bit-copy-path/composition/PRIVATE-MOVE-VALIDATION.json new file mode 100644 index 0000000000..6683b13ef9 --- /dev/null +++ b/research/bit-copy-path/composition/PRIVATE-MOVE-VALIDATION.json @@ -0,0 +1,10 @@ +{ + "checker_sha256": "9bb59aaae2a14a6775e8856014c7d38edbceb50c55e8cf7b907110b8f53a1a2c", + "default_nonmutating": true, + "normal_optimized_git_free_equal": true, + "portable_input_file_count": 9, + "python_version": "3.12.12 (main, Oct 9 2025, 11:07:00) [Clang 17.0.0 (clang-1700.6.3.2)]", + "result_bytes": 13773, + "result_sha256": "8d5a70da6dd9a675cbc8d3fb452b24954c3cb1549193220b58ff5e74a5372526", + "scope": "Exact local operator and fixed ordered-barrier controls; no compiler or supplier admission." +} diff --git a/research/bit-copy-path/composition/PRIVATE-MOVE-control.py.txt b/research/bit-copy-path/composition/PRIVATE-MOVE-control.py.txt new file mode 100644 index 0000000000..4f6ac09bff --- /dev/null +++ b/research/bit-copy-path/composition/PRIVATE-MOVE-control.py.txt @@ -0,0 +1,50 @@ +"""Finite literal-frame controller shared by execution and full-source checks.""" +from words import need + + +class LocalState: + def __init__(self,call): + self.call=call + self.private=len(call.owners) + self.frames=list(call.start)+[-1] + self.versions=[0]*(self.private+1) + self.opened=None + + def accept(self,position): + event=self.call.events[position] + op,role=event[:2] + private,frames,versions=self.private,self.frames,self.versions + need(0<=role2^80 throughout; constants and the fixed tape count may +depend on q and the selected word. We make no practical runtime claim. + +We supply a mathematical algorithm and finite interface checks. We have not +translated the entire recursive machine into transition tables or a proof +assistant. Reviewers must assess the composition argument below. The selected +finite certificate and the multiplication theorem have further interfaces, +including the complex supplier and analytic transfer; we leave their status +unchanged. + +## 1. Sources and the recursive contract + +We use DreamingOfClouds' selected PR589 word, building on eumemic's PR555 word, +and alephnan's PR1627 literal/chronological proof. We reuse the GL local-ring +packet integration of PR2195, the good adapters of PR2171, the fallback word +of PR2159/PR2207, and commdata2338's PR2188 weight grammar. Source527 already +gives the weighted factorization and outer borrowing method. PR1379 supplies +the unequal-buffer parking boundary. `SOURCES.json` pins the executable inputs; +`NOTICE.md` identifies the pending source commits and mathematical reuse. + +Fix an original radix-q width e, c=17/10000, and w=ceil(e^c). Put B=q^w and +R=Z/B. We keep w fixed throughout one coarse recursion. Let C_n reverse n +atom positions. For a diagonal unit vector A, we construct + + F_A(H,D) = (A C_n D, C_n A^-1 H). + +We quantify over any two ordered complete n-atom chunks, with arbitrary +complete preceding, intervening and following fields. This includes two +subchunks of one former bank, which we need for fallback reversals. We allow +arbitrary correlated dirty payload values. + +We generate A from a constant unit by the Good and FallbackPair expressions +below. A may read earlier frozen matrix fields; it may not read this node's +fresh family, matrix or unused stock. A unit child starts a new constant +expression but retains its global generation depth. We bound that depth by +the original root width, including after such a reset. + +At return we require the specified address permutation on I/O, pointwise +restoration of row and spectator coordinates, empty local work and restored +head positions. For a parking top h, we visit no cell below h-1 and write no +cell below h. We require a time bound uniform in parked contents and h. + +## 2. Actual residuals and full-ring preparation + +PR1627 reconstructs 636 rational frame projectors and proves the nesting +identities used by both selected literal programs. Their entries lie in +Z[1/2]. For an idempotent P0 of rational rank r, its characteristic polynomial +is T^(64-r)(T-1)^r. Reduction modulo an odd q preserves this identity and +idempotent diagonalizability, hence rank r, including when q divides r. + +Choose independent image columns of P0 and kernel columns of I-P0 modulo q. +Their lifts form a unit matrix B0 over R and satisfy + + P0 = B0 diag(I_r,0) B0^-1. + +We compute this identity in control arithmetic. We do not move payload through +B0. For the actual owner chart sigma and raw physical header g, the provider +passes g sigma B0 to the retained coordinate normalizer. It therefore computes +exactly P=g sigma P0 sigma^-1 g^-1. + +For MOVE, we use the positive nested frame difference and the actual owner +sigma. The common stage projector cancels. The reflected orientation gives +the same positive residual. For COPY, we first copy and align the source +packet to d=g sigma. Its private header is d and its projector label is P0; +we set its additional chart to I. Applying sigma again would change the +operator. ERASE retains the literal transported-source equality guard. + +For an idle remainder, we subtract the actual hosted slots from its pending +axes and use the fixed family chart. We skip the zero remainder. The provider +binds 28,278 canonical requests and 32 replicas, giving + + H_1=381024, H_2=400416, H_3=22752, H_4=26880, + H_5=29376, H_6=17376, H_7=26784, H_42=288. + +We obtain 541 distinct base labels and 1,235 assembled charts. The finite +checks verify image/kernel bases over Z/9 and Z/25 for each label, and 120 +full 128-coordinate operators from selected MOVE, COPY and idle requests. +These cases supplement the identities; they do not enumerate GL64(R). + +## 3. One schedule per complete low packet + +Write g=g0+qG with g0 in GL64(F_q) and G in [q^(w-1)]^4096. For a fixed +request, call g0 good when the first r northeast pivots of PC are units. +Equivalently, require the leading k principal minors of P to be nonzero +modulo q for 1<=k<=r. Unit tests depend only on g0. The full coefficients +still depend on G. + +For each k, multiply its minor by det(g)^k to obtain a polynomial of degree +at most 64k in the entries of g. Conjugating the fixed split rank-r projector +to diag(I_r,0) gives a nonzero witness. The union of these zero sets has +fraction at most 64r(r+1)/(2q) in all matrices. Since + + |GL64(F_q)|/q^4096 = product_(i=1)^64 (1-q^-i) > 1/2, + +the bad fraction for this request is at most 2*64^3/q<10^-16. We use this +bound separately for each paid edge. We do not require one matrix to be good +for every label or multiply the bound by the number of labels. + +On a good packet, unweighted northeast elimination gives unit-lower matrices +L,R0 and a pivot vector delta. Split rank and a unit rank-r minor force the +remaining Schur block to vanish over R itself. We retain q-multiples in the +arithmetic; reduction modulo q alone would not establish that conclusion. +The normal-form factorization in source527/PR2171 uses ordered head, tail +and cross ADDs, one weighted child, and the inverse adapters. For n=64f, + + A_child[k f+j] = A[k f+j] delta_k, 0<=k=1. Recompute on split and merge rather than retaining +an array indexed by the high fibre. Ripple counters have amortized linear +movement cost; initialize them once per scan and do not reread unrelated +spectator counters per fibre. Fixed low tables and finite labels depend on +q, not e. Thus these scans have the stated uniform bounds. For independently +chosen n,w,d, retain the preparation quotient instead of this absorption. + +The selected word has fixed length. Good and bad bodies use O(n) ordered +atom passes; header routes use O(Vw) each; selections, XORs, erasure and +parking use O(V) each. We include their finite q-dependent multiplicities +in a constant C. The local nonrecursive charge is C V(n+w), including the +remainder leaf. This is an asymptotic accounting, not an explicit optimized +finite invoice for the selected multiplier. PR2212 prices the separate +same-family route inventory and leaves further whole-invocation charges. + +Normalize by logical volume and bound the worst generated call by T(n,w). +For positive widths below 64, T<=Cw. The bad fraction is at most epsilon=10^-16 +for each request, and each fallback has at most 32*64^2 children of width at +most f=floor(n/64). We may overcount good calls on bad classes, obtaining + + T(n,w) <= sum_r (H_r/W) T(r f,w) + + epsilon F/W * max_(1<=j<=f) T(j,w) + C(n+w), + F=32*64^2*sum_r H_r=118606528512. + +We prove a monotone upper bound, so the maximum causes no extra factor. +Define mu_t=(sum H_r(r/64)^t + epsilon F(1/64)^t)/W. The two retained outward +rational engines independently give + + mu_1 = 2874298095721977/2890625000000000 < 199/200, + mu_(1-c) < 9999/10000. + +Induct on n and use f<=n/64. Increasing C to cover the leaf constants gives + + T(n,w) <= 200 C n + 10000 C n^(1-c) w. + +Indeed the first term leaves at least Cn after the recursive linear mass, +and the second leaves at least C n^(1-c)w>=Cw. We treat n=0 separately. +The resulting constants are independent of ancestor contents, stack top, +row multiplicity and original e. + +## 8. Internal borrowing and binary interchange + +Choose the sufficient stock S=K(w)^D(e) from the original width e, before +choosing a body width. Since M= 1` fibres. This closes one storage component, not the total caller reserve. + +The source is Jose Manuel/jmbarrancoml's [PR2697](https://github.com/CrocSwap/integer-mult-bounds/pull/2697) +at `a53fdf4bcbe33c6732f0bec1badc8c023380f9d1`. +[SNAPSHOTS.json](SNAPSHOTS.json) authenticates the 22 unchanged source exports. +The [Discussion2319 claim](https://github.com/CrocSwap/integer-mult-bounds/discussions/2319#discussioncomment-18955184) +takes the residual-lifetime handoff after Nick Petalas's +[PR2708 route bound](https://github.com/CrocSwap/integer-mult-bounds/pull/2708). +ywz65's [PR2688 directory bound](https://github.com/CrocSwap/integer-mult-bounds/pull/2688) +is another complementary component. Neither result is reproduced or claimed anew. + +## Contract and metric + +Count nonblank Boolean payload cells on distinct tapes. Reading a cell does not +erase it. Keep borrowed input cells in the count until the routine returns. +Exclude LEFT markers, prefix directories and fields, numerical/controller/count +tapes, and allocated Python backing capacity. Thus this is not a count of all +physical tape cells or host memory. + +Assume `q` is a fixed odd prime, `M = q^w`, `w >= 1`, `P >= 1`, and `R > 0`. +The input is a complete packet of equally sized fibres. Each fibre includes all +positive before/gap/suffix multiplicities and the full body domain supplied by +[Layout](vendor/layout.py). The trace is valid and completes successfully using +the pinned ADD, atom-swap and sign primitives and below-cutoff completed leaves. +Coefficient and observation callbacks allocate no payload tapes. Larger +recursive calls and arbitrary exception-path cleanup are outside this contract. + +## Lifetime proof + +### Primitive interiors + +The primitives preserve their borrowed source of `R` cells and return a new +`R`-cell output. Their bucket reads do not erase bucket content. + +* [ADD](vendor/vendor/stream.py) partitions the source into two buckets whose + combined size is `R`. Both buckets survive until the full output is written. + Its exact peak, including source, is `3R`. +* A