From 9aaf76f43493cd2392eaae248274c651f29bc28e Mon Sep 17 00:00:00 2001 From: CubeJerry <80299683+CubeJerry@users.noreply.github.com> Date: Fri, 30 Oct 2026 15:49:57 +0800 Subject: [PATCH] Prove the reader497 direct-retention cost floor Add a scoped all-dimension concavity proof and exact source-bound certificates with 13 rejection controls and a changed-incidence positive control. No new kappa. Prepared for @CubeJerry with substantial OpenAI Codex assistance. Retain Ali Romman, DreamingOfClouds, Nick Petalas, Matarener and the source lineage credit. --- .../CERTIFICATE.json | 6244 +++++++++++++++++ research/p104-reader497-cache-floor/PROOF.md | 264 + research/p104-reader497-cache-floor/README.md | 124 + .../p104-reader497-cache-floor/SOURCES.json | 42 + research/p104-reader497-cache-floor/TASK.json | 172 + .../p104-reader497-cache-floor/VALIDATION.md | 95 + .../p104-reader497-cache-floor/VERIFY.json | 35 + research/p104-reader497-cache-floor/verify.py | 414 ++ 8 files changed, 7390 insertions(+) create mode 100644 research/p104-reader497-cache-floor/CERTIFICATE.json create mode 100644 research/p104-reader497-cache-floor/PROOF.md create mode 100644 research/p104-reader497-cache-floor/README.md create mode 100644 research/p104-reader497-cache-floor/SOURCES.json create mode 100644 research/p104-reader497-cache-floor/TASK.json create mode 100644 research/p104-reader497-cache-floor/VALIDATION.md create mode 100644 research/p104-reader497-cache-floor/VERIFY.json create mode 100644 research/p104-reader497-cache-floor/verify.py diff --git a/research/p104-reader497-cache-floor/CERTIFICATE.json b/research/p104-reader497-cache-floor/CERTIFICATE.json new file mode 100644 index 0000000000..df80a70cbe --- /dev/null +++ b/research/p104-reader497-cache-floor/CERTIFICATE.json @@ -0,0 +1,6244 @@ +{ + "baseline_ranks": [ + 1, + 1, + 3, + 3, + 5 + ], + "baseline_truncated_sums": [ + 5, + 8, + 11, + 12, + 13, + 13 + ], + "certificate": [ + { + "dimensions": [ + 0, + 0, + 1 + ], + "geometric_realizations": 1, + "operations": { + "ADD": 3, + "CLEAR": 2, + "COPY": 2, + "INIT": 2, + "MOVE": 5 + }, + "ranks": [ + 1, + 3, + 6, + 6, + 6 + ], + "representative_bases": [ + [], + [], + [ + 52 + ] + ], + "truncated_gaps": [ + 0, + 1, + 2, + 4, + 6, + 9 + ] + }, + { + "dimensions": [ + 0, + 0, + 2 + ], + "geometric_realizations": 31, + "operations": { + "ADD": 3, + "CLEAR": 2, + "COPY": 2, + "INIT": 2, + "MOVE": 6 + }, + 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"truncated_gaps": [ + 0, + 0, + 0, + 0, + 0, + 0 + ] + } + ], + "contained_line_positive_control": { + "operations": { + "ADD": 3, + "CLEAR": 2, + "COPY": 2, + "INIT": 2, + "MOVE": 5 + }, + "ranks": [ + 1, + 1, + 2, + 3, + 6 + ], + "replacement_source_basis": [ + 30, + 42, + 74 + ], + "scope": "changed source frame, not a PR6167 implementation", + "strict_power_gain": "3^t+5^t-2^t-6^t > 0 for 0= phi(x+y)`, follows from concavity +and phi(0)=0. Compressing any unused intermediate stops can therefore only +decrease cost. After compression, sort `[0,a,b,f,7]` and take successive +positive gaps. The complete optimistic paid multiset is + + reader: consecutive positive gaps in sorted [0,a,b,f,7] + V cache: 3-a, if nonzero + B cache: 6-b, if nonzero + root: 7-f and f-1, omitting zero. + +The total transported rank is `22-a-b`; forgetting cache transport would +instead (incorrectly) make it 13 independently of the retained input frames. +Every rank is at most six. Longer monotone paths cannot undercut this compressed +bill. The actual input capture times may impose additional restrictions; ignoring +them enlarges the feasible class and is valid for the lower bound. + +## 4. A general proof of the minimum + +The minimum does not depend on enumerating binary subspaces. More generally, +let dim(S)=s, dim(U)=u, dim(H)=dim(R)=u+1, dim(L)=1, with +`1 <= s <= u-1`, `H intersect R=U`, `S,L <= U`, and `L` not contained in S. +The same two-cache model, over any field of frame coordinates, has minimum + + phi(s) + phi(u-s) + 2 phi(1) + phi(u-1). + +Here phi is increasing and concave with phi(0)=0; the assertion is about +rank tariffs, not a physical realization over every field. The source case is +`(s,u)=(3,6)`. Its compressed domain is `0<=a<=s`, `0<=b<=u`, `1<=f<=u`, `a=f, +the old reader tail plus B-cache cost is + + phi(b-f) + phi(u+1-b) + phi(u-b) + >= phi(u-f) + phi(1), + +by subadditivity on the first and third terms, then monotonicity on the second. +The right side is the tail after moving delivery to U. These transformations +are legal in the favorable model: remove the B-contact and retain its source +value at U until the reader reaches U; the other contacts stay unchanged. + +Thus assume b=u. The remaining exact compressed cost is + + phi(a)+phi(f-a)+phi(u-f)+phi(1) + +phi(s-a)+phi(u+1-f)+phi(f-1). + +For f>=2, both root segments u+1-f and f-1 are between 1 and u-1. Concavity +minimizes their fixed-sum cost at the extremes 1,u-1. Also + + phi(a)+phi(s-a) >= phi(s), + phi(f-a)+phi(u-f) >= phi(u-a) >= phi(u-s). + +Adding these inequalities proves the stated minimum. For f=1, a=0, and the +cost is `phi(s)+phi(u-1)+2 phi(1)+phi(u)`, which is at least the bound because +`phi(u)>=phi(u-s)`. This completes the all-length, all-dimension proof. + +For phi(r)=r^t, 00. At 2<=f0`, because the increments of x^t strictly +decrease. This is a **changed-geometry positive control**, not a way to replace +the actual source frame for free. It shows why a dimension-only argument +omitting the incidence hypothesis would be false. + +## 5. Exact concave-order certificate and quantitative gap + +For a paid multiset P define `M_k(P)=sum_{r in P} min(r,k)`, k=1,...,6. The +original multiset P0=(1,1,3,3,5) has + + (M_1,...,M_6)(P0) = (5,8,11,12,13,13). + +CERTIFICATE.json lists the paid ranks and all six integer differences +`M_k(P)-M_k(P0)` for every one of the 126 triples. Each difference is +nonnegative. The following table gives the coordinatewise minimum of those +differences at each f (the minima need not be attained by the same triple). + +| f | Number of recipes | k=1 | k=2 | k=3 | k=4 | k=5 | k=6 | +|---|---:|---:|---:|---:|---:|---:|---:| +| 1 | 7 | 0 | 0 | 0 | 1 | 2 | 3 | +| 2 | 14 | 1 | 2 | 1 | 2 | 2 | 2 | +| 3 | 21 | 1 | 3 | 2 | 2 | 1 | 1 | +| 4 | 28 | 1 | 2 | 2 | 1 | 0 | 0 | +| 5 | 28 | 1 | 2 | 1 | 1 | 0 | 0 | +| 6 | 28 | 0 | 0 | 0 | 0 | 0 | 0 | + +This is a complete finite integer certificate, not a failed heuristic search. +Its formula above and six checks per row provide a small standalone replay: + + for f = 1,...,6: + for a = 0,...,min(3,f-1): + for b = 0,...,6: + construct the paid multiset in section 3 + check its six capped sums against (5,8,11,12,13,13) + +To see why these finitely many integer inequalities imply a continuum of +tariff inequalities, put `d_k=phi(k)-phi(k-1)`, +`lambda_k=d_k-d_(k+1)` for k<6 and `lambda_6=d_6`. Increasing concavity makes +each lambda nonnegative, and direct telescoping gives, for 0<=r<=6, + + phi(r) = sum_{k=1}^6 lambda_k min(r,k). + +Multiplying the six capped-sum differences by these coefficients proves + + sum_{r in P} phi(r) >= 2 phi(1) + 2 phi(3) + phi(5). + +Only `(a,b,f)=(3,6,6)` has all six differences zero. For `phi(r)=r^t`, +0= 2*4^t - 3^t - 5^t > 0, 0 0` per local call. + +This answers one concrete continuation of Ali Romman's PR6167 and Matarener's +PR6214 handoff. The successful reader-451 retention rewrite does not transfer +directly: frame 737 contains the root's final direction, whereas reader 497's +first operand frame 519 does not. Retaining that earlier operand requires +paid transport; it cannot simply be made available at the new meeting frame. + +The new [proof](PROOF.md) first reduces arbitrary finite monotone paths and +arbitrary common frames to their contact dimensions. It then proves a general +minimum for endpoint dimensions `(s,u,u+1,1,u+1)`, for every +`1<=s<=u-1`, by moving the second delivery to its original frame and applying +subadditivity and concavity. The source-specific integer certificate gives +the quantitative gap and independently corroborates that proof. Here the +supplier's tariff is `H_r=r^t`, with `t=1-C` in its recursive moment notation. + +## Precisely what is covered + +The reader ascends from frame 3 to 914, the loaded owned root descends from +529 to 595, and the original V, B and -4Q additions are preserved. Each earlier +operand may have one separate owned copy captured at its original cut/frame, +transported through any descending chain, and erased after use. All reader, +root and cache MOVEs are charged. Additional intermediate frames need not +belong to the original list. All productive targets and other work are fixed. + +Mixed source representations, root cloning, public ascending carriers, changed +source prefixes or consumer cuts, new productive targets, and coordinated +changes to other paid work remain outside the result. The optimistic relaxation +grants free ordinary resources only to make rejection stronger; it is not a +claim that a physical cache has zero cost. The symbolic templates explicitly +record the copies, lifetime endpoints and erasure. No rounded implementation, +native phase replay, full caller admission or selected-frontier update follows. + +## Evidence and reproduction + +From the repository root, with CPython 3.12 and the standard library: + +```sh +python3.12 -B scripts/research_package.py --package research/p104-reader497-cache-floor +``` + +Expected result: `PASS_REGISTERED_RESEARCH_PACKAGE`, with both normal and +optimized