[mihir-s-05]

Scoped negative result — no new kappa

PR717 leaves open whether mixed pairing-rank totals can improve the P112 complex decoder. This package rules out one exact family: adding one total while retaining the eight coordinate reader columns cannot lower copy-rank cost below 64, even with arbitrary real coefficients and an arbitrary 112-component source correction outside the coordinate column space.

For X X^T = 2G, the family is

X X^T = sum_j x_j (x_j-a_j z)^T + (Xa) z^T,
a in R^8, z in R^112.

It includes every exact nine-term factorization retaining those reader columns, up to nonzero column scaling, and the transposed class. It does not cover arbitrary changes of the reader columns, masked completions with extra free terms, multiple additional totals, other geometries, or arbitrary physical delivery programs.

The proof uses annihilator dimensions. Exact premises cover 511 binary functionals and all 3,272 sign classes. At full eight-coordinate support, the linear system has rank 63 rather than 64: each of its 128 nonzero kernels forces contradictory correction values at 56 ports. That extra compatibility check is included explicitly. This is a written theorem with exact finite certificates, not an inference from failed searches or a proof-assistant multiplication theorem.

Reproduce and validation

python -B research/p112-single-lift/verify.py
python -B research/p112-single-lift/independent_check.py

Expected: PASS_P112_SINGLE_LIFT and PASS_P112_SINGLE_LIFT_INDEPENDENT.

• Standard library only; Windows/Python 3.12.0 normal and optimized checks pass.
• A relocated Git archive passes on Python 3.11.9 with -O -S -B; regenerated certificate is byte-identical.
• The independent implementation imports neither the main verifier nor its geometry module. It checks 511 integer minors with Bareiss elimination and repeats all binary ranks using set elimination.
• Nine mathematical negative controls, four independent arithmetic controls, and four additional source/certificate corruption controls pass.
• Original-base selected-record integrity and staged whitespace pass. upstream/, existing generators, and the selected frontier are unchanged.
• Repository-wide make verify, selected-supplier cold replay, and proof-assistant verification were not run. The new package has not run in hosted CI: the GitHub token lacks workflow permission, so the proposed scoped workflow was removed after a rejected push. Account permissions were not changed. Existing repository CI is separate from these local results.

At submitted head 125ce25bdae421627b56077adc14940c2d0aa52f, all five existing upstream workflows report action_required; Verify research artifacts has no passing result. No hosted CI pass is claimed. GitHub reports the PR mergeable. The exported patch also passed application checking against an isolated index at the exact base, and the final portable ZIP passed its verifier after fresh extraction.

Provenance, overlap and accounting

Exact base: 7ee02b7c96d85f22b38d28bd7ed49bb8c1aeb261. Seed 40, structural decoder research, drawn once from OS randomness. The local claim, full coverage, source pins and validation receipt are in the package. Only this isolated research package is added.

Credit Joel Pulikkan / GamingPuzzled's PR620 for the existing cost 64; DreamingOfClouds' PR717 at 4150829a2cb66b823f4282da98985959ee448a90 for the model/open problem, and PR446 at 309482849b6ddf34db998f53612d34c19f6e3233 for the inherited separate-total copy-tree interpretation. Their review scopes are preserved, and the original model/notice are retained. This adds a proof for a nine-term subclass beyond the prior eight-term/gauge and finite flip-walk evidence.

The final refresh inspected 639 open PRs through #842 and found no duplicate. Main advanced to f567884aeb8851b6e794af1184119a3bfbe1e631, selecting #799; that integration does not overlap these files or alter this pinned research model. There was no pinned coordination issue, so the claim is local.

Even granting all preparation, source-access, frame, restoration and recursive-child expenses zero extra cost, this mechanism cannot reduce the copy-rank sum. Improvements to other costs or to rank distributions at the same sum remain possible. All inherited all-size, physical, hosting, precision/tape and analytic assumptions remain; no new supplier, kappa, or runtime bound is claimed.

Prepared for Mihir Srivastava (@mihir-s-05) with substantial OpenAI Codex assistance in mathematics, implementation, verification and writing, under Apache-2.0. No independent human review is claimed.