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Prove the sharp COPY cost of the fixed P112 two-column completion - #2321
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…ient Extends PR #2321 from coefficient +1 to every nonzero complex lambda in K_lambda = G - (G e_0 - 4 e_0) e_0^T + lambda e_0 e_1^T at the fixed P112 ports (0,1). For every 0 < q <= 1 the sharp separate nine-term COPY-power minimum stays 7*8^q + 7^q + 1 (COPY sum 64), so changing only this coefficient cannot beat the eight-rank-eight benchmark 8*8^q. An invertible diagonal map carries the complete support catalogue to every nonzero lambda; the dual pairing 2 + 16/lambda vanishes at lambda = -8, where a separate compensation argument keeps the floor and an explicit exceptional attainer is checked. Negative result; no supplier or kappa change. Adds research/p112-coefficient-copy-floor/ only. Submitted by @commdata2338 with substantial AI assistance (OpenAI Codex research and implementation; packaged with Claude Code). Co-Authored-By: Claude Opus 5.5 (1M context) <noreply@anthropic.com> Claude-Session: https://claude.ai/code/session_01X86sCSXdoDd61SDFfC1PKG
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Takes PR #2321's released more-terms question. For the fixed P112 (0,1) completion with coefficient +1, over every finite complex factorization whose factors lie in its minimal column and row spaces, the separate COPY-power minimum stays 7*8^q + 7^q + 1 for every 0 < q <= 1 (COPY 64), as in the exactly-nine-term case. The argument uses projection duality for exactly eight mixed terms, a rank-three bound on the aggregate cost-seven contribution when there are more, and a compensation step that allows nonzero zero-COPY terms. Factors outside the minimal spaces are outside the theorem. Negative result; no supplier or kappa change. Adds research/p112-overcomplete-factor-spaces/ only. Submitted by @commdata2338 with substantial AI assistance (OpenAI Codex research and implementation; packaged with Claude Code). Co-Authored-By: Claude Opus 5.5 (1M context) <noreply@anthropic.com> Claude-Session: https://claude.ai/code/session_01X86sCSXdoDd61SDFfC1PKG
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PR2021 leaves open whether changing all nine factors of its displayed P112 two-column completion can improve COPY65. This package proves the sharp separate COPY-power minimum is
7*8^q + 7^q + 1for every0 < q <= 1, and supplies an exact rational factorization attaining COPY64. Forq < 1the optimum remains strictly worse than the original eight-rank-8 power bill; this is a fixed-completion result, with no κ or native-supplier claim.The proof uses a complete, source-bound character-kernel classification and dual-basis compensation cases. The standalone standard-library package includes deterministic certificates, the attaining witness, an optimized read-only verifier and failure controls. Credit: commdata2338/PR2021, Arham Tahir/PR1591, bitplane/PR1652 and the retained root/decoder lineage.
Validation: the written proof and exact witness received independent AI-agent review. The final 12-file patch applies cleanly to the original base and freshly fetched canonical main (both
0fdd7d82); the registered package replay passes with Git 2.47.1 / Python 3.12.13, including optimized-I -O -Bverification, 15 reason-specific rejection controls, detached read-only replay, invalid-write/no-output checks and a nonvacuous kernel-membership mutation. Certificate and witness hashes stayed unchanged through final corrections. No independent human review, proof-assistant formalization or hosted-CI result is claimed.Reproduce:
python3 scripts/research_package.py --package research/p112-two-column-copy-optimum.Canonical investigation: #2006 (comment)
Prepared for Hanhong Zhao (@Hope7Happiness) with substantial OpenAI Codex, Claude Code and Gemini assistance.