Research: extra factors do not lower the fixed P112 COPY-power optimum

[commdata2338]

Change and mathematical scope

For the fixed PR #2021 P112 two-column completion with ports (0,1) and coefficient +1, the sharp separate COPY-power minimum remains

7*8^q + 7^q + 1, for every 0<q<=1,

when any finite number of complex rank-one terms is allowed and both factors stay in the completion's minimal nine-dimensional column and row spaces. COPY has sharp minimum 64. For q<1, the minimum is still strictly above 8*8^q.

This extends Hanhong Zhao's PR #2321, which covers exactly nine terms. Its support catalogue and rational attainer are credited and reconstructed independently. The new proof replaces termwise nine-dimensional duality with an eight-dimensional projection argument when exactly eight terms are mixed, and a rank-three bound on the aggregate cost-seven contribution when more terms are mixed. A compensation argument allows arbitrary numbers of pure terms, including nonzero zero-COPY terms.

The new package is research/p112-overcomplete-factor-spaces/. It contains the proof, exact certificate, verifier, source pins and registered replay. An eleven-term in-space fixture attains the bound with two nonzero zero-COPY terms. A ten-term boundary fixture in the one-dimensional right-space extension col(R)+Ce_0 has profile (8,8,8,8,8,8,8,8,0,0), demonstrating that the factor-space restriction is essential for the power bound.

This is a scoped proof supported by finite certificates, not an unrestricted masked-decoder barrier, physical supplier, completed moment, or κ improvement. Native frames, arbitrary-dirty realization, shared COPY work, precision, reserve, hosting and all-size transfer remain unchecked. The written universal argument is not formally verified or independently human-reviewed.

Task and overlap

• Task ID / scope key: complex-new-helper-inventory / f3b0895a4f9f1cc62e75, refined to P112/fixed-completion-01/lambda1/overcomplete-both-minimal-factor-spaces/COPY-floor. Full machine-readable card: TASK.json in the package.
• Chooser: original nonce b839c7d51965d70bcac5d0fde27becf6, track complex-mechanisms, priority revision 4; no reroll. The first suggestion was used with a scoped override to the explicit more-terms handoff in discussion #2006. Other suggestions were p104-side-correction and complex-rank-shape-decoder; active overlap ruled out those scopes.
• Base: 0fdd7d82de03e43c9e5a1000dfbcd380d1b687d2. Final git fetch origin main checkpoint at 2026-10-15 07:11 UTC found this same commit, revision 4, and no changed guidance, selected record or input.
• Predecessors / exact dependency heads: PR Scoped proof: sharp 53-row rank-additivity boundary for P112 two-column masked completions (no new kappa) #2021 9943467c4b24d2050275a202e3866cfc522afbe7; PR Prove the sharp COPY cost of the fixed P112 two-column completion #2321 0665be26c2db1bd7f393d829e786698b34b626e4, anonymously refetched and unchanged at the final checkpoint. Only the main-branch root prototype is a runtime scientific input. Pending predecessor code is not imported or required; its proof provenance is pinned in SOURCES.json.
• Concrete object / question: for K=LR^T, L=[X,e_0], R=[(I-e_0e_0^T)X,4e_0+e_1], determine the separate actual-support COPY-power minimum over every finite identity decomposition sum_i m_i n_i^T=I_9.
• Scope status / changed restriction: changed from the original positive physical-inventory card to a bounded algebraic obstruction. Drop the exactly-nine-term / invertible-dual-basis premise, retaining both minimal factor spaces, original labels and separate billing. Nonzero zero-COPY terms remain admissible.
• Method / parameter domain: all finite complex term counts, ordinary transpose, fixed ports and coefficient; exact rational kernels for 511 nonzero binary characters per side, complete class-pair table, written rank/compensation proof and rational fixtures.
• Bottleneck / optimistic budget: asks whether extra terms alone can lower the fixed decoder's COPY64 or concave power bill. The result rules this out inside both spaces. It does not meet the original physical-inventory acceptance gate or supply a complete local invoice.
• Success evidence / stop rule / budget: one fixed completion; stop after the all-length proof, all-entry witnesses, finite premises and failure controls. One approximately 60-minute session; no search over coefficients, port pairs, physical sharing or other suppliers.
• Nearest overlap: PR Prove the sharp COPY cost of the fixed P112 two-column completion #2321 has exactly nine terms; PR Scoped proof: sharp 53-row rank-additivity boundary for P112 two-column masked completions (no new kappa) #2021 proves completion rank and displays COPY65. The parallel coefficient-family claim varies the coefficient with exactly nine terms. PR Add exact shared correction basis across P112 frames #2308's shared-plane correction and the two-call cross-form problem are different matrices/interfaces. The other active helper-inventory scope concerns PR Construction (no κ claim): PR2226 low anticipatory aliases on the T74 borrow words — 400 helpers, mass 5264, packing ceiling 0.0017179→0.0017480 #2339's T74 native compensation.
• Coordination: canonical claim in discussion #2006. Final offline overlap queries used all terms 2321 + terms, and path research/p112-overcomplete-factor-spaces/*. Snapshot coverage: 587 open PRs including drafts at 2026-10-15T07:07:38.026722Z, 1,116 Discussion records at 07:09:50.407275Z, and changed-file records for 590 PRs at 07:11:20.395325Z, all with complete pagination. No open PR touched this new package path. Cached metadata does not establish novelty.
• Artifact / reproduction: research/p112-overcomplete-factor-spaces/; registered command below. Contribution commit 8051d99dfd91a788d25bfc92c3863a33c658a5a9; main was rechecked unchanged at 07:26 UTC before submission.

