[petalas]

Summary

Scoped proof with exact finite corroboration. Every unit-diagonal binary pairing with at most one off-diagonal one per row, or per column, has a simultaneous rank-preserving integer lift. Every exact compatible decoder therefore costs at least v over every field, for any finite number of totals.

functional pairing B
  sign one edge of each odd successor cycle
  every minor of S is 0 or +/-1
  every rectangular rank agrees over all fields
  Hadamard/lift argument gives total binary COPY rank >= v

The bound is attained with v coordinate totals. Attainment with three totals is not asserted. Only research/functional-pairing-lift/ is added.

This answers a bounded part of DreamingOfClouds's #2111 Open next question 8 on further reductions. Credit #2111 for the model and lift framework, #1590 for the Hadamard/lift lemma, and #446/#595 for the cost model. Classical total unimodularity and leaf expansion retain their background status. This contribution was produced by Nick Petalas.

Evidence

Before, the retained reduction list did not identify this functional-support class. After, PROOF.md constructs one signing for every size and proves every rectangular rank identity and the all-field cost consequence. No predecessor examples or scientific replay are the deliverable.

The new checker enumerates all 288 functional pairings through order four, 67,332 rectangular rank comparisons, 18,486 square minors and 288 coordinate-decoder attainments. Larger cycles, paths, stars and disjoint pairs give 48 matrices and 27,680 further rectangular checks through order sixteen; 12 nontrivial two-total decoders are checked. Nine boundary/sign controls pass. Counts are generated, not handwritten receipts.

From the worktree root, with CPython 3.12.15 on Darwin arm64:

/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -I -W error -B research/functional-pairing-lift/verify.py
/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -I -W error -O -B research/functional-pairing-lift/verify.py
/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -B scripts/research_package.py --package research/functional-pairing-lift
/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -B scripts/check_research_commit.py --package research/functional-pairing-lift --check-source-history --check-modes

All pass locally. Scientific output is PASS_FUNCTIONAL_PAIRING_LIFT; the registered and committed-byte markers also pass. Each registered scientific command was timed locally: 0.88 seconds normally and 1.00 seconds optimized. These are check-budget observations, not a speedup claim.

The official eight-file package/input export passes integrity checking and both direct modes with empty PATH in a read-only Git-free tree. Exported bytes remain unchanged. Private source and receipt corruptions reject in both modes. RESULT.json binds all six proof/code/metadata files and the cited source by SHA256. No predecessor executable is imported.

Base and submission refresh are canonical main 3f0152a32eeebac0df11946dbd50cabe5a7f7839, priority revision 8. Guidance, selected record and cited source are unchanged. Advance CLAIM.

Limitations and unrun obligations

The hypothesis applies to the complete chosen pairing B. Excluding its functional subpairings does not exclude a better nonfunctional pairing. General pairings and the general three-total lift conjecture remain open. The API caps dense order at 256; the written proof is unbounded. No physical word, supplier invoice, new kappa or frontier change is claimed. Native all-column, ownership, compiler, precision/tape and analytic obligations remain separate. Selected cold replay, Lean, Docker, independent human and formal proof-kernel review were not run or claimed.

Open next questions

1. Specify one additional contact and determine whether its overlapping cycles still admit a simultaneous lift.
2. Integrate this exact exclusion into a new decoder search while retaining all nonfunctional candidates.

Related open work

No open PR on this functional-pairing lift was found. The broader decoder work below shares the cost framework or search context; this result computes none of its fixture values, and no extra comparison runs were made.

• P104 carrier code: the length-8 decoding code is unique (E8 root code, min weight 49) #6166 studies P104 ternary carrier-code uniqueness, not this sparse pairing class.

• Checkpoint symmetric two-star P112 supports and inconclusive COPY63 search #5088 studies masked two-star P112 supports; applicability of this sparse hypothesis is not asserted.

• Checkpoint eight masked-decoder basis searches (exploratory; no new K) #5064 searches masked decoder bases; it does not supply the present simultaneous lift.

• C(P112) >= 64: complete reduction to stated conjectures #3154 reduces the P112 cost floor to geometric conjectures; those general pairings remain outside this theorem.

• One-sided padding lemma: P112 to P120 at cost exactly +8 (L120 suffices for Delta >= n_nf) #3074 transfers one-sided P112 decoders to P120, a different operation on the shared cost model.

• Research: one-sided residue closures, part 2 — Lemma Q4, count lemma + E6 dim-cover and rerunnable span passes close 78 more of the 116 open residues (38 open; no kappa claim) #3049 closes one-sided geometric residue cells; no such cell is recomputed here.