single digit swap or complement has the same `3R` peak. In a + [multi-digit atom swap](vendor/vendor/bitstream.py), a later digit pass also + retains the original source while its current intermediate, buckets and next + output coexist. Its exact peak is `4R`. A one-digit atom swap needs `3R`. +* A sign first complements every digit, then calls `translate_one` before + clearing the complemented intermediate. During translation, the original + source, complemented current, translation buckets and complete output all + coexist. Its exact peak is `4R`, including when `w = 1`. + +All these counts are independent of the payload values. Empty bucket partitions +can change which bucket owns a cell, but not the combined bucket length. + +### Packet accumulation + +In [packet_engine.run](vendor/packet_engine.py), the first primitive uses the +borrowed input itself as `current`. After a primitive completes, `current` is a +new full packet. The original borrowed packet remains unchanged. The previous +owned `current` is cleared only after the next complete packet has been written. + +During the last fibre of a later primitive, the retained objects have sizes: + +| Object | Payload cells | +| --- | ---: | +| Original borrowed source | `v` | +| Distinct current packet | `v` | +| Previously completed fibres in the next output | `(P - 1)R` | +| Local source, intermediate, buckets and primitive output | at most `4R` | + +Their sum is `3v + 3R`. Earlier fibres have a smaller accumulated output. +After the primitive returns, copying its result to the packet output retains +only the local source and result, so that phase needs at most `3v + 2R`. +Cleanup and packet replacement cannot increase either bound. + +More precisely, numbering payload primitives from zero, their exact peaks are: + +| Primitive | First primitive | Any later primitive | +| --- | ---: | ---: | +| ADD or one-digit atom swap | `2v + 2R` | `3v + 2R` | +| Sign or multi-digit atom swap | `2v + 3R` | `3v + 3R` | + +An empty trace returns the source alias and has peak `v`. Entry/return records +allocate no payload. Every nonempty [completed leaf](vendor/vendor/trace.py) +contains a sign after earlier primitives. Consequently any trace containing +such a leaf attains `3v + 3R`; the result is not just an upper bound. + +This is a source-level lifetime proof for the stated contract. The finite checks +below corroborate the proof and instrument coverage; they do not replace it. + +## Retained-role corollary + +Let `K` distinct staged packets of size `v` remain live during a local call, +including any existing private packet. Count aliases by tape identity. +In [Driver.local](vendor/driver.py): + +* MOVE passes one of those packets as the engine's borrowed input. The staged + packets plus residual therefore need at most `(K + 2)v + 3R`. +* COPY first makes a distinct clone while all `K` staged packets remain live. + That clone is the engine's borrowed input. The bound is `(K + 3)v + 3R`. + The clone is cleared after return; the returned private packet remains until + its matching ERASE. + +For the source's 661 ordinary roles and at most one private role, both cases +are bounded by `664v + 3R`. This counts staged roles and the active residual +only. It does not include protected input, whole-family storage, directories, +routes, scatter output, peel/merge work, control tapes or markers. The corollary +is derived from source ownership and chronology; this checker does not execute +the full `Driver.local` schedule or certify its total reserve. + +## Reproduction and evidence + +From the repository root, with standard-library Python 3.12: + +```sh +python3.12 scripts/research_package.py --package research/selected-residual-live-volume --check-only +python3.12 scripts/research_package.py --package research/selected-residual-live-volume +python3.12 -B -O research/selected-residual-live-volume/verify.py +``` + +[verify.py](verify.py) authenticates sources before imports, instruments the +unchanged allocation/read/write paths, and checks a direct tape census against +the running meter. Instrumentation is restored in `finally`; it changes no +transform, cleanup order or allocation policy. Origin tags track every input +cell through actual copy operations. Expected address permutations use separate +ordinary modular arithmetic, and actual Boolean values are checked too. +Allocation-site classification is bound to the frozen source lines, not a +promise of compatibility with future versions. + +[RESULT.json](RESULT.json) records 17 fixtures and 231 primitive events. The +fixtures include an empty alias case, ADD/swap/sign over moduli 3 and 9 with +one and two prefixes, nontrivial spectators and gaps, an embedded layout, and +complete good/fallback traces with 1 and 19 child returns. The final fixture +rejects a missing child return. Positive fixtures cover 10,701 input cells with +individual origin tracking; these are complete small declared domains, not +selected GL64 execution or a complete high-header census. + +A sharp fixture has `P = 2`, `R = 972`, `v = 1944` and measured peak **8748**. +At that cut the census identifies both full packets, the partial next output, +local source, primitive intermediate, buckets and primitive output. Budgets +that omit the borrowed source or prematurely discard `current` give 6804; +omitting bucket/output overlap gives 7776. All are rejected by the actual census. +Source-byte mutation, incomplete manifests and borrowed-input mutation are also +rejected. Default replay compares the exact typed receipt and does not rewrite +it; `--write-result` refuses an existing result file. + +The registered replay and the standalone checker serve different provenance +roles. The runner validates historical catalogue provenance at the pinned Git +base. Standalone replay authenticates the frozen manifest and scientific source +exports plus the explicitly declared live selected-record input. Full Git +history is needed for the runner's historical provenance check, not for the +standalone scientific replay. + +## Open next questions + +Can the complete first-node caller fit a numerical reserve? Join this residual +component with the route and directory components at their actual overlapping +lifetimes, then include protected/family copies, count/controller tapes, markers +and disposal. A source-matched event ledger is the feasible first step, not +another independent maximum added blindly to all others. Larger recursive +returns remain a separate blocker. [TASK.json](TASK.json) records this handoff. + +Stop this contribution at the component result. It does not justify a larger +construction campaign by itself. No new supplier, kappa, selected-frontier +change, full selected cold replay, full GL64 execution, formal-kernel proof or +independent human review is claimed. The finite coefficient fixtures use small +explicit projectors rather than selected 64-dimensional callbacks. Numerical +storage, markers and arbitrary error-path cleanup remain unverified. + +## Credit + +Jose Manuel/jmbarrancoml supplies the unchanged PR2697 engine and first-node +caller. [vendor/NOTICE.md](vendor/NOTICE.md) retains its full stream, selected-word, +normal-form, controller, hosting and audit lineage. Nick Petalas supplies the +complementary route result and explicit residual-lifetime handoff; ywz65 supplies +the directory and earlier primitive/setup audits. + +New proof, checker and evidence prepared by starship-s with substantial Hermes +(`gpt-6-astra-900k`) and Codex (`gpt-6-astra`, max effort) assistance. Apache-2.0. diff --git a/research/bit-copy-path/composition/PROOF.md b/research/bit-copy-path/composition/PROOF.md new file mode 100644 index 0000000000..bc66dbecca --- /dev/null +++ b/research/bit-copy-path/composition/PROOF.md @@ -0,0 +1,314 @@ +# The R429 code in the actual nine-stage BIT caller + +This is a finite exact counterexample and two explicitly restricted repair +screens. It supplies no successful routed decoder, supplier admission or +positive kappa. The complete current coded word, its expected receipt and its +proof are frozen under `inputs/`; their original scientific pin is +`20ecf2891d80b5acf18d980b2164fb44ea7faa90`. + +The question is whether the three-COPY R429 word can replace the local calls +of the pinned `global_k.py` recipe while preserving the required public BIT +endpoint. The answer for direct substitution, even with a terminal free +decoder, is no. The failure is on public data, independent of helper dirt. +This conclusion does not exclude a changed caller with paid native repairs. + +## The exact local and caller contracts + +There are 111 arbitrary source fields, 111 arbitrary target fields and 429 +arbitrary dirty helper fields. One private field starts and ends zero. +The frozen forward word has binary VALUE endpoint + + F_A(x,y,h) = (A x, y+A x, h), + +and its literal inverse is + + F_A^-1(x,y,h) = (A^-1 x, y+x, h). + +Sources enter at their actual distinct rank-one port lines and finish at +FULL. Target p enters at zero and finishes at the cap orthogonal to port p. +Every surviving helper is restored in logical coordinates and makes its +declared full physical progress. These physical endpoints are part of the +contract; helpers are not restored to their original ADDRESS arrangement. +Over the integers the literal source code is unimodular and the target +contract is only asserted modulo two. No integer endpoint or rounded cleanup +bound is inferred from the binary formulas below. + +The checker replays every frozen record on all 651 independent ordinary +columns, including the private lifetime comparisons. It reconstructs A and +its actual inverse independently. The exact scalar facts are + + order(A)=28, rank(A-I)=9, rank(A^2-I)=8, + identity rows of A = {0,1,2}. + +The controller has four public pairs and uses the exact `recipe(4)` extracted +from the pinned source. With pair `(x_p,y_p)`, its local scalar actions are + + forward: (x,y) -> (A x, y+A x), + reverse: (x,y) -> (x+y, A^-1 y). + +Bridges and the final pair exchanges are unchanged. All 888 public scalar +columns are replayed. A second computation over the Laurent ring +`F2[a,a^-1]` constructs the entire 8-by-8 bank matrix; substituting `a=A` +must agree with the first computation entry by entry. The full Laurent +matrix and all nine chronological scalar endpoint hashes are in +`RESULT.json`. The final scalar defect has rank **63**. Setting A to I +restores all 888 columns. Replacing the literal inverse by A changes the +actual caller and is rejected as an equivalent endpoint. + +In particular, let only the initial target of the last pair, y3, be nonzero. +It is never read by an earlier local call. It reaches reverse stage 7 and +forward stage 8, followed by the backward bridge chain and final exchanges. +The resulting scalar output is + + x0=x1=x2=x3=(A+A^-1)y3, y3=A y3, + y0=y1=y2=0. + +These are complete arbitrary-field coefficient identities, not tests of a +single binary input. The first residual has rank eight. Already constant +VALUE arrays therefore disprove direct substitution irrespective of ADDRESS +routes or precision choices. + +## Actual chronological ADDRESS operators + +The following notation denotes operators on binary VALUE arrays induced by +ADDRESS permutations. It does not identify differences of VALUE permutation +operators with differences of their ADDRESS matrices. + +Let