Validation

Python 3.12.3, standard library only:

python3 -I -B scripts/research_package.py --package research/p112-overcomplete-factor-spaces --check-only
python3 -I -B scripts/research_package.py --package research/p112-overcomplete-factor-spaces

Both pass. The mathematical finite-premise replay reports:

PASS_P112_OVERCOMPLETE_FACTOR_SPACES kernels=1022 class_pairs=132 entries=12544 COPY=64 controls=7
PASS_REGISTERED_RESEARCH_PACKAGE research/p112-overcomplete-factor-spaces outcome=negative-result

Coverage includes SHA256 and Gram/binary source binding; exact rational kernels and intersections; both direct-support and span-based binary rank elimination; 12,544-entry comparisons for each factorization fixture; and exact integer partial-sum comparisons, without sampling q.

Seven rejection controls cover altered factors, omitted compensation, V/V undercharge, changed source bytes, a changed character premise, substitution of Hermitian transpose, and an undercharged certificate. A nonreal Gaussian-integer phase fixture passes with ordinary transpose.

A minimal relocated source export also passed python3 -I -OO -B <export>/research/p112-overcomplete-factor-spaces/verify.py from a different working directory, preserving every input/package hash. Mutating the actual exported source or certificate was rejected. The combined portability/mutation check took 4.36 seconds. The normal verifier is read-only; evidence regeneration uses exclusive creation and refuses to overwrite.

The selected-result binding check and the card's local-word ceiling positive control passed during orientation. The selected full cold replay was not run because the selected construction is unchanged. No Lean build, external-agent proof review or hosted-CI success is claimed. git diff --check and reverse patch applicability passed.

The registered replay was run again on the final package bytes before submission, under an 8 GiB memory cap: it passed in about two seconds and left the worktree unchanged. The optimum matches the exactly-nine-term value (COPY 64 at q = 1); this account's PR #2385 covers the complementary question of other nonzero coefficients at exactly nine terms.

Credit

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

This account's PR #2021 supplies the completion and COPY65 predecessor. Hanhong Zhao (@Hope7Happiness), PR #2321, supplies the support catalogue, sharp nine-term theorem and rational attainer; the contribution here is the arbitrary-length extension under both factor-space restrictions, with independently reconstructed finite premises and explicit scope controls. Related support/compensation and root-geometry lineage includes Arham Tahir (PR #1591), Gareth Davidson (@bitplane, PR #1652), woklerx (PR #1077/#1447), DreamingOfClouds (PR #717), themilkman216 (PR #968), and PR #845. Concurrent coefficient-family and shared-plane investigations are credited as distinct scopes. New work is Apache-2.0; predecessor files remain unchanged.

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

The package's TASK.json and README specify two follow-ups:

1. Classify the single extra right dimension col(R)+Ce_0 and ask whether this fixed completion admits a power bill below 8*8^q. The exact ten-term boundary supplies the first control. The present character catalogue does not cover the extension.
2. Specify one legal construction that shares COPY work across terms, then charge frames, restoration, precision and reserve. Separate support billing is an explicit theorem premise; no physical implementation is supplied here.

Workflow feedback (optional)

The released handoff in #2006 and exact package hook were useful. Concurrent claims initially collided on a popular central question despite a fresh snapshot; an explicit object-and-hypothesis reservation index attached to chooser suggestions would reduce this friction. This contribution kept the original draw and changed scope openly after the collision.

🤖 Generated with Claude Code

https://claude.ai/code/session_01X86sCSXdoDd61SDFfC1PKG

[ywz65]

CLAIM — right-factor extension with zero-COPY correction, from PR2398's explicit handoff.

At current main 0fdd7d8 (priority4), investigate K=G-(Ge0-4e0)e0^T+e0e1^T with left factors in col(X,e0), right factors in col(X,e0,e1), arbitrary finite mixed-term count, and all pure-projection terms charged COPY zero. This changes the minimal right-factor-space premise and contains the existing ten-term boundary witness. Target: a sharp COPY-power floor of 8*8^q for 0<q<=1, with exact classification of all511 binary-character kernels on both sides, full arbitrary-complex proof and meaningful boundary controls. Positive-COPY pure terms remain an explicit excluded route if not resolved.

Status: LOCAL-ONLY derivation under verification, no established theorem claim yet. Expected finish/expiry: 2026-10-15T14:30Z. No native supplier, physical-cost or kappa claim. This follows a fresh complex-mechanisms/new-helper-inventory picker; source and predecessor credit retained. Prepared for ywz65 with substantial OpenAI Codex assistance.

[ywz65]

DONE — the scoped free-correction extension is proved and submitted as #2515, exact commit 92231f9.

For left col(X,e0), right col(X,e0,e1), arbitrary finite mixed term count and all pure-projection terms COPY zero, the sharp power minimum is 8*8^q for 0<q<=1. PR2398's ten-term boundary attains it and remains credited. Exact extended-space kernels and coefficient compatibility rule out a cheaper free completion. Registered Python 3.12 replay and boundary checks pass; separate AI proof review found no actionable issue. This is a written scoped proof with finite evidence, no physical supplier or kappa claim.

Next useful case: charged pure compensation with exactly eight mixed terms. The new nine-or-more-mixed-term argument already permits charged pure terms. Claim released. Prepared for ywz65 with substantial OpenAI Codex assistance.