direct replays reporting `PASS_P104_READER497_CACHE_FLOOR`. +No network, optional solver or predecessor Python implementation is needed. +The direct command is `python3.12 -I -B research/p104-reader497-cache-floor/verify.py`. + +The checker authenticates the frozen source inputs, anchors all relevant +events, scans every use of reader 497 in the cut, and checks the actual binary +frame intersections. It enumerates all 2,825 subspaces of frame 520 and all +**52,964** allowed comparable contact triples, recovering exactly **126** +dimension recipes. It independently emits and symbolically interprets one +local template for every recipe: all arbitrary receiver/source values, +common charts, owned direction, copied values, cleanup and every MOVE rank. +The 2,006 template events do not constitute a full P104 multiplier replay. + +[CERTIFICATE.json](CERTIFICATE.json) records all 126 bills, six integer +concave-order inequalities per bill, geometric counts, representative bases, +the unique equality recipe and rejection controls. Additional finite checks +of the general normalization argument at `u=2,...,10` are explicitly separate +from its all-dimension written proof. Thirteen controls reject wrong root sign, +either missing operand copy, missing cleanup/transport, wrong ownership, +ascending owned transport, a changed source anchor or incidence, a widened +source-frame cap, a free-cache invoice and an impossible delivery order. +A convex-tariff counterexample demonstrates why concavity is required. +Separately, replacing the first operand frame with a same-dimensional frame +that contains the root's final direction gives a cheaper valid local template. +This positive control shows that the incidence hypothesis matters; changing +the actual source to that frame is not supplied for free. Checks use explicit +exceptions and remain active under `python -O`. + +Generate a candidate receipt outside the frozen package with +`verify.py --output /absolute/scratch/CERTIFICATE.json`; a normal run recomputes +and compares every canonical certificate byte. No mathematical equality depends +on regenerated gzip headers. [VALIDATION.md](VALIDATION.md) records executed +commands and their coverage. + +## Source, overlap and credit + +Base: `2485805a6d47f36ea83ddde6327d453ac82d44dc`, priority revision 8. +The selected record remains PR6030, `κ=781451/250000000=0.003125804`. +Research input: PR6167 head `564a076b113621a93edc4e39968ee4e4a8c9caf0`, +word SHA256 `68d1ff048bc6d30d85ceaeb38e2557adab87b20a9c67cf73948f95f54104575f`. +Exact retained blobs and hashes are in [SOURCES.json](SOURCES.json). + +PR6163 (Nick Petalas, `1743bbafb49ad2a351f6412825fb08d4a7c4f086`) instead +drops an operand and proves that receiver-only compensation fails in its +specified window. This package preserves both operands and proves a cost +minimum in a different, explicit model. PR6170's automated retention builder +does not supply this arbitrary-frame lower bound. PR6214 audits the source +word; PR6245 reconstructs its outer arithmetic. Neither result is claimed here. +The random draw, three candidates, bounded overlap and read-only Discussion +access limitation are recorded in [TASK.json](TASK.json). Lack of a matching +public search result is not a novelty certificate. + +New proof, certificate and checker: **Jerry Chen (@CubeJerry), with substantial +OpenAI Codex assistance**; no delegated agents, independent human review or +formal proof-kernel acceptance. Ali Romman retains PR6167/PR5827 credit; +DreamingOfClouds retains PR6144/PR6030/PR5907 and toolkit PR6170 credit; +Nick Petalas retains PR6163 credit; Matarener retains PR6214 credit. Retain +Mason Hale, Pierre-Loic Doulcet, Douglas Colkitt, Chris R / Clear1nsight and all +earlier native, caller, compiler, BIT and analytic attribution in those sources. +The capped-sum method is a standard concave-order certificate, not a newly +invented general method; the repository's recursive-invoice preflight and +mightyOil's T16 audit already use this order. New material is Apache-2.0. + +## Open next questions + +Can a changed source representation, public ascending carrier or changed +145/161 consumer remove the stop with a favorable complete bill? First supply +the exact changed value flow at the source and consumer cuts and count every +movement and productive-target debit. This proof supplies no such mechanism +and does not justify another search inside the excluded retention model. +All five selected framework assumptions remain inherited. diff --git a/research/p104-reader497-cache-floor/SOURCES.json b/research/p104-reader497-cache-floor/SOURCES.json new file mode 100644 index 0000000000..898509d413 --- /dev/null +++ b/research/p104-reader497-cache-floor/SOURCES.json @@ -0,0 +1,42 @@ +{ + "schema_version": 1, + "base_commit": "2485805a6d47f36ea83ddde6327d453ac82d44dc", + "files": [ + { + "path": "certificates/selected-result.json", + "usage": "provenance", + "git_blob": "d59b5035e7b4cbaa9d688a1d32dd0d6eb79c466a", + "sha256": "9b8d0eea95b3bf0dfcb7938a732a6439b87a2685dec2a5ab9e525b3b34debbfa" + }, + { + "path": "docs/research/agent-task-catalog.json", + "usage": "provenance", + "git_blob": "3404fbdd5e7529a0674eef26114b5071e4586cec", + "sha256": "ead6cdc9d0af6d58e8a49744d8ec8c7c15e13d2ba93c276c434bcf9b51b93a81" + }, + { + "path": "research/p104-retained-operand-descent/PROOF.md", + "usage": "input", + "git_blob": "71d6a77e7c9ea81097a0d11744c1bbc33a8188db", + "sha256": "941145209301df79d2b1a1c91dc6133435952c8d94e79b825ed19eee5988ceef" + }, + { + "path": "research/p104-retained-operand-descent/README.md", + "usage": "input", + "git_blob": "cbe390b9e7167b7aac019f52ed2c0c0707f8e7cc", + "sha256": "42704b4bf59b6f83ddc3a37c1e26d4c13d7e5f5da64fd6c1874e2391e6b7c4ff" + }, + { + "path": "research/p104-retained-operand-descent/WORD.json.gz", + "usage": "input", + "git_blob": "f2d3f98964f939e5ca652f9b1453a0604f15f829", + "sha256": "68d1ff048bc6d30d85ceaeb38e2557adab87b20a9c67cf73948f95f54104575f" + }, + { + "path": "research/p104-root2-receiver-obstruction/README.md", + "usage": "input", + "git_blob": "f6903dad589745b2de98da9126bd32612ce8c404", + "sha256": "2b005a550c76ab6f72d43a3f5438c8a091ea7ea18e5599958f4140d8961596c3" + } + ] +} diff --git a/research/p104-reader497-cache-floor/TASK.json b/research/p104-reader497-cache-floor/TASK.json new file mode 100644 index 0000000000..9d2038310c --- /dev/null +++ b/research/p104-reader497-cache-floor/TASK.json @@ -0,0 +1,172 @@ +{ + "schema_version": 1, + "task_id": "p104-side-correction", + "priority_revision": 8, + "base_commit": "2485805a6d47f36ea83ddde6327d453ac82d44dc", + "original_card": { + "id": "p104-side-correction", + "question": "Can joint helper bases and a legal causal split improve the complete P104 productive/paid invoice?", + "starting_points": [ + "docs/research/current-priorities.md", + "docs/research/research-family-status.md", + "certificates/selected-result.json", + "docs/research/pr-intake-evidence/pr5599-completion/PROOF-REVIEW.md", + "research/p104-helper-subspace-splits/README.md", + "research/dyadic-helper-channel-capacity/README.md" + ], + "methods": [ + "Construct and independently check one source-bound instance of the stated first experiment.", + "Seek an exact counterexample or target-specific complete-budget obstruction to that same proposed change." + ], + "acceptance_gate": "An exact arbitrary-input identity, source/target frame availability and restoration at the chosen cut, plus the complete channel+correction+idle/hosting bill or an explicit favorable bound. The native channel alone is not a supplier. 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", + "scripts/research_selected_baseline.py" + ], + "expected": "PASS_RESEARCH_BASELINE_BINDINGS_ONLY", + "inspect": "Read revision 8, the selected review and live exact-source handoffs. Then follow this card's bounded first_experiment.", + "scope": "Source-bound orientation only, not a supplier replay or theorem proof." + }, + "first_experiment": "Start with the selected #5599 emitted word and review pending #5718 distinct input/output bases plus #5717/#5719 coverage before claiming work. Pick one uncovered source word, helper group and cut region. Prove the arbitrary-input relation and legal entry/exit corrections, then compare full target and paid profiles. Reject if completion, owned lifetimes or consumer costs repay the gain. A static pairing rank alone is not a physical split.", + "bottleneck_link": "Judge the actual current supplier contract and complete signed target/paid bill; frozen-profile ceilings require an applicability proof.", + "allowed_variations": [ + "One exact source head, helper/cut region or quantified proposition and changed hypothesis; check live handoffs first. Different seeds are not different scopes." + ], + "scope_change_triggers": [ + "Changing source head, helper/cut domain, operation alphabet or caller boundary without recording the new scope.", + "Treating an archived proposal or local numerical pass as complete supplier admission." + ], + "overlap_terms": [ + "P104", + "helper basis", + "helper split", + "5718", + "dyadic capacity" + ], + "existing_checks": [ + { + "scope": "PR #4133: literal-slot defect", + "evidence": "research/p104-literal-slot-defect/README.md", + "remaining": "Independently validate the stated scope or change its producer/consumer premise and pay the complete boundary.", + "status": "author-checked", + "source_commit": "88666f1dee93fbd85b31fd5d6f91a219e111b5f4", + "selected_record_sha256": "bb2eaa58647b353f88d0aec2eafb9d30df1be53589231b23e12ceef3e4bad129" + }, + { + "scope": "PR #4136: uniform commutation gap", + "evidence": "research/p104-uniform-commutation-gap/README.md", + "remaining": "Independently validate the stated scope or change its producer/consumer premise and pay the complete boundary.", + "status": "author-checked", + "source_commit": "281ce601a8be9db470de630759179c23e1dfaa88", + "selected_record_sha256": "bb2eaa58647b353f88d0aec2eafb9d30df1be53589231b23e12ceef3e4bad129" + }, + { + "scope": "PR #4140: uniform encoding gap", + "evidence": "research/p104-uniform-encoding-gap/README.md", + "remaining": "Independently validate the stated scope or change its producer/consumer premise and pay the complete boundary.", + "status": "author-checked", + "source_commit": "de25c6525aa866962309df5d107b32d55e6e5628", + "selected_record_sha256": "bb2eaa58647b353f88d0aec2eafb9d30df1be53589231b23e12ceef3e4bad129" + }, + { + "scope": "PR #4153: source-chart shortcut exhausts its optimistic allowance", + "evidence": "research/p104-source-chart-shortcut/README.md", + "remaining": "Independently validate the stated scope or change its producer/consumer premise and pay the complete boundary.", + "status": "author-checked", + "source_commit": "9123241fea53df77e51ffe30d9aca1625d4bf3a8", + "selected_record_sha256": "bb2eaa58647b353f88d0aec2eafb9d30df1be53589231b23e12ceef3e4bad129" + }, + { + "scope": "PR #4154: fiber-tag seam mismatch", + "evidence": "research/p104-fiber-tag-seam/README.md", + "remaining": "Independently validate the stated scope or change its producer/consumer premise and pay the complete boundary.", + "status": "author-checked", + "source_commit": "4ffcdaee5d6255a9cea1fe53a5424435068089d0", + "selected_record_sha256": "bb2eaa58647b353f88d0aec2eafb9d30df1be53589231b23e12ceef3e4bad129" + }, + { + "scope": "Selected PR5599: complete helper-subspace construction and independent caller replay; five framework assumptions remain.", + "evidence": "docs/research/pr-intake-evidence/pr5599-completion/PROOF-REVIEW.md", + "remaining": "The next experiment must identify uncovered source-specific work. Successor archive intake does not transfer this admission.", + "status": "reviewed", + "source_commit": "09dd0965711ca7eeb169057a420abab1c9052137", + "selected_record_sha256": "2a0220e866417774606e0f991e41b9c1d26b1f43b97857238cf478dbc7d330b5" + } + ] + }, + "scope_status": "refined", + "actual_scope": "Exact concave-cost minimum for reader497 and root772 on PR6167, allowing one independently transported retained copy of each original operand, arbitrary binary frames and monotone intermediate chains.", + "scope_change_reason": "Take PR6167/PR6214 released reader497 handoff. Unlike PR6163, preserve both original operands; change their delivery charts through paid owned caches and quantify all frames, rather than dropping an operand and repairing receiver161.", + "bottleneck_link": "Complex currently binds. Decide whether the successful reader451 retention mechanism can also eliminate root772 frame520 without changing productive targets or other callers.", + "budget_minutes": 120, + "stop_rule": "Stop at a rigorously checked optimistic cost obstruction or a legal favorable schedule. No wider helper search; reserve final 30 minutes for verification/publication.", + "outcome": "completed", + "evidence": "PROOF.md: general all-dimension direct-retention floor; CERTIFICATE.json: exact source case, 126 rank recipes, 52,964 frame triples, 2,006 local symbolic events, 13 rejection controls and a changed-incidence positive control. Full physical/native/selected replays are not claimed.", + "author": "Jerry Chen (@CubeJerry), with substantial OpenAI Codex assistance", + "ai_assistance": "Substantial OpenAI Codex assistance for research, implementation, proof checking and packaging; no subagents.", + "open_questions": [ + { + "question": "Can reader497 avoid frame520 by a changed source representation, a public ascending carrier, or a changed 145/161 consumer rather than two direct descending caches?", + "why_it_matters": "A favorable complete paid profile on this binding complex supplier could improve the conditional result.", + "first_step": "Exhibit the exact changed value flow at the source and consumer cuts and price every move and any productive-target debit.", + "blocked_by": "This package does not permit mixed source caches, changed prefix/other helpers, extra productive inputs, or changed consumer boundaries." + } + ], + "seed": 3398684913100150128, + "track_draw": 28, + "track": "complex supplier", + "scope_key": "pr6167-reader497-direct-retention-concave-floor", + "claim": { + "status": "COMPLETED / LOCAL-ONLY", + "created_utc": "2026-10-30T07:13:19.254295+00:00", + "expiry_utc": "2026-10-30T08:54:00Z", + "expected_deliverable": "One