Research: uniform selected-k8 frame optimum and optimal-profile rigidity

[RossRichardson]

PR #1671 certifies the selected k8 word's frame optimum at two exponents. This contribution audits its unchanged symbolic duals and proves that the same paid histogram minimizes the frame LP for every 0 <= a <= 1/32. For 0 < a <= 1/32, every optimal assignment has exactly that histogram, although its frames can differ. The interval includes the 1/99 planning target.

This is one scoped proof extension and arithmetic audit, no new kappa or physical supplier. Only research/selected-k8-uniform-frame-audit/ is added. The source word, event/domain model, endpoints, scalar chronology, centre cut and cleanup/cone contract remain fixed. The original physical decoder construction gate is not claimed satisfied.

Exact evidence

• Reconstructs all 23,004 LP columns, 17,507 rows and 135 component objectives; a second column assembly works directly from the shared event graph and agrees on every cost and residual.
• The 239 distinct rational residual vectors comprise one zero vector (16,564 columns), one exact square (10 columns), and 237 vectors (6,430 columns) proved positive uniformly by rational degree-eight Bernstein coefficients. No floating-point or solver output certifies the interval.
• Strict positivity outside the identically-zero columns plus symbolic objective equality proves optimal-profile rigidity. Rank-one counts follow from the invariant local mass 5,586. A distinct feasible model assignment at gate G616 retains the full histogram.
• Exact boundary controls show this particular dual fails for min(r,2) and at a=1/20. Neither control supplies a cheaper feasible assignment or contradicts Research: exact complete frame ceiling for the selected k8 word #1671. The interval is not claimed maximal.

The written argument is in PROOF.md; RESULT.json retains every distinct residual/multiplicity and sign certificate. Canonical receipt: 859fcb8c40601ce3a1d453b5703f20268c5bfab2228aac8484e1cbba47aea703.

Reproduce and validation

python3.12 -I -B scripts/research_package.py --package research/selected-k8-uniform-frame-audit

Passed on macOS 26.5.2 arm64 / CPython 3.12.12, standard library only:

• Registered replay, metadata/source pins and all eight test groups, including integer-root arithmetic at three sample exponents, exact Bernstein basis identities, dual/objective/column/domain mutations, damaged sources/receipts, and optimized read-only relocation.
• Optimized mathematical replay from a clean Git export of contribution commit cd696b3df479cd221ca812f2eb0804027a44da2f reproduces the same receipt.
• make selected-record-check passes separately.
• The pinned PR1671 package's separate checker was also replayed unchanged: PASS_K8_FRAME_INDEPENDENT components=135 columns=23004 root_calibrations=72 graph_and_duality_match=True.

Both new column assemblies share the inherited event/domain model; this is not an independent implementation or proof of the physical interface. Full selected cold replay, native/hosting admission, all-size/compiler, common-label/mutable-source, precision/tape, analytic/bit H1-H5 obligations, Lean and human review are unclaimed. Local checks do not establish hosted-CI success or maintainer acceptance.

Scope, pins and credit

• Base: 418e40eb2572d930b58c1cf9b82b41c0fc7ee08d.
• Exact predecessor: Research: exact complete frame ceiling for the selected k8 word #1671 at ce130d291f5ab6dd801544886a16f62d42d86118, rechecked open and unchanged. Its DUALS.json.gz is retained byte-for-byte with attribution and source pins.
• Chooser nonce 4071338634dbc7c5d72c8e5f2a5a4633, revision 3, complex-rank-shape-decoder; TASK.json declares the changed scope to an independent audit/extension after overlap found Research: exact complete frame ceiling for the selected k8 word #1671.
• Coordination claim. Final overlap at 06:45 UTC covered 407 open PRs including drafts, 332 Discussion records, and 710 local records; complete remote pagination and no warnings. This is metadata coverage, not a novelty guarantee. Current main 5e554ff58d01921747d27177045c0061f31e1ad1 preserves the selected record, instructions, catalogue and all three live inputs.
• Discussion #1674 changes helper endpoints and is complementary, outside this theorem. Research: sharp rank-one block budgets and per-block overhead obstruction #1605 studies abstract reblocking/overhead rather than this fixed-word LP.

Credit Gareth Davidson / bitplane for #1671's unchanged symbolic duals and selected-word theorem; Aaron Wang / navinor for the retained X4 frame model/method; and Jeremy Kothe, Rohan Arun, Hao Yan, icekylinx, DreamingOfClouds, eumemic, the earlier frame-method authors, Sagar Thakrar and Douglas Colkitt for the retained construction/coverage lineage. NOTICE.md and SOURCES.json preserve details.

Prepared for Ross Richardson with substantial OpenAI Codex assistance in the mathematics, implementation, tests and writing. No subagents or independent human review. Apache-2.0. Selected evidence and predecessor files remain unchanged.

The concrete next step is one explicit scalar-dependency or gate-order change with a new dirty-state contract and complete cost screen. The old fixed-word certificate cannot certify that changed construction.

[Mister-Meeseeks]

Batch review at cd696b3df479cd221ca812f2eb0804027a44da2f.

Merged to main, preserving the original contributor commits and package attribution.

Reviewed scope: Uniform exact sign proof for PR1671 symbolic duals on0<=a<=1/32 using a square and rational Bernstein coefficients; strict signs force the same optimal paid histogram for positive a. Inherits event/domain interface; negative dual outside range is not a cheaper construction.

The pinned scoped command passed; the batch ledger contains its command, input hashes, environment and log. This incorporates the research/tooling contribution and does not change the selected κ or discharge the inherited multiplication-theorem assumptions.

Batch review, attribution and evidence. Thank you for the contribution.