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..08940db493 --- /dev/null +++ b/research/bit-copy-path/README.md @@ -0,0 +1,173 @@ +# The BIT COPY floor for closed snapshot paths + +**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 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 +\(14-r\geq8\). + +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 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. + +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. +The [validation record](VALIDATION.md) separates the retained mathematical +replays, publication review and hosted CI. + +## Open next questions + +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. + +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 +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..dc125ae0b5 --- /dev/null +++ b/research/bit-copy-path/RELATED-WORK.md @@ -0,0 +1,119 @@ +# 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. 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. +- 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. + +## 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) +(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 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 + +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. 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{ + "bytes": 5758, + "path": "verify.py", + "sha256": "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..7fc5ee84ec --- /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 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. 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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. See [NOTICE.md](NOTICE.md). diff --git a/research/bit-copy-path/bounds/inputs/SELECTED-ASSEMBLY.json b/research/bit-copy-path/bounds/inputs/SELECTED-ASSEMBLY.json new file mode 100644 index 0000000000..5fd676df06 --- /dev/null +++ b/research/bit-copy-path/bounds/inputs/SELECTED-ASSEMBLY.json @@ -0,0 +1,345 @@ +{ + "status": "PASS_GAUGE44_NATIVE_EXACT_GLOBAL_ASSEMBLY", + "kappa": "836330098538459/500000000000000000", + "kappa_decimal": "0.001672660197076918", + "complex_lower": "67018507072324140917/40000000000000000000000", + "complex_decimal": "0.001675462676808103522925", + "bit_lower": "1733427039589844234749/1000000000000000000000000", + "bit_decimal": "0.001733427039589844234749", + "binding": "complex", + "gain_over_pr635_percent": "2.225753072846734155686237293080565286991751430561837568524869952894006", + "complex_profile": { + "1": 349604, + "2": 60192, + "3": 62585, + "4": 20448, + "5": 11808, + "7": 7, + "8": 1152, + "48": 4866 + }, + "provider_counts": { + "gauge44": 144 + 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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 0000000000..672c415346 Binary files /dev/null and b/research/bit-copy-path/bounds/inputs/SOURCE-CODE-WORD.json.gz differ 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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"interval": "after inverse C, before forward call", + "pair": 0, + "stage": 2 + }, + { + "bank": 3, + "interval": "after inverse call, before C", + "pair": 1, + "stage": 3 + }, + { + "bank": 2, + "interval": "after inverse C, before forward call", + "pair": 1, + "stage": 4 + }, + { + "bank": 5, + "interval": "after inverse call, before C", + "pair": 2, + "stage": 5 + }, + { + "bank": 4, + "interval": "after inverse C, before forward call", + "pair": 2, + "stage": 6 + }, + { + "bank": 7, + "interval": "after inverse call, before C", + "pair": 3, + "stage": 7 + }, + { + "bank": 6, + "interval": "after inverse C, before forward call", + "pair": 3, + "stage": 8 + } + ], + "literal_code_A_3_0": 1, + "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", + "rank_one_requests_per_whole_port_sweep": 111, + "restricted_bank_sweep_lower_bound": 9, + "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.", + "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 0000000000..672c415346 Binary files /dev/null and b/research/bit-copy-path/composition/inputs/CODED-WORD.json.gz