N be the diagonal rank-one exchange on the actual port roots. Let G0 +exchange each port's original seven-dimensional cap, and let Gs, s=1,...,8, +exchange the fresh seven-dimensional coordinate block of stage s. Then + + Fs=N Gs, T=N G0 G1 ... G8, + +where T is the required full 64-dimensional parent exchange. The Gs and N +commute. For s>0 the actual source route rho_(s,p) fixes the port root and +interchanges its core cap with fresh block s. Its cap isometry has columns +from the pinned native chart. Write + + A_s = diag(rho_(s,p)) A diag(rho_(s,p))^-1. + +A_s is HEAD/TAIL-separated. It commutes with Fs and with every Gt, t != s. +The latter statement includes G0: conjugating the old cap by rho_s makes it +the same fresh coordinate block for every port. A_s generally does not +commute with N. These are the native route relations already distinguished +in the credited C511 interface proof; no order-511 fact is transferred to +the order-28 code. + +The actual physical forward and role-swapped inverse are + + B_s = [[Fs A_s N, 0], [Gs A_s N, Gs]], + R_s = [[Gs, Fs], [0, Gs A_s^-1]]. + +The inverse formula uses Fs A_s=A_s Fs. No step moves A_s through N. + +Let L6=G0...G6. On the untouched entering last target, the state before +stage 7 is `(0,L6 y3)`. Applying the two local calls gives + + R7: (F7 L6 y3, G7 A7^-1 L6 y3), + B8: (T A8 y3, (G0...G8)(A8+A7^-1)y3). + +For example, `N F7=G7`, and A8 commutes with G7 and L6. These are the +commutations needed in this calculation. The final backward bridges copy +the second component to all four source banks. After final exchanges and +factoring off the required full parent T, the last-target column is + + x0=x1=x2=x3=N(A8+A7^-1)y3, y3=A8 y3, + y0=y1=y2=0. (1) + +This is the native counterpart of the preceding Laurent column. + +## One integral native impulse and the free-decoder obstruction + +The actual roots at ports 0 and 3 are `2e0` and `2e3`, and the reconstructed +code has `A[3,0]=1`. Give the initial last target at port 0 one VALUE impulse +with + + HEAD=e59, TAIL=0. + +Axis 59 is the third coordinate of fresh stage-8 block 57,...,63. +The checker reconstructs the actual two port charts from the frozen full +120-root geometry using the upstream Weyl breadth-first route convention. +It verifies their cap isometries and rho squares exactly. The actual +root-zero chart has cap columns e1,...,e7, so + + rho_(8,0)e59=e3, rho_(8,3)e3=e3. + +Every stage-7 chart fixes e59. Consequently the A8 term in (1), at output +port 3, sends the input to HEAD=e3; the following N sends it to + + output port 3: HEAD=0, TAIL=e3, VALUE coefficient 1. (2) + +The A7^-1 term keeps its entire input on HEAD=e59 and TAIL=0, so it cannot +cancel (2). There is only one A8 coefficient path from the chosen input +port to output port 3. Earlier stages cannot contribute another path: +the initial y3 was untouched until stage 7. + +This witness uses only integral displayed coordinates and survives in every +odd ADDRESS field. It is also stable under cap-gauge changes: replace e59 +by the fresh-block vector `a_0^T e3`, where a_0 is the root-zero cap +isometry. Then `rho_(8,0)(a_0^T e3)=e3` and the same argument applies. +Changing latent bases for the factorization A-I does not change A[3,0]. + +Let U denote VALUE arrays supported on TAIL=0 across all public banks and +ports. Scalar mixing and HEAD/TAIL-separated ADDRESS routes preserve U. +Every invertible decoder made from these operations, and its inverse, +preserves U, so it induces an invertible map on the quotient by U. +The nonzero quotient output (2) cannot be erased by such a decoder. +The decoder may mix every bank and port; invertibility is what prevents +a nonzero quotient class from disappearing. + +This is a terminal free-alphabet obstruction. It does not prohibit a +decoder that makes further native exchanges, accesses another observation +cut or changes the earlier caller word. + +## Why scalar involutivity does not fix the inverse-kernel mismatch + +Appending the literal source inverse at local FULL costs 1,499 additional +scalar ADDs and no additional native query rank. It makes the physical +source-preserving local shear + + B'_s = Gs [[I,0],[A_s N,I]]. + +Its literal inverse with exchanged public roles is + + R'_s = Gs [[I,N A_s],[0,I]]. + +At one fixed stage, involutivity of A_s would make `N A_s` the inverse of +`A_s N`. Consecutive caller stages use A7 and A8, which are differently +routed operators. The last-pair column of the source-restored variant, +after factoring off T, is instead + + x0=x1=x2=x3=N(A8 A7+I)y3, y3=A7 y3. (3) + +Choose y3 so that its port VALUE vector is `A^-1 e0`, all at HEAD=e59 and +TAIL=0. A7 acts as the scalar A on this fresh block and hence produces +the single port-zero impulse above. The identity contribution in (3) +stays on the fresh block and has zero TAIL. The A8 contribution again +gives (2). This argument needs only invertibility, so it also excludes +the proposed scalar-involutive repair of this unchanged caller whenever +the code retains that off-diagonal edge. + +It does not exclude a newly built physical inverse kernel which uses the +previous stage's routed operator. Such a kernel is not supplied by the +same local scalar code at the next fresh stage and needs its own complete +literal word and cost. + +## Two priced, restricted repair screens + +First consider restoration to the original local shear before entering the +unchanged caller, using exactly nine independent scalar latent readers and +one common-cap completion per reader. Reconstructing the original retained +totals gives the actual rank-nine R and K. The R429 code satisfies + + A-I=W'R, W'=K with physical rows 0,1,2 set to zero. + +Both W' and R have rank nine. Thus every minimal nine-reader factorization +has nine independent output directions in the column space of W'. The +checker enumerates all 511 nonzero directions over F2 and computes their +actual root-support rank over the rationals: + +| Root-support rank | Number of directions | +| --- | ---: | +| 7 | 12 | +| 8 | 499 | + +The twelve rank-seven coefficient vectors are + + 135,185,205,208,238,336,349,415,425,437,503,507. + +The original nine rows of RK are an independent rank-seven basis. In the +declared common-cap tariff, where one completion must cover the root span +of its output direction, every reader therefore costs at least seven: +the repair rank is at least **63 per restored local call**. This is a +necessary price, not an emitted endpoint decoder. Actual capped-target +access and restoration can cost more. For a positive power exponent, +`Phi_b(7)=7(64/7)^b > 7`, so the tariff only increases. + +The parent budget screen leaves less than 58.383 tariff units per coded +call at the requested 10-percent target. A per-call repair in this exact +minimum-basis common-cap class cannot meet that budget. This is not a +bound for a single shared decoder after all nine calls, overcomplete +readers, ADDRESS-dependent readers or a different native query alphabet. + +Second, suppose one changes the source code to a routed code C commuting +with N, while retaining the clean map J on every target row outside a +small set P. Each unchanged off-diagonal entry `J[p,q]=I`, p != q, would +force + + Np C[p,q] = C[p,q] Nq, hence Np=Nq. + +The actual port projectors are all distinct. The checker supplies one +noncommuting off-diagonal edge and an exact unequal projector entry for +**each of the 111 rows of J**. Thus no row of J can remain unchanged in +an N-commuting code. In the prior one-snapshot-per-repaired-row interface, +this requires at least **111 rank-one snapshots**. In particular the +three-row repair does not survive a free replacement of its off-diagonal +identity coefficients by root-intertwining ADDRESS routes. + +This second count covers that physical-row repair interface. A combined +repair with different output directions or a changed clean factory is a +different construction and is not lower-bounded by 111 here. + +## Construction search boundary and exact local inverse decoder + +The cheapest concrete exact endpoint repair found algebraically for one +physical coded call is the decoder identity + + B_clean B_A^-1 = [[A^-1,0],[N(A^-1+I),I]]. + +It separates the source inverse from the missing target operation but +does not implement the latter cheaply. Its rank-nine scalar factorization +does not make nine native queries sufficient at rank one: N acts at each +destination's different cap. Conversely, merely rebasing the nine latent +readers cannot lower the preceding common-cap support floor below seven. + +The requested next step is therefore a new, explicitly charged routed +realization of that operator or a changed coupled caller. No favorable +upper bound was found. The calculation must not be turned into either a +63-unit universal lower bound or a zero-cost scalar decoder. + +One tempting chart-sharing mechanism has an already known paid failure. +For `H_p=rho_(t,p)rho_(s,p)`, one has `H A_s H^-1=A_t`, but H also +transports the completed fresh block Es to future block Et. Therefore + + H^-1 D_Et H D_Es = D_Es D_Es = I, + +where the required combined progress is `D_(Es+Et)` of rank fourteen. +The equality follows from orthogonal cap isometries: rho_s sends a vector +in Es to the core cap and rho_t sends that core vector to Et. Thus the +specific alignment/query/unwind word loses both seven-dimensional advances +on each affected family. It cannot be billed as sharing two completed +native advances. This is the prior R429 calculation retained in +`inputs/R429-PHYSICAL-CODE-REFERENCE.py.txt`, whose word hash matches this +package; it is credited rather than claimed as new. Its larger commutant +and orbit searches are not run here or used as proof premises. + +## Reproduction, controls and remaining obligations + +Run from any working directory: + + python3.12 -I -B -Werror /absolute/path/to/composition/check_composition.py + +The checker needs only the standard library. It authenticates all runtime +inputs by SHA256, reads the frozen coded word, executes the pinned recipe +and compares `RESULT.json` without writing. `--output PATH` explicitly +emits a recomputed receipt. It rejects an incorrect inverse and deletion of +the actual code edge used by the native witness. The identity-code control +restores all 888 public scalar columns. Normal and optimized runs should +emit identical receipts. + +The finite all-column replays, all-511 rational census and exact native +witness are separate from the written operator and free-decoder arguments. +There is no formal-kernel proof. The original local helper/COPY restoration +is retained; no new chronological owner schedule, rounded child-error +bound, physical source/tape implementation or joint complex/BIT transfer +is supplied. A positive result still needs those obligations. + +Credit the retained R438 word and transport/rehosting lineage, the frozen +R429 three-source-snapshot construction, and the C511 mutable-source +interface/observation work for its physical inverse and freshness lessons. +The historical R429 physical-code source supplies the chart-alignment +coverage-debt predecessor. The old sources are copied as references, not +executed or treated as certificates for the changed caller. This note contributes the exact +nine-stage R429 column, fresh-stage last-pair witness and restricted repair +screens. Prepared