exact source-bound cost obstruction with executable certificates and negative controls.", + "posting_limitation": "No pinned sprint issue; GitHub plugin exposes no Discussion mutation; cloud browser initialization failed. Public HTML #1163, #1166 and #5645 and board read successfully; HTTP session is unauthenticated. No claim was posted.", + "result": "General direct-retention cost floor, source-specific exact certificate and changed-incidence positive control; no new kappa. This local scope is released.", + "artifact": "https://github.com/CubeJerry/integer-mult-bounds/tree/research/cubejerry-reader497-meeting-20261030-s28/research/p104-reader497-cache-floor", + "completion_update": "Local handoff only. No coordination post was possible with the available signed-in tools; browser was signed out." + }, + "candidate_tasks": [ + "469/471 alternate rung carrier: close prior recodings/obstructions already exist", + "497 retained operands and arbitrary monotone meeting frames: selected released handoff", + "Pooled dyadic helper split: common-reader obstruction requires a broader new producer/consumer contract" + ], + "scientific_result": "No new kappa. In the specified two-independent-owned-cache model, original reader497/root772 schedule minimizes every increasing concave rank tariff. For r^t, 0> vector) & 1, "independent basis") + space = extend(space, vector) + return space + + +def inside(left, right): + return left & right == left + + +def dimension(space): + size = space.bit_count() + need(size > 0 and size & (size - 1) == 0, "subspace cardinality") + return size.bit_length() - 1 + + +def basis(space): + result, generated = [], 1 + for vector in elements(space): + if not (generated >> vector) & 1: + result.append(vector) + generated = extend(generated, vector) + need(generated == space, "representative basis reconstruction") + return result + + +def source_contract(word): + need(word["public"] == 770 and word["owned_fields"] == 29, "source role inventory") + need(len(word["program"]) == 22821, "source event count") + for index, expected in ANCHORS.items(): + need(word["program"][index] == expected, "source anchor " + str(index)) + touches = [] + for i in range(12888, 15573): + e = word["program"][i] + if e[1] == 497 or (e[0] in ("ADD", "COPY") and e[2] == 497): + touches.append(i) + need(touches == [12888, 12933, 12934, 12935, 13486, 15462, 15572], + "complete receiver interval") + frames = {str(i): span(word["frames"][i]) for i in (3, 519, 520, 529, 595, 737, 914)} + zero, s, u, h, line, end = [frames[str(i)] for i in (3, 519, 520, 529, 595, 914)] + need([dimension(x) for x in (zero, s, u, h, line, end)] == [0, 3, 6, 7, 1, 7], + "source frame dimensions") + need(inside(s, u) and inside(line, u) and h & end == u and s & line == zero, + "source frame intersections") + need(inside(line, frames["737"]), "reader451 contrasting containment") + actual_ranks = sorted(abs(dimension(frames[str(e[2])]) - dimension(frames[str(e[3])])) + for e in ANCHORS.values() if e[0] == "MOVE") + need(actual_ranks == BASE_RANKS, "baseline paid ranks from source events") + return frames + + +def rank_recipe(a, b, f): + need(type(a) is int and type(b) is int and type(f) is int, "integer dimensions") + need(0 <= a <= 3 and 0 <= b <= 6 and 1 <= f <= 6 and a < f, "dimension domain") + cuts = sorted([0, a, b, f, 7]) + receiver = [y - x for x, y in zip(cuts, cuts[1:]) if y != x] + caches = [x for x in (3 - a, 6 - b) if x] + root = [x for x in (7 - f, f - 1) if x] + return sorted(receiver + caches + root) + + +def truncated_sums(ranks): + return [sum(min(r, k) for r in ranks) for k in range(1, 7)] + + +def proof_row(a, b, f): + ranks = rank_recipe(a, b, f) + gaps = [x - y for x, y in zip(truncated_sums(ranks), truncated_sums(BASE_RANKS))] + need(all(x >= 0 for x in gaps), "concave tariff domination") + need(any(gaps) or (a, b, f) == (3, 6, 6), "unique equality dimensions") + if f < 6: + need(gaps[3] >= 1, "strict rank-four gap for deleted stop") + need(sum(ranks) == 22 - a - b, "complete transported rank mass") + return {"dimensions": [a, b, f], "ranks": ranks, "truncated_gaps": gaps} + + +def all_subspaces(universe): + """Closure enumeration in literal source vectors, not a dimension formula.""" + points = tuple(elements(universe)) + seen, queue = {1}, deque([1]) + while queue: + old = queue.popleft() + for vector in points: + if old >> vector & 1: + continue + new = extend(old, vector) + if new not in seen: + seen.add(new) + queue.append(new) + return sorted(seen, key=lambda s: (dimension(s), s)) + + +def geometry(frames): + s, u, line = (frames[k] for k in ("519", "520", "595")) + spaces = all_subspaces(u) + need(Counter(map(dimension, spaces)) == {0: 1, 1: 63, 2: 