• Non-dense fitting B: every E8 term with x·y ≠ 0 has rank ≥ 9; square root-code floors E8 ≥ 72, P112 ≥ 64, P104 ≥ 56, D4 ≥ 12 for any fitting B (no κ claim) #3045 proves root-code floors for non-dense fitting pairings; this theorem instead restricts pairing support and allows arbitrary factors.

• Two-sided deficiency budget: Theorem BX for b_Q <= 4 (conditional frame, exact residue, no κ claim) #3029 studies a two-sided deficiency budget; that budget is not used here.

• Research: one-sided residue closures — Circuit Lemma, relative root counting and plane lemma close 480 of the 603 d>=4 residues (116 open; no kappa claim) #2985 closes one-sided residue cells; the present reduction uses no root-code geometry.

• KB5L: tau >= 1 box floor PROVEN at v = 5 - shareable class narrowed - no kappa #3031 studies a small-size F-star box floor; this theorem bounds actual cost for a specified pairing class.

• F-star family gap: the box floor FAILS minimally at v = 3 - surviving tau >= 1 form exact (no kappa) #3012 studies a failure of box completeness; this proof uses a signed lift instead of box certificates.

• F-star family gap: Lemma BPP's without-loss question answered NO - box-completeness fails at tau >= 1, minimal counterexample v = 3 (no kappa) #2978 studies the box-completeness question; no box-completeness conclusion is claimed here.

• F-star family gap: the twelve tied_min v = 11 sub-decoders settled exactly - gap_named = 2 EXISTS below v = 12 (no kappa) #2963 computes F-star quantities on restricted tied_min configurations; those configurations are not replayed here.

• F-star family gap: the v = 7..11 census, the first gap-2 instance below v = 12, and the box-completeness attack (no kappa) #2936 studies F-star census and certificate gaps; this theorem uses a structural pairing hypothesis.

• F-star family gap: the sub-anchor-box decomposition and a fixture-density sub-decoder census (no kappa) #2914 studies the F-star family gap; the present result does not settle its unrestricted certificate problem.

• F-star residue (i-prime): cancellation lines - tie lemma, insertion-finite mixing, fixture settlement, and the corrected dyadic corollary (no kappa) #2901 studies cancellation-line residues; this proof requires no valuation or tie assumptions.

• F-star residue (ii): the chi(Z) >= 4 case - structural lemmas, exact anchors, and a smallest-first search with exact obstructions (no kappa) #2890 studies the chi(Z) residue; this proof instead restricts the selected binary pairing.

• Prove exact two-channel P104 support 72 over F3 and Q (no new κ) #6173 computes two-channel support in the P104 root code; no code-support optimum is computed by this functional-pairing theorem.

• Prove stable joint-bank Gram spectral obstruction (no new κ) #6181 proves a joint-bank Gram spectral obstruction; its endpoint-code and correction-rank contract differs from the pairing-support hypothesis here.

• Prove a sharp 1/4 error floor for contractive P104 mode deletion (no κ) #6183 bounds coefficient error for contractive P104 mode deletion; this theorem concerns exact decoder cost and computes no approximation-error optimum.

• Certify rectangular dirty circuits with k+1-coordinate witnesses (no κ) #6180 certifies rectangular dirty circuits; its program-equivalence certificates are distinct from the simultaneous submatrix-rank lift here.

The completion refresh read this account's open PRs created since the CLAIM: #6173, #6174, #6175, #6176, #6177, #6179, #6180, #6181, #6182, #6183 and #6184. The newly published related scopes above contain no competing computation of functional-pairing decoder cost. No extra computations were run to check those contributions.

CI notes

Both hosted checks passed for exact head c651f5a1180ab5611f16533a1837c8a386f1ab50 in run 40880938069: plan in 4m49s and replay (research/functional-pairing-lift) in 1m38s. The check list was read once; run metadata confirms completion, success and the exact contribution head.

Hosted replay certifies execution of the registered normal and optimized package checks. All recorded local scientific, publication bytes/modes/source-history, export-integrity and read-only Git-free checks also passed. The written theorem, its sparse pairing hypothesis and all unrun physical/full-supplier obligations retain their stated limits. Independent human and formal proof-kernel review remain unclaimed.

Merge danger

Door: two-way

Blast radius: isolated

Seven new package files. Shared code, frozen predecessors and the selected record are unchanged.