with substantial OpenAI Codex assistance; Apache-2.0. diff --git a/research/bit-copy-path/composition/RESULT.json b/research/bit-copy-path/composition/RESULT.json new file mode 100644 index 0000000000..f267d79460 --- /dev/null +++ b/research/bit-copy-path/composition/RESULT.json @@ -0,0 +1,32553 @@ +{ + "A": [ + 1, + 2, + 4, + 576460752303423479, + 2596069508172130947062333349822703, + 309485009244884317495099359, + 10140895326260170442259558318015, + 2586086145551572543258365643440255, + 2589398147263233519358262897489001, + 2592804859121274158248396097259625, + 2592757506467882867942327749505385, + 2589350696924854570449116603662953, + 2592757607046729626097202769225065, + 2589398046688157573239251990943081, + 1734130175483521454113919338159274, + 872127581635430275692856302967978, + 1734082824753018225619070127219626, + 872080229941564978246942081967018, + 1734082638102400486742802049377194, + 872080416603461748079412450059178, + 1734129989784902589838982619225258, + 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"one_snapshot_per_changed_row_native_rank_lower_bound": 111, + "rank7_basis": [ + 507, + 135, + 238, + 349, + 185, + 437, + 205, + 503, + 425 + ], + "scope": "Nine independent scalar latent readers, each using one common-cap completion whose rank is at least the span of its output port roots. This is a necessary bound, not an emitted legal decoder or a bound for overcomplete or ADDRESS-dependent readers.", + "support_rank_histogram": { + "7": 12, + "8": 499 + } + }, + "no_new_kappa": true, + "rank_A_minus_I": 9, + "rank_A_squared_minus_I": 8, + "source_code_order": 28, + "source_pins": { + "composition/inputs/CODED-WORD.json.gz": "1a6841fccba51fa922993c77b4b296e659c2c538252e5e06113d0438c5edb870", + "composition/inputs/FULL-E8-GEOMETRY.json.gz": "1fb726ea3818fe0837cceb0dd22a71a0814f064e5e781628104610d6468e0e55", + "inputs/GEOMETRY.json.gz": "a353999bceac4ac957f540adfd3e5f8768297138b27132eeb83332e2ea4b1022", + "inputs/global_k.py.txt": "6368196a393ac25bb2918880e4c4bd31ed733b762c27b6873ab391cbc0c5c3c8" + }, + "status": "PASS_R429_NINE_STAGE_ENDPOINT_DEFECT_AND_BOUNDED_REPAIR_FLOOR" +} diff --git a/research/bit-copy-path/composition/VALIDATION.json b/research/bit-copy-path/composition/VALIDATION.json new file mode 100644 index 0000000000..ebc8ae02d9 --- /dev/null +++ b/research/bit-copy-path/composition/VALIDATION.json @@ -0,0 +1,7 @@ +{ + "normal_optimized_git_free_equal": true, + "result_sha256": "905ee73a1175320a7c4c39a7b665a67f57316def616b29ffdc5d64e100151303", + "result_bytes": 483798, + "python_version": "3.12.12 (main, Oct 9 2025, 11:07:00) [Clang 17.0.0 (clang-1700.6.3.2)]", + "scope": "Exact scoped script replay only, not supplier admission." +} diff --git a/research/bit-copy-path/composition/check_composition.py b/research/bit-copy-path/composition/check_composition.py new file mode 100644 index 0000000000..0264f84213 --- /dev/null +++ b/research/bit-copy-path/composition/check_composition.py @@ -0,0 +1,452 @@ +#!/usr/bin/env python3 +"""R429 coded endpoint in the pinned nine-stage BIT recipe. + +Finite exact scalar and rational witnesses only. No positive supplier tariff. +Apache-2.0; substantial OpenAI Codex assistance. +""" +from collections import Counter +from fractions import Fraction +from hashlib import sha256 +from pathlib import Path +import argparse +import ast +import gzip +import json + +HERE = Path(__file__).resolve().parent +ROOT = HERE.parent +PINS = { + "composition/inputs/CODED-WORD.json.gz": + "1a6841fccba51fa922993c77b4b296e659c2c538252e5e06113d0438c5edb870", + "inputs/GEOMETRY.json.gz": + "a353999bceac4ac957f540adfd3e5f8768297138b27132eeb83332e2ea4b1022", + "inputs/global_k.py.txt": + "6368196a393ac25bb2918880e4c4bd31ed733b762c27b6873ab391cbc0c5c3c8", + "composition/inputs/FULL-E8-GEOMETRY.json.gz": + "1fb726ea3818fe0837cceb0dd22a71a0814f064e5e781628104610d6468e0e55", +} + + +def need(ok, message): + if not ok: + raise ValueError(message) + + +def canonical(value): + return json.dumps(value, sort_keys=True, separators=(",", ":")).encode() + + +def read(relative): + path = ROOT / relative + raw = path.read_bytes() + need(sha256(raw).hexdigest() == PINS[relative], "Source bytes changed") + return json.loads(gzip.decompress(raw)) if path.suffix == ".gz" else raw.decode() + + +def bits(value): + while value: + bit = value & -value + yield bit.bit_length() - 1 + value ^= bit + + +def xor(values): + result = 0 + for value in values: + result ^= value + return result + + +def product(left, right): + return [xor(right[j] for j in bits(row)) for row in left] + + +def rank2(rows): + pivots = {} + for row in rows: + while row: + p = (row & -row).bit_length() - 1 + if p in pivots: + row ^= pivots[p] + else: + pivots[p] = row + break + return len(pivots) + + +def inverse(rows): + n = len(rows) + rows = [row | (1 << (n + j)) for j, row in enumerate(rows)] + for j in range(n): + p = next((p for p in range(j, n) if rows[p] >> j & 1), None) + need(p is not None, "Singular code") + rows[j], rows[p] = rows[p], rows[j] + for p in range(n): + if p != j and rows[p] >> j & 1: + rows[p] ^= rows[j] + return [row >> n for row in rows] + + +def rankq(rows): + basis = {} + for row in rows: + row = list(map(Fraction, row)) + for p, old in sorted(basis.items()): + c = row[p] + if c: + row = [x - c*y for x, y in zip(row, old)] + p = next((p for p, x in enumerate(row) if x), None) + if p is not None: + c = row[p] + basis[p] = [x/c for x in row] + return len(basis) + + +def literal_endpoint(word, reverse=False): + n = word["public_registers"] + rows = [1 << i for i in range(n)] + [0] + live = None + records = list(reversed(word["records"])) if reverse else word["records"] + for op, dst, src, coefficient, _frame, _phase in records: + if op == 1: + if coefficient & 1: + rows[dst] ^= rows[src] + elif op == (3 if reverse else 2): + need(live is None and rows[n] == 0, "Private field already live") + rows[n] = rows[dst] + live = dst + elif op == (2 if reverse else 3): + need(live == dst and rows[n] == rows[dst], "Snapshot changed") + rows[n] = 0 + live = None + else: + need(op == 0, "Unknown literal opcode") + need(live is None and rows[n] == 0, "Unclosed private field") + return rows + + +def recipe(source): + tree = ast.parse(source) + selected = [node for node in tree.body + if isinstance(node, ast.FunctionDef) and node.name == "recipe"] + need(len(selected) == 1, "Unique source recipe") + module = ast.Module(body=selected, type_ignores=[]) + namespace = {} + exec(compile(module, "", "exec", dont_inherit=True, optimize=0), + namespace) + events = namespace["recipe"](4) + need(sum(e["kind"] == "local" for e in events) == 9, "Nine local stages") + return events + + +def controller(events, code, code_inverse, source_restored=False): + v = len(code) + rows = [1 << i for i in range(8*v)] + snapshots = [] + for event in events: + if event["kind"] == "local": + p = event["pair"] + x = rows[2*p*v:(2*p+1)*v] + y = rows[(2*p+1)*v:(2*p+2)*v] + if event["reverse"]: + if source_restored: + x = [a ^ b for a, b in zip(x, product(code, y))] + else: + x = [a ^ b for a, b in zip(x, y)] + y = product(code_inverse, y) + else: + ax = product(code, x) + x = x if source_restored else ax + y = [a ^ b for a, b in zip(y, ax)] + rows[2*p*v:(2*p+1)*v] = x + rows[(2*p+1)*v:(2*p+2)*v] = y + snapshots.append({"stage": event["stage"], "pair": p, + "reverse": event["reverse"], + "endpoint_sha256": sha256(canonical(rows)).hexdigest()}) + elif event["kind"] == "bridge": + d, s = event["dst"], event["src"] + for p in range(v): + rows[2*d*v+p] ^= rows[2*s*v+p] + rows[(2*s+1)*v+p] ^= rows[(2*d+1)*v+p] + else: + need(event["kind"] == "exchange", "Unknown recipe event") + for bank in range(4): + for p in range(v): + a, b = 2*bank*v+p, (2*bank+1)*v+p + rows[a], rows[b] = rows[b], rows[a] + return rows, snapshots + + +def laurent_controller(events): + rows = [[{0} if i == j else set() for j in range(8)] for i in range(8)] + + def add(d, s): + rows[d] = [a ^ b for a, b in zip(rows[d], rows[s])] + + def scale(d, power): + rows[d] = [{x + power for x in row} for row in rows[d]] + + for event in events: + if event["kind"] == "local": + p = event["pair"] + if event["reverse"]: + add(2*p, 2*p+1) + scale(2*p+1, -1) + else: + scale(2*p, 1) + add(2*p+1, 2*p) + elif event["kind"] == "bridge": + d, s = event["dst"], event["src"] + add(2*d, 2*s) + add(2*s+1, 2*d+1) + else: + for p in range(4): + rows[2*p], rows[2*p+1] = rows[2*p+1], rows[2*p] + return rows + + +def actual_two_charts(geometry, full): + """Reconstruct the upstream Weyl BFS routes at ports zero and three.""" + roots = full["lines"] + need(len(roots) == 120, "Full E8 source geometry") + parent = {0: None} + queue = [0] + for p in queue: + for q in range(120): + inner = sum(x*y for x, y in zip(roots[p], roots[q])) + if q not in parent and abs(inner) == 2: + parent[q] = (p, inner//2) + queue.append(q) + need(len(parent) == 120, "Complete actual Weyl graph") + result = [] + for port in (0, 3): + label = geometry["ports_geom"][port] + need(roots[label] == geometry["lines"][port], "Actual retained port label") + path = [] + at = label + while parent[at] is not None: + previous, sign = parent[at] + path.append((previous, at, sign)) + at = previous + g = [[Fraction(i == j) for j in range(8)] for i in range(8)] + for previous, current, sign in reversed(path): + v = [x-sign*y for x, y in zip(roots[previous], roots[current])] + norm = sum(x*x for x in v) + read = [sum(v[i]*g[i][j] for i in range(8)) for j in range(8)] + g = [[g[i][j]-Fraction(2*v[i], norm)*read[j] for j in range(8)] + for i in range(8)] + cap = [row[1:] for row in g] + root = geometry["lines"][port] + p = [[Fraction(x*y, 4) for y in root] for x in root] + need(all(sum(cap[i][k]*cap[j][k] for k in range(7)) + == int(i == j)-p[i][j] for i in range(8) for j in range(8)), + "Actual cap columns span the root complement") + rho = [p[i]+cap[i] for i in range(8)] + rho += [[cap[j][i] for j in range(8)] + [Fraction(0)]*7 for i in range(7)] + need(all(sum(rho[i][k]*rho[k][j] for k in range(15)) == int(i == j) + for i in range(15) for j in range(15)), "Actual rho squared") + result.append(rho) + return result + + +def rational_fresh_witness(geometry, code, charts): + roots = geometry["lines"] + need(roots[0] == [2,0,0,0,0,0,0,0], "Actual source root") + need(roots[3] == [0,0,0,2,0,0,0,0], "Actual destination root") + need(code[3] & 1, "Actual nonzero A[3,0]") + head = [Fraction(i == 59) for i in range(64)] + + def apply_chart(rho, vector, stage): + axes = list(range(8)) + list(range(7*stage+1, 7*stage+8)) + out = vector[:] + for i, a in enumerate(axes): + out[a] = sum(rho[i][j]*vector[b] for j, b in enumerate(axes)) + return out + + middle = apply_chart(charts[0], head, 8) + need(middle == [Fraction(i == 3) for i in range(64)], "First actual route") + routed = apply_chart(charts[1], middle, 8) + need(routed == middle, "Actual destination route fixes its root") + need(all(apply_chart(rho, head, 7) == head for rho in charts), + "Actual stage7 routes fix fresh stage8 input") + p = [[Fraction(x*y, 4) for y in roots[3]] for x in roots[3]] + tail = [sum(p[i][j]*routed[j] for j in range(8)) + if i < 8 else Fraction(0) for i in range(64)] + outhead = [x-y for x, y in zip(routed, tail)] + need(outhead == [0]*64 and tail == [int(i == 3) for i in range(64)], + "Actual uncancelled TAIL impulse") + # Stage7 routes are identity on fresh stage8 coordinates; every inverse + # coefficient therefore stays on HEAD=e59, which N also fixes. + return { + "input_bank": 7, "input_port": 0, "input_HEAD_axis": 59, + "input_TAIL": 0, + "output_banks_after_parent_full_factor": [0, 2, 4, 6], + "output_port": 3, "output_HEAD": 0, "output_TAIL_axis": 3, + "VALUE_coefficient": 1, + "ADDRESS_domain": "Every odd field; the displayed coordinates are integral", + "route_reason": "rho8,0 e59=e3; rho8,3 e3=e3; rho7,p e59=e59", + "actual_two_rho15_sha256": [ + sha256(canonical([[str(x) for x in row] for row in rho])).hexdigest() + for rho in charts], + } + + +def supports(geometry, code): + n = 2*geometry["v"] + geometry["R"] + rows = [1 << i for i in range(n)] + for phase in ("D", "A"): + for _frame, dst, sources in geometry[phase]: + rows[dst] ^= xor(rows[src] for src in sources) + mask = (1 << 111) - 1 + r = [rows[i] & mask for i in geometry["ret"]] + k = [sum(1 << i for i in indices) for indices in geometry["scatter"]] + w = [0 if p < 3 else row for p, row in enumerate(k)] + need(rank2(w) == rank2(r) == 9, "Full nine-channel defect") + need(product(w, r) == [row ^ (1 << p) for p, row in enumerate(code)], + "Literal source defect equals W' R") + census = [] + histogram = Counter() + for q in range(1, 512): + support = [p for p, row in enumerate(w) if (row & q).bit_count() & 1] + dimension = rankq(geometry["lines"][p] for p in support) + census.append({"query_output_vector": q, "support": support, + "support_rank": dimension}) + histogram[dimension] += 1 + need(histogram == {7: 12, 8: 499}, "Complete rational direction census") + chosen = product(r, k) + need(rank2(chosen) == 9 and all(census[q-1]["support_rank"] == 7 for q in chosen), + "Nine independent rank-seven directions") + clean = [row ^ (1 << p) for p, row in enumerate(product(k, r))] + commutator_edges = [] + for p, row in enumerate(clean): + q = next((q for q in bits(row) if q != p), None) + need(q is not None, "Every clean-map row has an off-diagonal identity") + pp = [[Fraction(x*y, 4) for y in geometry["lines"][p]] + for x in geometry["lines"][p]] + pq = [[Fraction(x*y, 4) for y in geometry["lines"][q]] + for x in geometry["lines"][q]] + changed = next(((i, j) for i in range(8) for j in range(8) + if pp[i][j] != pq[i][j]), None) + need(changed is not None, "Distinct root projectors in actual edge") + i, j = changed + commutator_edges.append({ + "row": p, "column": q, "projector_entry": [i, j], + "destination_projector_value": str(pp[i][j]), + "source_projector_value": str(pq[i][j]), + }) + return { + "R": r, "W_prime": w, + "complete_output_direction_census": census, + "support_rank_histogram": dict(sorted(histogram.items())), + "rank7_basis": chosen, + "basis_model_native_rank_lower_bound": 63, + "N_commuting_code_unchanged_row_obstructions": commutator_edges, + "N_commuting_code_required_changed_physical_rows": 111, + "one_snapshot_per_changed_row_native_rank_lower_bound": 111, + "scope": ( + "Nine independent scalar latent readers, each using one common-cap " + "completion whose rank is at least the span of its output port roots. " + "This is a necessary bound, not an emitted legal decoder or a bound " + "for overcomplete or ADDRESS-dependent readers." + ), + } + + +def main(): + parser = argparse.ArgumentParser() + parser.add_argument("--output", type=Path, + help="Write a recomputed receipt to this explicit path.") + args = parser.parse_args() + word = read("composition/inputs/CODED-WORD.json.gz") + geometry = read("inputs/GEOMETRY.json.gz") + full_geometry = read("composition/inputs/FULL-E8-GEOMETRY.json.gz") + charts = actual_two_charts(geometry, full_geometry) + events = recipe(read("inputs/global_k.py.txt")) + forward, backward = literal_endpoint(word), literal_endpoint(word, True) + code, code_inverse = forward[:111], inverse(forward[:111]) + identity = [1 << i for i in range(111)] + need(backward[:111] == code_inverse, "Literal inverse source code") + for p in range(111): + need(forward[111+p] == (1 << (111+p)) ^ code[p], "Coded forward target") + need(backward[111+p] == (1 << (111+p)) ^ (1 << p), "Literal inverse target") + for p in range(222, 651): + need(forward[p] == backward[p] == 1 << p, "Arbitrary dirty helper return") + rows, snapshots = controller(events, code, code_inverse) + scalar = laurent_controller(events) + powers = {-1: code_inverse, 0: identity} + for i in range(1, 5): + powers[i] = product(code, powers[i-1]) + rebuilt = [] + for blockrow in scalar: + for p in range(111): + rebuilt.append(xor(xor(powers[e][p] for e in polynomial) << (111*j) + for j, polynomial in enumerate(blockrow))) + need(rows == rebuilt, "Independent Laurent and bit-row caller products") + last_mask = ((1 << 111)-1) << 777 + residual = [a ^ b for a, b in zip(code, code_inverse)] + for bank in range(8): + expected = residual if bank % 2 == 0 else code if bank == 7 else [0]*111 + need([(row & last_mask) >> 777 for row in rows[111*bank:111*(bank+1)]] + == expected, "Exact untouched-last-target block") + clean, _ = controller(events, identity, identity) + need(clean == [1 << i for i in range(888)], "Identity-code positive control") + wrong, _ = controller(events, code, code) + need(wrong != rows, "Wrong inverse-code mutation changes the caller") + rational = rational_fresh_witness(geometry, code, charts) + mutant = code[:] + mutant[3] ^= 1 + try: + rational_fresh_witness(geometry, mutant, charts) + except ValueError: + pass + else: + raise ValueError("Deleted actual code edge was not rejected") + output_support = supports(geometry, code) + power = identity + order = None + for i in range(1, 29): + power = product(code, power) + if power == identity: + order = i + break + need(order == 28, "Actual source-code order") + output = { + "status": "PASS_R429_NINE_STAGE_ENDPOINT_DEFECT_AND_BOUNDED_REPAIR_FLOOR", + "no_new_kappa": True, + "source_pins": PINS, + "A": code, "A_inverse": code_inverse, + "source_code_order": order, + "rank_A_minus_I": rank2([a ^ b for a, b in zip(code, identity)]), + "rank_A_squared_minus_I": rank2( + [a ^ b for a, b in zip(product(code, code), identity)]), + "identity_source_rows": [p for p, row in enumerate(code) if row == 1 << p], + "caller_laurent_exponents": [[sorted(p) for p in row] for row in scalar], + "caller_scalar_defect_rank": rank2([row ^ (1 << p) for p, row in enumerate(rows)]), + "caller_endpoint_sha256": sha256(canonical(rows)).hexdigest(), + "chronological_scalar_endpoints": snapshots, + "last_pair_residual_rank": rank2(residual), + "actual_native_last_pair_witness": rational, + "local_terminal_basis_floor": output_support, + "controls": ["identity code restores all 888 public scalar inputs", + "using A as its inverse changes the full caller", + "deleting actual A[3,0] invalidates the native impulse"], + "limits": [ + "No all-parameter or formal-kernel verification is claimed.", + "Native last-pair derivation retains the actual fresh-block route contract.", + "No legal cheap routed decoder, whole-owner schedule, precision or outer transfer.", + "The 63-unit floor covers only the stated minimum-basis common-cap model.", + ], + } + receipt = json.dumps(output, indent=2, sort_keys=True) + "\n" + if args.output is not None: + args.output.write_text(receipt) + else: + need((HERE / "RESULT.json").read_text() == receipt, + "Saved composition receipt differs from exact recomputation") + print(json.dumps({k: output[k] for k in + ("status", "caller_scalar_defect_rank", "rank_A_minus_I", + "rank_A_squared_minus_I", "no_new_kappa")}, sort_keys=True)) + + +if __name__ == "__main__": + main() diff --git a/research/bit-copy-path/composition/inputs/C511-PROOF-INTERFACE.md b/research/bit-copy-path/composition/inputs/C511-PROOF-INTERFACE.md new file mode 100644 index 0000000000..7e3bbc02b3 --- /dev/null +++ b/research/bit-copy-path/composition/inputs/C511-PROOF-INTERFACE.md @@ -0,0 +1,242 @@ +# The fixed C511 interface and its five-call invoice + +The scientific pin is `20ecf2891d80b5acf18d980b2164fb44ea7faa90`. +The fixed local word is `BIT-THREE-COPY-INPUT-WORD.json.gz`, with gzip SHA256 +`ac8a6dcd5957f49eefd6fc945332b1c31afa26c48fc301d37bfb1338beea6847`. +It uses111 source registers,111 target registers and438 ordinary helpers: +660 arbitrary independent ordinary inputs. One additional private register +is required initially zero and is erased after each of three copied totals. +The private register is not a661st arbitrary input. Its zero row is included +in every endpoint and roundtrip check. + +The scalar code C has binary rank111, defect rank `rank(C-I)=9` and exact +order511. Its literal signed source code has1487 ADDs. This is the older +C511 word; it is not the later R429 order28 code with1499 ADDs. + +## Exact local scalar contract + +In logical frame coordinates the binary VALUE endpoint is + + (x,y,h,0) -> (C x, y+C x, h,0). + +The literal inverse has binary endpoint + + (x,y,h,0) -> (C^-1 x, y+x, h,0). + +The independent checker carries all660 arbitrary ordinary input columns +as dense integer rows, plus the zero private row. It verifies both endpoints, +every old-target identity coefficient, every helper row, all private +COPY/ERASE cuts and the full signed literal-inverse roundtrip. It also +reconstructs the1487 signed source ADDs separately and compares their +entire integer matrix with the source endpoint. + +Over Z the precise endpoint has the form + + (x,y,h,0) -> (C_Z x, y+A_Z x+E_Z h, h,0), + +where `C_Z` is the literal unimodular signed code, `A_Z=C mod2` and +`E_Z=0 mod2`. The integer target block is not inferred to equal `C_Z`. +Reversing all ADDs with opposite signs and reversing the typed COPY/ERASE +word restores every ordinary integer input. Forward and raw-inverse signed +prefix row-L1 maxima are64400 and1405988; these are finite scalar bounds, +not rounded precision or native-child error bounds. + +At each COPY the source is at actual FULL and the private field is at its +actual rank-seven cap, costing rank1. The copied source remains unchanged +until ERASE, where the private and source rows are compared over Z. All +three episodes reuse the same zero private field. An omitted ERASE is +rejected when the word next attempts to modify the still-live copied source. +Changing one source-code sign preserves the full binary endpoint but changes +the integer source matrix, providing a separate signed-control witness. + +The complete coded local histogram is + + H1=1325, H2=1342, H3=79, H4=44, H5=102, H6=11, H7=9. + +It has native rank mass5061 and13283 scalar ADDs. These counts include all +three private rank-one COPY queries. The same positive rank invoice applies +to the literal inverse, whose frame path runs in the opposite direction. + +The code order is checked independently by `C^511=I`, `C^73!=I` and +`C^7!