Research: P112's nine-term COPY floor holds for every nonzero coefficient

[commdata2338]

Change and mathematical scope

For the fixed P112 ports (0,1), every nonzero complex coefficient in
K_lambda=G-(G e_0-4e_0)e_0^T+lambda e_0 e_1^T has sharp separate nine-term COPY-power minimum
7*8^q+7^q+1, for every 0<q<=1. This extends PR2321's coefficient +1 result; it does not claim its already established COPY64 optimum as new. Changing only this coefficient cannot beat the eight-rank-eight COPY-power benchmark.

The precise new obligation is the parameter transfer and its exceptional duality case. An invertible diagonal row/coefficient map preserves the complete support catalogue for every nonzero complex coefficient. The pairing 2+16/lambda vanishes at lambda=-8, where a valid exceptional basis defeats the generic exclusion used at +1. A separate compensation case retains the sharp floor, and both a uniform attainer and the exceptional attainer are checked. The fixed-coefficient predecessor theorem remains intact.

Result kind: scoped written proof with an exact finite certificate, author-checked. The package is research/p112-coefficient-copy-floor/. No native supplier, full recursive moment, physical COPY sharing or kappa improvement is claimed. Frames, dirty restoration, global hosting/owners, precision/tape, all-size and analytic admission remain unchecked.

Task and overlap

Base and final refreshed main: 0fdd7d82de03e43c9e5a1000dfbcd380d1b687d2, priority revision 4. Contribution commit: 3fa1c93c8c66ada108336b99add7db927ea9fa1f; main was rechecked unchanged at 07:12 UTC before submission.

Task card:

• Task ID / scope key: complex-rank-shape-decoder / cad3ba853d7f324f16fa; refined to P112-01/nonzero-complex-coefficient/nine-term-COPY-power. Saved chooser nonce b85a2c0fbfaf5aec5e4b172d8785e44d, track complex-mechanisms.
• Exact predecessors: PR2321, Hanhong Zhao, 0665be26c2db1bd7f393d829e786698b34b626e4; this account's merged PR2021, 9943467c4b24d2050275a202e3866cfc522afbe7. No pending executable is imported. Both used factor-space catalogues are rechecked from the main root input.
• Object and question: the displayed one-parameter family, fixed ports and labels, all exact nine-term complex ordinary-transpose factorizations; determine its sharp separate COPY power.
• Scope: changed from a positive physical block-climb construction to an algebraic cost screen. The original five-percent supplier gate is not met. The concrete PR2321 handoff was chosen after central P104 handoffs were already actively claimed.
• Changed restriction: drop the coefficient +1 restriction; retain nine factors and separate actual-support billing. Complex coefficients are not reduced modulo two.
• Method and domain: all lambda in C*, all 0<q<=1, GL(9,C) dual bases; 511 binary characters per side, exact annihilators and nonzero integer minors, symbolic parameter transport, compensation cases and concavity.
• Nearest work / delta: PR2021 classified completion rank and gave COPY65; PR2321 proved the sharp COPY64/power result at +1. PR1591/1652 supply related one-anchor methods. The adjacent overcomplete claim fixes +1 and changes factor count, a distinct question.
• Budget and observable: compare COPY power with 8*8^q before physical synthesis. The proved floor closes this coefficient-only nine-term route; it is not a complete supplier cost bound.
• Success / falsification / cap: parameter-universal proof or an exact cheaper factorization; check the exceptional case and actual supports. One bounded session, stopped after the proof, attainers and controls.
• Artifact / reproduction / credit: README proof, complete deterministic certificate and registered standard-library replay; source and predecessor details below and in TASK/SOURCES.

Canonical claim.
The final overlap snapshot contains 583 open PRs including drafts (2026-10-15 06:55:23 UTC) and 1,111 Discussion records (06:57:32 UTC), with complete pagination and expected offline-cache warnings. Searches included the task ID, 2321, two-column, COPY-power, coefficient family and coefficient-copy; the board and adjacent claims were reread. This is metadata coverage, not a novelty guarantee. Canonical main and PR2321 were anonymously re-fetched at 07:00 UTC and were unchanged.