651, 3: 1395, + 4: 651, 5: 63, 6: 1}, + "complete binary six-space census") + a_spaces = [x for x in spaces if inside(x, s)] + f_spaces = [x for x in spaces if inside(line, x)] + need((len(a_spaces), len(f_spaces)) == (16, 374), "source subspace intervals") + counts, representatives = Counter(), {} + for a_space in a_spaces: + for f_space in f_spaces: + if not inside(a_space, f_space): + continue + for b_space in spaces: + if not ((inside(b_space, a_space)) or + (inside(a_space, b_space) and inside(b_space, f_space)) or + (inside(f_space, b_space))): + continue + triple = tuple(map(dimension, (a_space, b_space, f_space))) + counts[triple] += 1 + representatives.setdefault(triple, (a_space, b_space, f_space)) + expected = {(a, b, f) for a, b, f in product(range(4), range(7), range(1, 7)) if a < f} + need(set(counts) == expected and len(expected) == 126, "all geometric dimension recipes") + return counts, representatives + + +def emit(frames, meeting): + """Local normalized-chart template, not a full physical supplier word. + + Dirty receiver A, preserved V/B, preloaded owned Q, and two owned caches. + Captures precede root arrival; arbitrary retention is granted. + """ + a_space, b_space, f_space = meeting + s, u, h, line, end = (frames[k] for k in ("519", "520", "529", "595", "914")) + events = [("INIT", 4, s), ("COPY", 4, 1, s), ("INIT", 5, u), ("COPY", 5, 2, u)] + + def move(role, old, new, owned): + if old != new: + events.append(("MOVE", role, old, new, owned)) + + move(4, s, a_space, True) + move(5, u, b_space, True) + move(3, h, f_space, True) + current = 1 + for at, donor, coefficient in sorted([(a_space, 4, 1), (b_space, 5, 1), (f_space, 3, -4)], + key=lambda x: (dimension(x[0]), x[1])): + move(0, current, at, False) + events.append(("ADD", 0, donor, coefficient, at)) + current = at + move(0, current, end, False) + move(3, f_space, line, True) + events.extend([("CLEAR", 4, a_space), ("CLEAR", 5, b_space)]) + return events + + +def execute(frames, events): + """Independent event ledger and all-input rational linear interpretation.""" + zero_row = (Fraction(0),) * 4 + rows = {r: tuple(Fraction(int(j == r)) for j in range(4)) for r in range(4)} + chart = {0: 1, 1: frames["519"], 2: frames["520"], 3: frames["529"]} + owned, ranks, operations = {3}, [], Counter() + for event in events: + op, role = event[:2] + operations[op] += 1 + if op == "INIT": + need(role in (4, 5) and role not in chart, "clean cache allocation") + chart[role], rows[role] = event[2], zero_row + owned.add(role) + elif op == "MOVE": + _, _, old, new, private = event + need(chart.get(role) == old, "move source chart") + need(type(private) is bool and private == (role in owned), "move ownership") + need(old != new and (inside(new, old) if private else inside(old, new)), "monotone move") + ranks.append(abs(dimension(old) - dimension(new))) + chart[role] = new + elif op == "COPY": + donor, at = event[2:] + need(role in owned and rows[role] == zero_row, "copy into clean owned cache") + need(chart[role] == chart[donor] == at, "copy common chart") + rows[role] = rows[donor] + elif op == "ADD": + donor, coefficient, at = event[2:] + need(chart[role] == chart[donor] == at, "addition common chart") + rows[role] = tuple(x + coefficient * y for x, y in zip(rows[role], rows[donor])) + elif op == "CLEAR": + need(role in (4, 5) and role in owned and chart[role] == event[2], "cache erasure") + rows[role] = zero_row + del chart[role] + owned.remove(role) + else: + raise ValueError("unknown local opcode") + need(rows[0] == (1, 1, 1, -4), "arbitrary-input receiver endpoint") + need(all(rows[r] == tuple(int(j == r) for j in range(4)) for r in (1, 2, 3)), "preserved source values") + need(chart == {0: frames["914"], 1: frames["519"], 2: frames["520"], 3: frames["595"]}, + "complete exit charts and cache cleanup") + need(owned == {3}, "no leaked cache lifetime") + return sorted(ranks), dict(sorted(operations.items())) + + +def reject(action, reason): + try: + action() + except ValueError as error: + need(str(error) == reason, "control rejected for wrong reason: " + str(error)) + return reason + raise ValueError("negative control accepted: " + reason) + + +def controls(word, frames, representatives): + results = {} + original = emit(frames, representatives[(0, 0, 1)]) + bad = deepcopy(original) + i = next(i for i, e in enumerate(bad) if e[:3] == ("ADD", 0, 3)) + bad[i] = (*bad[i][:3], 4, bad[i][-1]) + results["root_sign"] = reject(lambda: execute(frames, bad), "arbitrary-input receiver endpoint") + for cache, label in ((4, "missing_V_capture"), (5, "missing_B_capture")): + changed = [e for e in original if not (e[0] == "COPY" and e[1] == cache)] + results[label] = reject(lambda: execute(frames, changed), "arbitrary-input receiver