=I`. Since511=7*73, these exclude every proper divisor of511. The +five-call proof below does not use finite order511; it uses only invertibility +of C and the actual intertwiners. + +## Physical local blocks + +Let F be the local full rank-eight head/tail exchange, N the diagonal +rank-one source-line exchange and `G=F N` the rank-seven cap exchange. +F and N are commuting involutions. The source code acts separately on HEAD +and TAIL and therefore commutes with F. It does not commute with N. + +Converting the verified logical endpoint to the declared physical frames +gives, over binary VALUE operators, + + B_C = [[F C N, 0], [G C N, G]]. + +Its inverse is + + B_C^-1 = [[G C^-1,0], [F,G]]. + +For example, the lower-left inverse entry is +`G*(G C N)*(N C^-1 F) = F`; the signs disappear in characteristic two. +Both complete block products are the identity using only +`F^2=N^2=I`, `FN=NF`, `FC=CF` and `C C^-1=C^-1 C=I`. +No step commutes C with N. + +These operator identities are statements about binary VALUE arrays. They +are distinct from the signed integer lift above. Ordinary helpers are +restored in logical frame coordinates. Forward helper frames run ZERO to +FULL; inverse helper frames run FULL to ZERO. For an arbitrary incoming +physical helper, interpreting its logical symbol in the declared entrance +frame gives the rank-eight physical residual in either direction. This +interpretation does not execute a free entrance query, nor does it establish +a chronological caller owner schedule matching those boundaries. + +## The actual routed charts + +The route matrices use the deterministic upstream Weyl-BFS tree on the +actual120 full E8 root lines. The independent checker selects that tree +from the root adjacency graph, reconstructs each reflection path over the +dyadic rationals and verifies all111 resulting rho hashes against the +frozen receipt. No producer geometry function is imported. + +For a port root q of norm four, put `P=q q^T/4` and `K=I-P`. +The Weyl route supplies an8-by7 matrix A with `A A^T=K` and `A^T A=I7`. +On the common core8 and one fresh7 block the chart is + + rho = [[P,A],[A^T,0]]. + +It is symmetric and involutive. With `G0=diag(K,0)`, `E=diag(0,I7)` and +`F_s=diag(P,I7)`, the exact projector identities are + + rho G0 rho = E, rho F_s rho = diag(I8,0). + +The chart also preserves the source-line projector. The checker evaluates +all these equalities at every port. A code coefficient between ports q and p +is routed by `rho_p rho_q`, acting separately on HEAD and TAIL. Consequently +each of the6465 nonzero binary C coefficients intertwines the corresponding +old-cap operators and local full operators. Thus C_s commutes with G0 and +its local F_s. Every other fresh G_t is disjoint from this chart's active +coordinates and commutes with C_s. A common full parent exchange commutes +with every HEAD/TAIL-separated route and with its literal inverse. + +The noncommutation with common N is concrete. For actual code edge(0,1), +the `(0,8)` entry of `P_0 rho_0 rho_1` is1, whereas the same entry of +`rho_0 rho_1 P_1` is0. This is an exact dyadic witness, not a formal naming +distinction. It remains nonzero in every odd ADDRESS field. The projector +equations likewise descend to odd ADDRESS fields because all denominators +are powers of two. The old local native MOVE/COPY implementations and +their guards remain inherited premises. + +## Five calls and a paid terminal decoder + +Normalize each local call by factoring out its fresh cap exchange G. +The two CLEAN orientations and the coded forward call become + + F_clean = [[I,0],[N,I]], + B_clean = [[I,N],[0,I]], + Q_C = [[N C N,0],[C N,I]]. + +The universal identity uses two arbitrary banks. Execute CLEAN forward, +CLEAN backward, CLEAN forward, and then an ordinary bank swap. Since N is +an involution, the normalized output is `diag(N,N)`. Next execute the coded +forward call and one CLEAN forward call. Acting on that diagonal gives + + F_clean Q_C diag(N,N) = diag(N C,N). + +One direct check starts with `(N x,N y)`. The coded call gives +`(N C x, C x+N y)`; the final CLEAN call adds `N(N C x)=C x` to its target, +leaving `(N C x,N y)`. This proves the identity for every invertible C; +no commutation of N and C is used. + +Let the five cap exchanges be `G0,G1,G2,G3,G4`, with C coded at stage3. +The actual chart intertwiners allow all other fresh exchanges and G0 to +pass C3. Their product is Gamma. The parent has width +`8+4*7=36`, and `Gamma N=T_parent`. Therefore the physical five-call word +is + + T_parent diag(C3,I). + +Because the full parent exchange commutes with the actual routed code, +the literal1487-ADD inverse on bank0 removes C3 at the final full parent +cut. The bank swap costs333 scalar ADDs. The word is restored on its two +arbitrary public banks in the binary VALUE/native sense, with all helper +occurrences retaining their declared logical return and physical residual +contracts. + +The independent operator oracle uses normal forms in the group ring over +F2 of `C2(F) x (C2(N) * Z(C))`. It checks both local physical inverse +products and executes the five normalized bank updates on both input +columns. This is a symbolic all-operator identity, not a sample of small +numeric matrices. The actual source/geometry binding remains finite and +separate. A control replacing the first of the three seed calls by the +coded call gives `[[0,N],[C N,0]]`, exposing the wrong-side `C N` rather +than `N C`. The actual noncommutation witness makes this distinction physical. + +## Complete failed cost invoice + +One CLEAN local call has24248 ADDs and histogram + + H1=1322, H2=1342, H3=79, H4=44, H5=102, H6=11, H7=18, + +including nine rank-seven private copies. The independent check reconstructs +that literal CLEAN word from the frozen R438 phases and verifies its +all-column endpoint and complete typed invoice. + +Four CLEAN calls plus one coded call give + + H1=6613, H2=6710, H3=395, H4=220, H5=510, H6=55, H7=81. + +The native rank mass is25545. Public progress is `2*111*36=7992`; +2190 independent ordinary helper occurrences contribute `2190*8=17520`. +The one reused private field incurs total COPY rank `4*63+3=255`. +The uncovered deficit is + + 7992 + 17520 - 25545 = -33. + +The complete scalar ADD count, including terminal decoding and the swap, is + + 4*24248 + 13283 + 1487 + 333 = 112095. + +The local five-call construction therefore fails already at zero rate. +This invoice does not become favorable by omitting the COPYs, the final +decoder or the target/source frame progress. Further owner padding and +physical costs are additional obligations, not improvements to this fixed +invoice. No positive-rate moment or exponent is supplied. + +## Source-consuming inverse + +Precode the inverse's source by C and then decode its target by C^-1. +For arbitrary source x and old target y, the exact binary physical map is + + (x,y) -> (G x, F x + C^-1 G y). + +Here the inverse first gives `(G x,F C x+G y)`; the target decoder commutes +with F but not necessarily G. Consequently the desired full output `F x` +requires all111 caller-supplied target fields to be zero. The source is +consumed into Gx. These two linked outputs come from the same111 inputs +and do not supply two independent families of public query credit. + +Crediting only the FULL output and the ordinary helper progress gives + + 111*8 + 438*8 - 5061 = -669, + +before charging the111 zero-output fields. The two1487-ADD codes and the +13283-ADD inverse total16257 scalar ADDs. Calling these codes free means +they add no native recursive query rank; it does not mean zero arithmetic, +source access or routing cost. The internal private field is separately +required zero and erased. No restored-source or restored-zero-output +interface follows from this construction. + +## Covered domain and unresolved implications + +The result comprises the exact finite C511 scalar word and lifecycle, +source-bound dyadic routes, a universal noncommutative binary composition, +and complete negative local invoices. It covers arbitrary ordinary inputs +with the stated zero-private/zero-output premises. It does not turn logical +helper restoration into physical identity restoration. No chronological +whole-owner allocation, rounded child-defect theorem, all-size source or +tape bound, formal-kernel proof or improved multiplication exponent is +claimed. The separate historical million-pair searches, rolling words and +quotient obstructions are not premises of this bounded interface result. diff --git a/research/bit-copy-path/composition/inputs/C511-PROOF-OBSERVATION.md b/research/bit-copy-path/composition/inputs/C511-PROOF-OBSERVATION.md new file mode 100644 index 0000000000..0829522b30 --- /dev/null +++ b/research/bit-copy-path/composition/inputs/C511-PROOF-OBSERVATION.md @@ -0,0 +1,229 @@ +# Actual rolling residual and a common scalar observation floor + +Scientific pin: `20ecf2891d80b5acf18d980b2164fb44ea7faa90`. +This note concerns one source-bound raw three-call word and a precisely +specified scalar observation contract. It proves an exact counterexample and +a111-reader information lower bound. It proves no native tariff, unrestricted +routed-reader bound or new multiplication exponent. + +## Actual source and coefficient fields + +The source is the THREE-COPY C511 word, gzip SHA256 +`ac8a6dcd5957f49eefd6fc945332b1c31afa26c48fc301d37bfb1338beea6847`. +Its1487 source-only ADDs determine a binary111-by-111 matrix C with +`rank(C)=111`, `rank(C−I)=9`, and exact order511. This is distinct from the +later pruned order28 code. The independent checker executes each source input +column separately, reconstructs all entries, and checks C^511=I while C^7 and +C^73 are nonidentity. Since511=7·73, these checks establish its exact order. +The finite-order relation is not used in the three-call expansion below. + +VALUE coefficients are over F₂. The counterexample's ADDRESS coordinates are +over the odd field F101. These fields play different roles: VALUE impulses +cancel modulo two, while ADDRESS routes use dyadic matrices reduced modulo101. + +The111 retained root lines and their port order come from the pinned R438 word +(`a353999bceac4ac957f540adfd3e5f8768297138b27132eeb83332e2ea4b1022`). +The complete120-line geometry is pinned by +`1fb726ea3818fe0837cceb0dd22a71a0814f064e5e781628104610d6468e0e55`. +The actual upstream breadth-first Weyl routes determine an eight-dimensional +orthogonal matrix g_p for each retained port. With root q_p, define + +`P_p = q_p q_p^T /4`, `A_p = g_p[:,1:8]`, +`rho_p = [[P_p,A_p],[A_p^T,0]]` on15 coordinates. + +These rho_p are the actual source charts; they are not newly chosen gauges. +The independent checker reconstructs the breadth-first tree with a FIFO queue +and applies each Householder reflection as an exact rank-one update. It matches +all111 saved rational rho hashes and verifies rho_p²=I. This retains source +signs and routes before reducing to F101. + +## The raw three-call residual + +Let N be the common rank-one ADDRESS exchange, N²=I. After the declared +source precode, the normalized source-consuming inverse call at stage i has +the two-bank matrix + +`S_i = [[I,0],[N C_i,I]]`. + +Both public banks are arbitrary. No initially zero target or restored-source +condition is imposed. The actual local native identity and fresh-prefix +commutations justifying this normalization belong to the source interface +proof; the following computation then occurs in the noncommutative operator +ring over F₂. + +Use S_0 forward, its bank-swapped stage1 counterpart backward, and S_2 forward. +Writing `a_i = N C_i N`, direct multiplication gives + +`R3 = [[I+a_1 C_0, N C_1], + [N(C_2+C_2 a_1 C_0+C_0), I+a_2 C_1]]`. + +The independent polynomial checker reduces only adjacent N N to I. It matches +every saved coefficient of this matrix without assuming the C_i commute or +using C_i^511=I. After factoring off the required parent FULL exchange, its +bank0 diagonal block is + +`Phi = N(I+a_1 C_0) = N+C_1 N C_0`. + +Thus the scalar rank of C−I does not identify the rank or observation needs of +this