[Matarener]

Requested by Mark

Change and mathematical scope

Proof with an exact finite certificate; two Lean-checked lemmas; conjecture with search evidence. No new κ.

Before: for the full 56-plane decoder of the strict 112-port mask (PR1825 matching, PR1889 mask), the only global COPY-mass floor valid for arbitrary coefficients was the fitting rank 49. The floor 224 was proved only for 2-integral coefficients and characteristic 2 (Philippe Van Velzen), and for decoders split into disjoint plane pairs (#4365). Decoders with even denominators were open between 49 and 224.

After: every finite rank-one decoder, over any field, has additive full-plane COPY mass at least 18233853/131072 = 139.113, hence at least 140. Under #4365's contract this excludes the κ-doubling budget (about 128) for every coefficient field and narrows the break-even window to [140, 164], open only in characteristic 0.

How: on a clique of pairwise non-orthogonal planes the mask is an identity block, so at least 2|Q| terms touch it on both sides over every field. Weights on 443 cliques that are feasible against every tariff give the bound by weak duality. Feasibility is a finite exhaustive check over all 39,087² ordered pairs of plane-generated subspaces.

The package also states the conjecture that 224 holds over every field, with structural conditions an escape must meet (essential halves, 2-adic layering, a spectral condition on cliques), and a real-coefficient search on the open local cases of the three- and four-plane rank duals that found no escape. That part is evidence, not proof.

New package: research/plane-port-global-rank-dual/. No shared code or frozen package changes.

Remaining obligations: the conjecture; the optimum of the full clique programme (a floating-point estimate of at most about 167 is not certified); native realization and hosting (#4365's caveats).

Task and overlap

• Base 13be6ac685845c1ff671f5d26e7166165abd998e. Card complex-rank-shape-decoder, scope changed (see TASK.json): the coefficient question for the plane-port decoder instead of block children.
• Nearest overlap: two-plane-copy-floor (Prove the sharp two-plane COPY floor for arbitrary scalar factorizations #4365), three-plane-rank-dual (Extend #4365's rank dual to every plane triple: three-plane COPY floor 12 over every field on all 27,720 triples (no new κ) #4421), four-plane-rank-dual, four-plane-integral-shadow, the C5 packages (c5-dyadic-decoder-gate, c5-gram-rank-cover, c5-asymmetric-cost-floor, c5-residual-weighted-cover, c5-global-charging-*), paired-plane-minrank-bracket-audit, plane-port-copy-budgets. Those are local (2-5 planes) or rank results. This package is the first global all-field COPY-mass bound for the whole decoder; it reuses their rank-dual functional on cliques of the full mask. The c5-global-charging-* ceilings concern restricted-tariff charging of C5 floors, a different functional.

Validation

• python3 -B research/plane-port-global-rank-dual/verify.py: bounded CI mode, about 1 minute. It checks the closure census, 20,000 tariffs against Gaussian elimination, all 443 clique columns, the bound, three rejecting controls (overweight column, orthogonal pair as clique, wrong bound) and feasibility on every 97th row.
• --full: all 1,527,793,569 ordered pairs, min slack 0, 1,732 tight pairs (FULL-CHECK.json). Run locally on Windows and again on Kaggle with the same result.
• scripts/research_package.py --package research/plane-port-global-rank-dual: PASS.
• Lean v4.21.0 + pinned Mathlib (lean/check.sh): V4.identity_block_needs_terms, V4.weak_duality, V4.decoder_cost_ge; axioms only propext, Classical.choice, Quot.sound. The finite feasibility check is not formalized.
• Search evidence (search/, scipy; EVIDENCE.md): no exact real decoder below 4 per plane on the 6 open triple classes of three-plane-rank-dual (all reach 12), 40 random open quadruples of four-plane-rank-dual (30 reach 16), the 60 most degenerate quadruples and 69 C5-based groups of 5-7 planes. Heuristic, not a proof; weak on 5-7 plane groups where the solver often stops above the baseline.
• Not run: the full repository test suite. Locally tests.test_research_package_diagnostics and tests.test_lean_sources pass; tests.test_open_pr_lean fails only on a Windows console encoding error in formal/open-prs/drift_A.py, unrelated to this package.

Credit

Contract and two-plane theorem: CubeJerry (#4365). Rank-dual functional: pinterbanget (Ryanson Jonathan; #4421 and the four-plane extension). 224 for 2-integral decoders: Philippe Van Velzen (Discussion 1587, #2532). Fitting rank 49: ywz65 (PR2471), Sagar Thakrar (PR2484). C5 results: Tekalign, Sagar Thakrar, Nick Petalas. Budgets: josie. Labels and matching: Cinna Davis (PR1825/2126). Prepared by Mark (Matarener) with substantial Claude Code assistance; no independent human review claimed.

Open next questions

In the package README: (1) does 224 hold over Q (first step: induction on the top 2-adic valuation); (2) can a functional stronger than the clique programme certify more than 164; (3) is there an exact rational cost-15 decoder on an open four-plane class.

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