differ diff --git a/research/bit-copy-path/composition/inputs/FULL-E8-GEOMETRY.json.gz b/research/bit-copy-path/composition/inputs/FULL-E8-GEOMETRY.json.gz new file mode 100644 index 0000000000..8bec1751f7 Binary files /dev/null and b/research/bit-copy-path/composition/inputs/FULL-E8-GEOMETRY.json.gz differ diff --git a/research/bit-copy-path/composition/inputs/R429-PHYSICAL-CODE-REFERENCE.py.txt b/research/bit-copy-path/composition/inputs/R429-PHYSICAL-CODE-REFERENCE.py.txt new file mode 100644 index 0000000000..9070fa2c56 --- /dev/null +++ b/research/bit-copy-path/composition/inputs/R429-PHYSICAL-CODE-REFERENCE.py.txt @@ -0,0 +1,226 @@ +#!/usr/bin/env python3 +"""Actual repaired R429 source code, routed conjugacy and native-cut tests. + +The exact native intertwiner is constructive; its simplest use loses fresh +coverage. 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 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"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 credited proof reference; no imported proof checker." + }, + { + "path": "FULL-E8-GEOMETRY.json.gz", + "original_path": "/Users/ross/git/math/integer-mult-bounds/research/c511-mutable-source-boundary/frozen/upstream/research/e8-bit-carrier-skeleton/data/bit/bitE8-r865.json.gz", + "sha256": "1fb726ea3818fe0837cceb0dd22a71a0814f064e5e781628104610d6468e0e55", + "scope": "Actual native chart geometry or credited R429 route/coverage-debt reference." + }, + { + "path": "R429-PHYSICAL-CODE-REFERENCE.py.txt", + "original_path": "/Users/ross/git/math/integer-mult-bounds/research/c511-mutable-source-boundary/frozen/changed/bit-native-caller/physical_code.py", + "sha256": "1188b1884a1d8b6873ea118d39f17f939f9e78ca778fa2a821d8a8fe2fd3d043", + "scope": "Actual native chart geometry or credited R429 route/coverage-debt reference." + } + ] +} diff --git a/research/bit-copy-path/construction/ACQUISITION-RELAXED.json b/research/bit-copy-path/construction/ACQUISITION-RELAXED.json new file mode 100644 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"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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103, + 104, + 107, + 108 + ] + }, + { + "centre": 8, + "exact_projected_rank": 7, + "modular_pivot_ports": [ + 0, + 2, + 5, + 7, + 9, + 10, + 18, + 59 + ], + "snapshot_row": "1560874275157996115690798614896583152874299071332485575429578479812685869409882810060153051531745985579913465561313554715590246120137719243860309", + "source_frame": 518, + "source_normal": [ + [ + 0, + 1 + ], + [ + -1, + 1 + ], + [ + 0, + 1 + ], + [ + 1, + 1 + ], + [ + 0, + 1 + ], + [ + 1, + 1 + ], + [ + 0, + 1 + ], + [ + 1, + 1 + ] + ], + "source_register": 479, + "source_row": "610243267866258280493576590715221", + "support_rank": 8, + "target_support": [ + 0, + 2, + 5, + 7, + 9, + 10, + 11, + 13, + 18, + 19, + 20, + 21, + 22, + 24, + 26, + 28, + 31, + 33, + 35, + 37, + 39, + 40, + 42, + 45, + 46, + 47, + 50, + 51, + 52, + 54, + 55, + 57, + 59, + 60, + 62, + 63, + 66, + 68, + 70, + 72, + 73, + 74, + 75, + 76, + 77, + 78, + 79, + 80, + 81, + 82, + 84, + 86, + 89, + 91, + 93, + 95, + 96, + 99, + 100, + 103, + 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..e4e5f4b138 --- /dev/null +++ b/research/bit-copy-path/independent/REVIEW.md @@ -0,0 +1,321 @@ +# 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 tails", + "coded429_allowance_per_call": [ + "11676535802398904937/200000000000000000", + "29191339505997262343/500000000000000000" + ], + "free_bad_direct_histogram_excess": [ + "590298056111308193819/1000000000000000000", + "29514902805565409691/50000000000000000" + ], + "free_bad_nine_rank7_repair_excess": [ + "280790565058008966091/200000000000000000", + "175494103161255603807/125000000000000000" + ], + "free_bad_rank5_replacement_excess": [ + "1643220711311315591/125000000000000000", + "13145765690490524729/1000000000000000000" + ], + "one_COPY7_allowance_per_call": [ + "4977907425421188143/1000000000000000000", + "311119214088824259/62500000000000000" + ], + "scope": "fixed-stock raw owner LP premise; charged