routed operator. The bank0 residual contains C_1 N C_0, with N between +codes belonging to different native stages. + +### Exact neighboring cancellation is excluded under the stated freshness + +Suppose one tries to cancel `a_(i+1) C_i` exactly. This requires +`C_(i+1)=N C_i^(-1) N`. Let G be the next fresh exchange. The actual native +family supplies `[C_i,G]=0`, `[N,G]=0`, and +`[C_(i+1),F]=0` for `F=NG`; G is invertible. Substitution into the last +commutation relation gives + +`N C_i^(-1) G = G C_i^(-1) N = C_i^(-1) N G`. + +Cancel G to obtain `[C_i^(-1),N]=0`, equivalently `[C_i,N]=0`. This contradicts +the actual source coefficient. Independently reconstructed routed edge(0,1) +has `C[0,1]=1`; the two projector-route products differ at entry(0,8), with +values1 and0. The same local code family at fresh stages retains this +noncommutation. At the initial scalar stage, distinct root projectors already +give a nonzero commutator on this off-diagonal edge. + +This excludes that exact cancellation mechanism with those freshness +relations. It excludes neither arbitrary intermediate gates nor other coupled +bank words. The earlier millions-of-pairs bounded searches are not premises +of this proof and are not independently replayed here. + +## Actual ADDRESS impulse + +Use15 ADDRESS coordinates: the common eight plus the seven fresh coordinates +used by the routed stage1 code. Pad the remaining seven fresh coordinates by +zero to embed this example in the22-coordinate three-call parent. + +At a port p, the partial exchange N acts on HEAD/TAIL pair(h,t) by + +`(h,t) -> ((I−P_p)h+P_p t, P_p h+(I−P_p)t)`. + +C_0 is the literal scalar source code. The stage1 coefficient from middle port +k to destination p is `C[p,k]` times the head/tail-separate push by +`rho_p rho_k`. For an entering HEAD vector v and TAIL0, put + +`h_k = (I−P_k)v`, `t_k = P_k v`. + +The contribution of C_1 N C_0 along a source q, middle k, destination p path +has VALUE coefficient `C[p,k] C[k,q]` and ADDRESS + +`(rho_p rho_k h_k, rho_p rho_k t_k)`. + +Add the direct N contribution at destination q and cancel equal output +impulses modulo two. For q=0 and v=e_0, the result has exactly3131 nonzero +output impulses. Exactly one has nonzero TAIL: + +`output port0, HEAD=0, TAIL=e_0, VALUE coefficient1`. + +The complete ordered output digest is +`ae0d4019d97be344e97730f586c84152eaea4fb16e83f541edadf0e22dc76eb0`. +The independent reconstruction obtains every output, rather than checking +only this witness or its total count. + +Let U be the subspace of VALUE arrays supported on TAIL0, across all ports +and banks. The admitted free alphabet uses scalar port/bank mixing and +head/tail-separate linear ADDRESS routes, so it preserves U. Its invertible +elements and their inverses preserve U, hence induce invertible maps on the +quotient by U. The displayed nonzero quotient output cannot be erased by +such an invertible terminal free decoder. This remains true when the decoder +mixes both banks: a nonzero class cannot become zero under an invertible +quotient map. This is a counterexample to the actual raw word, not a claim +about every ADDRESS operator or caller. + +## Complete common scalar readout bound + +For each j=0,...,7, give every source port q the same input ADDRESS impulse +`HEAD=e_j, TAIL0`, with an arbitrary binary VALUE coefficient x_q. For each +output port and ADDRESS with nonzero TAIL, the corresponding VALUE output is +one scalar linear form `w·x`, with `w ∈ F₂^111`. + +The independent checker constructs these coefficient rows by following each +individual source column through the actual C_0, N and routed C_1. Equal +ADDRESS paths cancel modulo two. It then sorts the rows in the saved canonical +order. The complete census is: + +| HEAD sample | Nonzero projected coefficient rows | Accumulated binary rank | +|---|---:|---:| +|e_0|3268|63| +|e_1|3051|90| +|e_2|3054|105| +|e_3|3127|105| +|e_4|3127|111| +|e_5|3125|111| +|e_6|3128|111| +|e_7|3018|111| + +There are24898 rows in total; every sample uses all111 source columns. The +complete row digest is +`607fd1d55202d616c7d50a913511c205ebe83aeb1dfec49be000beb598c725f3`. +All111 saved original witness rows are looked up by their exact sample, +destination and ADDRESS, compared entrywise, and independently shown to have +rank111 with low-bit Gaussian pivots. This independent rank engine differs +from the producer's high-bit echelon construction. + +**Theorem.** Suppose the residual admits a factorization + +`Phi = H_free + sum_(i=1)^t A_i (c_i^T tensor I_ADDRESS)`, + +where H_free preserves U, each A_i is an F₂-linear response map, and each +`c_i ∈ F₂^111` is one fixed scalar source +row, independent of ADDRESS. Each reader combines whole source arrays at +matching addresses; the same c_i must work for all inputs. The response maps +A_i may depend on ADDRESS and route their already collected reader arrays. +Then `t ≥ 111`. + +**Proof.** Restrict the input to sample j. Reader i returns +`(c_i·x) delta_(e_j,0)`. Apply A_i and project one tail-nonzero output impulse. +Its scalar output coefficient is a multiple of `c_i·x`. H_free contributes +zero to that projection because the input lies in U. Therefore every observed +coefficient row w is in `span{c_1,...,c_t}`. The complete observed row space has +dimension111, so t is at least111. This is a finite witness to an exact +necessary statement for all factorizations in the declared model. ∎ + +The readers counted here are whole-array observations with scalar port +coefficients, not individual bits and not native child queries. The theorem +does not price their acquisition, storage, ADDRESS access or restoration. In +particular, it gives no native rank111 tariff. Routed/ADDRESS-dependent reader +rows, direct access to public inputs, extra native source queries, other cuts +or a changed word alter the factorization model and remain outside this bound. + +## Controls and verification boundary + +Setting both codes to identity makes `Phi=N+N=0` identically. The independent +checker verifies cancellation on all111·8 sampled impulses using the actual +rho_p rho_p adapters. This is a positive cancellation control, not merely an +expected rank string. + +Two actual changes affect the reconstructed map. Removing the existing bit +`C_0[0,1]` while retaining C_1 and all ADDRESS routes changes a projected +coefficient row at output port3; the row's source1 bit flips and the full first +sample digest changes. Omitting the direct N term flips the source0 coefficient +of the unique tail-nonzero boundary impulse from1 to0. Neither changed word is +claimed to be an admitted construction. + +The original failed-candidate invoice had rank mass15183 and a hypothetical +deficit213 if both banks reached the required parent endpoint. The actual +counterexample prevents that coverage from being credited. The number213 is +only a failed-candidate diagnostic. It is not a positive supplier moment. + +Normal and optimized execution of the independent checker compare identical +receipts. This review does not replay the large bounded cut searches, and it +does not substitute for the root's primary producer or portable-package checks. +The five-call restored interface is a separate construction with its own +native proof and invoice. Chronological whole-helper owners, arbitrary dirty +restoration at new caller cuts, rounded errors, all-size recursion, physical +tape/access cost and an outer multiplication theorem remain separate. + +## Credit + +The source word and geometry retain their E8/R438 ancestry and source notices, +including Sagar Thakrar's transported R438 word and the predecessor THREE-COPY +source-code construction. The local mutable-source lane supplied the C511 +interface, actual chart receipt, raw residual and scalar quotient certificate. +This review independently reconstructs the source code, Weyl routes, impulse +paths, coefficient rows and noncommutative expansion. It claims no new geometry +or generic observation theorem. Prepared with substantial OpenAI Codex +assistance; Apache-2.0. Exact finite evidence and the written argument are not +formal-kernel evidence or independent human review. diff --git a/research/bit-copy-path/composition/inputs/CODED-EXPECTED.json b/research/bit-copy-path/composition/inputs/CODED-EXPECTED.json new file mode 100644 index 0000000000..1ad8df8143 --- /dev/null +++ b/research/bit-copy-path/composition/inputs/CODED-EXPECTED.json @@ -0,0 +1,54 @@ +{ + "source_word": "research/bit-r438-transport-rehost/words/r438-t154.json.gz", + "source_sha256": "a353999bceac4ac957f540adfd3e5f8768297138b27132eeb83332e2ea4b1022", + "canonical_word_sha256": "ebc33ee086f610c1fd540fa56bb99b2b96a768e4aed4a18defcb0e1f507fca89", + "frozen_gzip_sha256": "1a6841fccba51fa922993c77b4b296e659c2c538252e5e06113d0438c5edb870", + "semantic_word_sha256": "e5fa1dbcb80f61c7c728bad7cc166f368ec73c673828273bea091caba8708892", + "six_copy_semantic_sha256": "f44b26376769580295293b1e782568d81ed297647106450a13bd82ecf086dd1b", + "unsigned_section_semantic_sha256": "a50fcbbec51830f84f7368bc6153fe3340ef345849a6cf9ea287d624a753cea4", + "original_six_copy_canonical_sha256": "6448fb2b201c6be722087fc547064ec7c3135af5a72ed363434822212a9b2afc", + "normalized_fields": [ + "source_word", + "signed_from", + "parent_word_gzip_sha256" + ], + "normalization_contract": "The final word omits only checked source location and validated same-run parent compressed identity in semantic comparison. Signed ancestry also normalizes its validated same-run parent hash. All schemas, frames, signed records, deleted physical indices, register counts and original source hashes are compared completely. Immutable frozen inputs retain exact byte hashes.", + "profile": { + "1": 1298, + "2": 1342, + "3": 79, + "4": 44, + "5": 93, + "6": 11, + "7": 9 + }, + "counts": { + "0": 2873, + "1": 13223, + "2": 3, + "3": 3 + }, + "rank_mass": 4989, + "rank_saving": 132, + "source_matrix_sha256": "0ab35dade762a3050b6fc15f653041134f8d5a8015cbdbdc049be0528659240b", + "source_code_ADDs": 1499, + "prefix_row_l1": { + "forward": 34285, + "inverse": 5839860 + }, + "support_semantic_sha256": "c281439e7e78c74c8223e5adfebf34c2be945f308b0f7e53b928c700151cbd74", + "support_canonical_sha256": "3541eab8057c8cebadbd46775ff2d8e759fd592d8a1b9aebde834821d96c051b", + "public_registers": 651, + "dirty_helpers": 429, + "deleted_original_families": [ + 471, + 472, + 473, + 474, + 475, + 476, + 477, + 478, + 479 + ] +} diff --git a/research/bit-copy-path/composition/inputs/CODED-PROOF.md b/research/bit-copy-path/composition/inputs/CODED-PROOF.md new file mode 100644 index 0000000000..1fbd2cad0e --- /dev/null +++ b/research/bit-copy-path/composition/inputs/CODED-PROOF.md @@ -0,0 +1,202 @@ +# Three copied source pivots for the actual R438 local word + +This is a finite, source-pinned local construction over F2, an exact signed +integer lift, and a conditional native cap calculation. It is neither a formal +kernel proof nor a complete recursive supplier or multiplication theorem. + +## Registers, frames and endpoint + +The final word has 111 sources x, 111 arbitrary targets y, 429 arbitrary dirty +helpers h and one owned private copied field. All 651 public input columns are +independent. Its construction first uses the 438-helper, 660-column source +word and then deletes nine literally unread physical families as proved below. The private field is zero outside