endpoint") + changed = [e for e in original if not (e[0] == "CLEAR" and e[1] == 4)] + results["uncleared_cache"] = reject(lambda: execute(frames, changed), "complete exit charts and cache cleanup") + changed = [e for e in original if not (e[0] == "MOVE" and e[1] == 4)] + results["unmoved_cache"] = reject(lambda: execute(frames, changed), "addition common chart") + changed = deepcopy(original) + i = next(i for i, e in enumerate(changed) if e[0] == "MOVE" and e[1] == 4) + changed[i] = (*changed[i][:4], False) + results["wrong_cache_ownership"] = reject(lambda: execute(frames, changed), "move ownership") + changed = deepcopy(original) + changed[i] = (*changed[i][:3], frames["914"], True) + results["ascending_owned_cache"] = reject(lambda: execute(frames, changed), "monotone move") + mutant = deepcopy(word) + mutant["program"][12933][3] = "-1" + results["changed_source_read"] = reject(lambda: source_contract(mutant), "source anchor 12933") + results["source_frame_cap"] = reject(lambda: rank_recipe(4, 6, 6), "dimension domain") + paid, _ = execute(frames, original) + free = list(paid) + free.remove(3) + free.remove(6) + results["free_cache_transport"] = reject( + lambda: need(all(x >= y for x, y in zip(truncated_sums(free), truncated_sums(BASE_RANKS))), + "concave tariff domination"), "concave tariff domination") + results["read_V_after_root_at_595"] = reject( + lambda: need(inside(frames["595"], frames["519"]), "historical V cannot follow root"), + "historical V cannot follow root") + results["convex_tariff_is_outside_theorem"] = reject( + lambda: need(sum(r * r for r in rank_recipe(3, 6, 4)) >= sum(r * r for r in BASE_RANKS), + "concavity is necessary"), "concavity is necessary") + mutant = deepcopy(word) + mutant["frames"][519] = basis(frames["595"]) + basis(frames["519"])[:2] + results["changed_source_incidence"] = reject(lambda: source_contract(mutant), "source frame intersections") + return results + + +def contained_line_example(frames): + """Changed-hypothesis positive control, explicitly NOT the source geometry.""" + altered = dict(frames) + altered["519"] = span(basis(frames["595"]) + basis(frames["519"])[:2]) + need(dimension(altered["519"]) == 3 and inside(altered["595"], altered["519"]), + "contrasting incidence") + events = emit(altered, (altered["519"], altered["520"], altered["595"])) + ranks, operations = execute(altered, events) + need(ranks == [1, 1, 2, 3, 6], "contrasting paid bill") + gaps = [x - y for x, y in zip(truncated_sums(ranks), truncated_sums(BASE_RANKS))] + need(all(g <= 0 for g in gaps) and gaps[2] < 0, "contrasting concave improvement") + return {"scope": "changed source frame, not a PR6167 implementation", + "replacement_source_basis": basis(altered["519"]), "ranks": ranks, + "operations": operations, "truncated_gaps": gaps, + "strict_power_gain": "3^t+5^t-2^t-6^t > 0 for 0= normalized >= old, "general delivery-normalization inequalities") + comparisons += 2 + rows += 1 + return {"u_range": [2, 10], "recipes": rows, "integer_inequalities": comparisons, + "scope": "finite corroboration only; all-u result uses written subadditivity/concavity proof"} + + +def run(): + word, hashes = authenticate() + frames = source_contract(word) + counts, representatives = geometry(frames) + certificate, total_events = [], 0 + for triple in sorted(counts): + row = proof_row(*triple) + events = emit(frames, representatives[triple]) + ranks, operations = execute(frames, events) + need(ranks == row["ranks"], "emitted moves equal independently priced recipe") + need(operations["INIT"] == operations["COPY"] == operations["CLEAR"] == 2, + "charge two full cache lifetimes") + row.update(geometric_realizations=counts[triple], representative_bases=[ + basis(s) for s in representatives[triple]], operations=operations) + certificate.append(row) + total_events += len(events) + equality = [r["dimensions"] for r in certificate if not any(r["truncated_gaps"])] + need(equality == [[3, 6, 6]], "unique equality recipe") + return { + "status": STATUS, "source_hashes": hashes, + "scope": "Monotone reader/root paths, two direct retained operands, fixed sources and consumer; no source mixing or target changes.", + "source_anchors": {str(k): v for k, v in ANCHORS.items()}, + "frame_dimensions": {k: dimension(v) for k, v in frames.items()}, + "baseline_ranks": BASE_RANKS, "baseline_truncated_sums": truncated_sums(BASE_RANKS), + "dimension_recipes": len(certificate), "geometric_realizations": sum(counts.values()), + "symbolic_template_events": total_events, "equality_recipes": equality, + "min_gaps_by_root_dimension": { + str(f): [min(r["truncated_gaps"][j] for r in certificate if r["dimensions"][2] == f) + for j in range(6)] for f in range(1, 7)}, + "strict_deleted_stop_lower_bound": "2*4^t-3^t-5^t for 0