displayed-primal allowances", + "target": "9199631083923049/4990800368916076951" + }, + "checker_sha256": "66f19765e975bc98e2b62ed2ce1a91a951c86f6a96682e8fc4f036353855aaba", + "coded_local_histogram": { + "1": 1298, + "2": 1342, + "3": 79, + "4": 44, + "5": 93, + "6": 11, + "7": 9 + }, + "credited_direct_floor": { + "forced_minor_rows": [ + 1, + 4, + 2 + ], + "lower_bound_histogram": { + "63": 2, + "64": 9, + "65": 40, + "66": 144, + "67": 214, + "68": 93, + "69": 10 + }, + "prior_result": "63 direct scatter floor; not a new lower bound", + "ten_or_more_floor": "6*n+3", + "unit_subset_cases": 512 + }, + "exact_geometry": { + "coded_monotone_moves": 2873, + "coded_rank7_directions": [ + 135, + 185, + 205, + 208, + 238, + 336, + 349, + 415, + 425, + 437, + 503, + 507 + ], + "frame_dimension_histogram": { + "0": 1, + "1": 111, + "2": 153, + "4": 70, + "6": 180, + "7": 120, + "8": 1 + }, + "ordinary_monotone_moves": 2904, + "rational_closed_frames": 636, + "same_total_branch_theorem_finite_premises": [ + { + "centre": 0, + "projected_rank": 7, + "root_rank": 8, + "source_normal": [ + 0, + 1, + 1, + 0, + -1, + 0, + 0, + 1 + ], + "target_count": 63 + }, + { + "centre": 1, + "projected_rank": 7, + "root_rank": 8, + "source_normal": [ + 0, + 1, + -1, + 0, + -1, + 0, + 0, + -1 + ], + "target_count": 55 + }, + { + "centre": 2, + "projected_rank": 7, + "root_rank": 8, + "source_normal": [ + 0, + 1, + 1, + 0, + 0, + -1, + -1, + 0 + ], + "target_count": 55 + }, + { + "centre": 3, + "projected_rank": 7, + "root_rank": 8, + "source_normal": [ + 0, + 1, + -1, + 0, + 0, + 1, + -1, + 0 + ], + "target_count": 55 + }, + { + "centre": 4, + "projected_rank": 7, + "root_rank": 8, + "source_normal": [ + 0, + 0, + 0, + 0, + 1, + 1, + 1, + 1 + ], + "target_count": 63 + }, + { + "centre": 5, + "projected_rank": 7, + "root_rank": 8, + "source_normal": [ + 1, + 1, + -1, + 1, + 0, + 0, + 0, + 0 + ], + "target_count": 63 + }, + { + "centre": 6, + "projected_rank": 7, + "root_rank": 8, + "source_normal": [ + 1, + 0, + 0, + 1, + 1, + 0, + 0, + 1 + ], + "target_count": 55 + }, + { + "centre": 7, + "projected_rank": 7, + "root_rank": 8, + "source_normal": [ + 0, + 0, + 1, + 1, + 0, + 0, + 1, + -1 + ], + "target_count": 55 + }, + { + "centre": 8, + "projected_rank": 7, + "root_rank": 8, + "source_normal": [ + 0, + 1, + 0, + -1, + 0, + -1, + 0, + -1 + ], + "target_count": 63 + } + ], + "support_histograms": { + "coded_defect": { + "7": 12, + "8": 499 + }, + "source": { + "7": 9, + "8": 502 + }, + "target": { + "7": 9, + "8": 502 + } + } + }, + "input_pins": { + "bounds/inputs/SELECTED-ASSEMBLY.json": "1122cb3a575f8c788345fc1bf4c108ac7c3250fb81a5988671f5eb04b3ab3045", + "composition/inputs/CODED-WORD.json.gz": "1a6841fccba51fa922993c77b4b296e659c2c538252e5e06113d0438c5edb870", + "composition/inputs/FULL-E8-GEOMETRY.json.gz": "1fb726ea3818fe0837cceb0dd22a71a0814f064e5e781628104610d6468e0e55", + "inputs/GEOMETRY.json.gz": "a353999bceac4ac957f540adfd3e5f8768297138b27132eeb83332e2ea4b1022", + "inputs/WORD.json.gz": "c46ccd7d3eedabc6a7efbb1a91ec13f2928bad8ee0bac57c6e593a092d13d96c", + "inputs/global_k.py.txt": "6368196a393ac25bb2918880e4c4bd31ed733b762c27b6873ab391cbc0c5c3c8" + }, + "late_factory_separators": { + "cases": [ + { + "centre": 0, + 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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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Chronological whole-owner availability, paid ADDRESS adapter execution and source/tape costs are separate obligations.", + "source": { + "/Users/ross/git/math/integer-mult-bounds/research/e8-bit-carrier-skeleton/data/bit/bitE8-r865.json.gz": "1fb726ea3818fe0837cceb0dd22a71a0814f064e5e781628104610d6468e0e55" + }, + "status": "PASS_THREE_DYADIC_ORTHOGONAL_RANK4_TERMINAL_CHARTS" +} diff --git a/research/bit-copy-path/inputs/WORD.json.gz b/research/bit-copy-path/inputs/WORD.json.gz new file mode 100644 index 0000000000..aa9ec3c341 Binary files /dev/null and b/research/bit-copy-path/inputs/WORD.json.gz differ 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