its immutable snapshot +lifetimes. Sources enter at their actual port lines P_p, targets and helpers at +frame zero. Sources and helpers finish at FULL; target p finishes at its actual +cap K_p=P_p-perp. These are the original R438 boundary frames. + +For the explicit invertible binary matrix A constructed below, the new forward +endpoint is `(x,y,h) -> (Ax,y+Ax,h)`. Its literal inverse is +`(x,y,h) -> (A^-1 x,y+x,h)`. The source endpoint is changed. Original theorems +requiring an unchanged source cannot be applied without proving the new caller +contract. Complemented physical backward-call interpretation and the complete +compiler remain separate obligations. + +## Delete centres and rebuild the zero-frame compensation + +Replay D then A of the actual pinned word. The nine retained helper totals are +`R x+T h`, with R of shape 9-by-111. Let K be the actual 111-by-9 scatter matrix. +No original gate reads a target. The complete row masks of M=RK are +`507,135,238,349,185,437,205,503,425`. Both M and R have rank 9, while +N=I+M has rank 6. + +Delete the nine centre COPY/ERASE episodes and their target scatters. Over F2, +this changes the original target `y+x` by `KR x+KT h`. Replace its original +zero-frame compensation by `D_new=D_old+KT`. Duplicate helper-to-target gates +cancel at frame zero: 5,758 ADDs replace 18,030. Every original public frame +visit is retained, including empty-source visits. The resulting clean-centre +word has endpoint `(x,y,h) -> (x,y+Jx,h)`, where J=I+KR and rank(J)=108. +The verifier derives R,K,J from the literal original word and checks every +public input column; it does not assume a model scatter matrix. + +## An invertible source code with three actual repaired target rows + +Let C0 be a basis of the left nullspace of N, with emitted row masks +`39,193,272`. Let P select physical target coordinates 0,1,2 and S=RP. +The 3-by-3 matrix C0 S is invertible. Define + +``` +B0 = (C0 S)^-1 C0. +``` + +Thus B0 S=I and B0 N=0. Let D be rows 0,1,2 of K. Delete these rows +from K to form W=K+PD, and define + +``` +Y = N + S D, A = I + W R, Q = P B0 R. +``` + +The actual row masks of Y are `10,373,282,165,176,405,356,145,166`, of rank 9. +A fixes kernel(R) and induces the invertible map Y on the quotient identified +by R: RA=YR. Thus A is invertible. Equivalently, if Ax=0 then YRx=0, so +Rx=0 and Ax=x=0. Both matrix inverse identities are also checked finitely. +Rows 0,1,2 of A are exactly their identity rows. Since B0Y=D, + +``` +QA = P B0 R A = P B0 Y R = P D R, +(I+Q)A = I + (K+PD)R + PDR = I+KR = J. +``` + +After the full-frame source code, z=Ax and the target is y+Jx. Adding Qz +therefore gives y+Ax over F2. Q has exactly three nonzero physical rows, +0,1,2. The actual source-code rank is rank(A-I)=9; small target support does +not imply small source-code rank. + +Three repaired target rows are minimal in this declared class. For any +invertible A, rank(A+J) >= rank(A)-rank(J)=3. A correction supported on r +physical target rows has rank at most r. This excludes r<3 for this fixed +clean-centre J and invertible source recoding, not arbitrary suppliers. + +## Literal source code and one snapshot per repaired row + +Normalize R on actual source coordinates `0,1,2,4,8,9,14,30,59`. Commuting +CNOTs replace those coordinates by a query basis. In this basis A has an +invertible 9-by-9 query block and fanout to the other source coordinates. +Gaussian elimination synthesizes the small block with 47 CNOTs. The ordered +code uses `474 prepare + 504 fanout + 47 quotient + 474 inverse prepare`, +or 1,499 ADDs, all at FULL. It uses no extra helper or private field. + +For each nonzero row q=Q[p], choose a source index t with q[t]=1. All source +fields are at FULL. Add each z_i with i!=t and q[i]=1 to z_t. The prepared +field is exactly `sum_i q[i] z_i` over the integers: its original z_t is one +of the required terms, not an unknown extra offset. Copy this actual source +into the cleared private field at target p's cap, add it to target p, and +ERASE the copy while the physical source is unchanged. Undo the preparation +in reverse order with negated ADD coefficients. Every source is restored +exactly to its value before this episode. + +The chosen pivots are 0,1,2. Their preparations use 59,53,60 ADDs; their inverse +preparations use the same counts. Including three scatters gives 347 repair +ADDs. Only one private snapshot is live at a time. No dirty helper is touched +by these three repair episodes. The six-copy dirty-helper variant is retained +only as explicit reconstruction ancestry; it is not the final construction. + +## Integer and native boundaries + +The signed word negates the actual factory cleanup, source unpreparation and +repair unpreparation ADDs. Literal reversal negates every ADD coefficient. +The complete 651-column integer replay establishes source `A_Z x`, every +helper exactly h and the private field zero. The target is +`y+T_Z x+E_Z h`, where `T_Z mod2=A` and E_Z is even. Target correctness is +asserted modulo 2, not as integer equality to y+A_Z x. The reverse source is +`A_Z^-1 x`. Full integer roundtrip is checked separately; A is not assumed +involutive. Wrong unpreparation signs preserve parity but change the integer +source, and are rejected independently. + +For each actual rational E8 line vector u with squared norm 4, the checker +constructs P=uu^t/4 and K=I-P. It binds the actual entrance frame to P and +target frame to K, including all 222 bindings. For +`D_U=[[I-U,U],[U,I-U]]`, exact rational multiplication proves +`D_I D_P=D_K` and `D_P^2=I` on all 16 columns, for all 111 ports (3,552 checked +columns). Hence the FULL-to-cap copied-total residual is D_P of rank one, +under the inherited native ADDRESS/copied-total interface. Source and target +port labels need not agree: the physical source is FULL at the cut. + +The independent ordered-word audit checks common frames for every ADD, +monotone containment and rank for every MOVE, actual source COPY, immutable +source throughout each snapshot, exact ERASE and endpoint frames, in both +directions. These local identities do not establish a full physical native +compiler, rounded implementation or recursive hosting theorem. + +## Delete nine literally unread physical families + +In the signed three-copy word, precisely the old helper indices 471 through +479 are never read as ADD sources or COPY/ERASE sources. The independent +checker discovers this set by scanning every record. The only operations on +these fields are 72 ADD destinations and 36 MOVEs, with paid profile +`27 e1+9 e5`. The same no-read property holds in the literal inverse. + +Delete these nine physical input/output families, discard their destination +operations, and renumber every remaining operand in order, including the +private field from 660 to 651. No remaining value or frame depends on a +removed family. Induction through the literal word therefore identifies the +new execution with projection of the original execution, forward and inverse. +The constructor also checks exact integer endpoint projection, and the +independent checker compares the entire projected record list and both +boundary arrays before replaying all 651 independent columns. Every one of +the remaining 429 arbitrary helpers is restored exactly. The deleted families +are absent from the contract; they are not retained with unpaid endpoints. +A mutant that reads a discarded field is rejected before projection. + +This removes another 72 local rank units and 72 ADDs. It does not establish a +72-unit hosted saving: under the incumbent width-8 whole-family accounting, +returning nine families to idle-owner children adds `9*8=72` rank units per +local call at beta=0 (648 across nine calls). Thus that returned idle obligation +cancels the pruning reduction at this mass level. Changed chronological owners, +stock and complete higher moments remain outside this package. The local +132-unit reduction below must not be read as a net global gain. + +## Complete local invoice + +| Item | Pinned R438 | This word | +|---|---:|---:| +| Sources / targets / dirty helpers | 111 / 111 / 438 | 111 / 111 / 429 | +| Owned private fields / maximum live copies | 1 / 1 | 1 / 1 | +| ADDs | 24,248 | 13,223 | +| MOVE children | 2,909 | 2,873 | +| COPY / ERASE pairs | 9 | 3 | +| COPY ranks | 9 times 7 | 3 times 1 | +| Paid children | 2,918 | 2,876 | +| Rank mass | 5,121 | 4,989 | + +The complete profile is `{1:1298,2:1342,3:79,4:44,5:93,6:11,7:9}`. +Its exact change is `-9 e7-24 e1-9 e5`, saving 132 rank units per local invocation. +The ADD invoice is `5758 compensation + 5619 clean factory/cleanup + 1499 +source code + 347 repair`. No chronological owners, native allocation, +private reservation, code restoration or final decoder cost is removed from +a global bill by this local calculation. + +The exact signed prefix row norms are 34,285 forward and 5,839,860 inverse. +They are checked local arithmetic facts only. A new global perturbation/rounding +proof and precision reserve are required before using them in a supplier. + +## Verification and remaining conditions + +`reconstruct.py` regenerates the complete ordered word from the two pinned live +sources through the declared section and signed ancestry. The registered +verifier compares the full decompressed payload with `WORD.json.gz`. Only +validated source-location metadata and validated same-run compressed-parent +identities in signed/pruned ancestry are normalized. Every record, sign, frame, schema and register count +is compared. Immutable source and frozen word files retain exact byte hashes. + +`check.py` imports no producer. Its separate sparse integer interpreter checks +all 651 columns in both directions and the roundtrip, actual R/K/J, source +inverse, all dirty restoration, all target parity, complete profile and exact +rational native caps. Eleven ancestry fault checks and seven independent controls +cover missing compensation/unpreparation, mutable snapshots, wrong inverse +code, wrong integer signs, FULL-copy rank, a corrupt section query and a read of a deleted family. + +Open global obligations include composition of the changed source interface, +terminal code removal at the actual caps, physical helper/owner/access and +storage schedules, rounded cleanup and precision, recursive caller contracts, +and a full joint complex/BIT outer transfer. This local result supplies none +of those admissions and claims no new kappa. diff --git a/research/bit-copy-path/composition/inputs/CODED-WORD.json.gz b/research/bit-copy-path/composition/inputs/CODED-WORD.json.gz new file mode 100644 index 0000000000000000000000000000000000000000..672c41534608d0d812b8020e496177d7c41c8527 GIT binary patch literal 59514 zcmZ6y2|QHM8$WC}wrLo+WhqJ4Y!y)m@6qq~`~Tnft@*h3KKGn+&pqck&v~Bj^SuXBAS`?T*mf76j`c^k zqa zSkB6;&ElAiZS&ook18>{+jYCE=DSZ~{_fn_{d00>;lS067sWAK565>lJ{Ir(`Mvvd zd}j%?@lz#c^Y^ozk4`aL-3NA7W4_s3{bo}av$Q!>7ZbSsZlTy=w`=Y4&XCj2kB{GD zHoGq;Ni6LaNgQ}Mp%Sxw-8Z{NISa*(yNjF0YGYQ#yo(pc*B<__nOytU zsnJ72$}OS2Yo}|S0=Kgka_yf@Z5|p@X}S6zQDCpb+L6HC-a~S^h4z8}=Om){+o3zT zg^q#$pA*mAPw$~qKHP!L>@B0Q_DlHMP^7JP#$D-Thrr1!soYPGrn(LtQ*H_9O*?(b z{y(B8Q!|H-skDUmzCL})F>p2OR_-St`g`hR@q$y!&ie(Y)}6J5V*A}+o1R0@0=8SH z7>8_?TLOD?PS@H8PG=p?wQu>4D5CfM=~~CYwXD0jpVm(N+T0%zz3*vTlH)kB?mKPh z_rOBr+<(P4R-Wx{jqh%M-`#n(`@KMSXXPQd0&_Bx?zFo;JHGQ*<=Jno-RbYoc5m!H 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