The sole live scientific input is research/direct-delivery504-global/inputs/hamming/PROTOTYPE.json, Git blob 9efe30ee107b9c67f15f1c726e1a1cb5cef03929, SHA256 327d2a868043e846381aa58a33053c24425e8cad522ff401b10ec8af1d649a3f. Frozen packages and selected records are unchanged.

Validation

CPython 3.12.3, standard library; Git 2.43.0. Passed:

python3 -I -B research/p112-coefficient-copy-floor/verify.py
python3 -I -O -B research/p112-coefficient-copy-floor/verify.py
python3 -I -O -B research/p112-coefficient-copy-floor/test_controls.py
python3 scripts/research_package.py --package research/p112-coefficient-copy-floor --check-only
python3 scripts/research_package.py --package research/p112-coefficient-copy-floor

Primary result: PASS_P112_COEFFICIENT_COPY_FLOOR kernels=1022 lambda=nonzero_complex exception=-8 copy_sum=64 controls=8 no_new_kappa.
Registered result: PASS_REGISTERED_RESEARCH_PACKAGE ... outcome=negative-result.
Canonical receipt SHA256: ca9b5b26f4571c08ec9f3163c9daf9c29ec9556be0b79797a220d203fb5bb289.

The finite evidence includes 1,022 complete character kernels with annihilating bases and exact nonzero integer minors, 132 actual support-pairing ranks, five compensation-case profiles, and 25,088 polynomial endpoint coefficient entries. The coefficient argument is symbolic, not a finite sampling inference. The uniform and exceptional rational bases are checked against complete endpoints and support ranks.

Eight mathematical control groups cover the parameter boundary, wrong support transport/kernel, compensation omission, ordinary versus Hermitian transpose, the exceptional legal basis, and zero-cost duality. Nine CLI groups cover an optimized read-only Git-free export, source/receipt/type/schema/witness corruption and occupied-output preservation; a fresh generated receipt is semantically identical. All code uses explicit checks that survive -O. Patch whitespace checks pass.

Primary replay takes about one second; registered replay including controls about eleven seconds on the checked environment. The full selected cold replay, hosted CI, Lean, independent human review and all physical/all-size obligations were not run or established.

The registered replay was run again on the final package bytes before submission, under an 8 GiB memory cap: it passed in about ten seconds and left the worktree unchanged. I also checked by hand that the exceptional pairing 2+16/lambda vanishes exactly at lambda=-8, that the floor gives COPY sum 64 at q=1, and that for q strictly between 0 and 1 the floor exceeds the benchmark 8*8^q, because 8^q is less than 7^q+1 there.

Credit

Hanhong Zhao / @Hope7Happiness / PR2321 supplied the fixed-coefficient optimum, complete support classification, duality and compensation method, and the uniform attaining basis. This account's PR2021 supplied the masked completion and rank classification. Arham Tahir / PR1591 and Gareth Davidson / @bitplane / PR1652 supplied related support/duality methods. Retain credit to woklerx / PR1447 and PR1077, DreamingOfClouds / PR717, themilkman216 / PR968 and the PR845/523/514/507/425/433 root-data and decoder lineage. New material is Apache-2.0; no predecessor executable or frozen artifact is modified.

Prepared for commdata2338 (@commdata2338), with substantial OpenAI Codex assistance.

AI assistance: submitted by @commdata2338 with substantial AI assistance. OpenAI Codex (GPT-6 Astra) did the derivation, the implementation, the checking and the writing; it was packaged and submitted with Claude Code. No independent human review is claimed.

Open next questions

Can redundant factors lower this family's COPY power? The smallest specific follow-up is 9x10 coefficient matrices M,N with M N^T=I9 at lambda=-8, where termwise dual-basis exclusions disappear. Classify that compatibility before any physical synthesis. Coordinate with the active lambda=+1 overcomplete claim linked above; its premise is distinct. README and TASK give the first step and blockers. No favorable redundant-factor or physical candidate is claimed.

Workflow feedback (optional)

A concise chooser summary flag would help retain the nonce, track, task IDs and scope keys when full task cards and orientation reads exceed tool-output limits. The saved full JSON and a short summary should remain tied to the same draw.

🤖 Generated with Claude Code

https://claude.ai/code/session_01X86sCSXdoDd61SDFfC1PKG