[petalas]

Result

Exact finite certificate for a scoped lower bound. No new κ. Take the strict 112-port, 56-plane mask (PR1825/PR1889) with CubeJerry's additive full-plane COPY tariff (#4365). Every finite rank-one decoder of this mask, over any coefficient field, costs at least

36732955/262144 = 140.1251...,   hence  >= 141.

This raises the all-field floor of Mark (Matarener)'s merged #5560 (research/plane-port-global-rank-dual) from 140 to 141. It answers #5560's Open next question 2, "Can a stronger functional beat the clique programme?", with a better point of the same clique programme. The plane-port break-even window narrows from [140, 164] to [141, 164].

Claim type: an exact finite certificate (425 integer clique weights over 2^20). The mathematical argument is #5560's Lemmas 1-3 (identity block on a clique, weak duality, reduction to span pairs), reused unchanged with credit. The new contribution is the weight vector, the generator that found it and an independently written verifier. This is not a multiplication theorem, a supplier admission or a selected-frontier change.

Evidence

research/plane-port-clique-floor-141/verify.py was written separately from the generator. It checks in exact arithmetic:

• 56 nondegenerate planes from the pinned archive and matching, with 109 orthogonal pairs;
• an independent closure census by reduced echelon bases, giving 39,087 generated subspaces;
• the tariff dim A - dim(A meet R^perp), compared with Gaussian elimination on 4,000 random pairs;
• that every column is a clique with a positive integer weight;
• feasibility on all 1,527,793,569 ordered span pairs. The minimum slack is 0, and it occurs exactly on the 1,732 tariff-0 pairs (asserted);
• the recorded bound 36732955/262144 > 140;
• four rejecting controls: an overweight column, an orthogonal pair as a clique, a wrong recorded bound, and Prove a global all-field COPY floor 140 for the 56-plane port decoder (no new κ) #5560's own certificate, which is feasible but reaches only 139.11.

generate/ holds the row-and-column generation (scipy HiGHS master LP, local-search clique pricing, exhaustive separation every round) and the exact rounding step. The proof does not need it. On the machine that produced the certificate, a rerun from a clean cache reproduced CERTIFICATE.json byte for byte (14 rounds, about 6 minutes).

Reproduce

From the repository root, with CPython 3.12 and numpy 2.2.6 (the VERIFY.json pin):

python3.12 -B research/plane-port-clique-floor-141/verify.py      # full check, about 30 s; prints PASS_PLANE_PORT_CLIQUE_FLOOR_141
python3.12 -B scripts/research_package.py --package research/plane-port-clique-floor-141
python3.12 -B scripts/check_research_commit.py --package research/plane-port-clique-floor-141 --check-source-history

Local results: the registered replay passed (PASS_REGISTERED_RESEARCH_PACKAGE ... outcome=completed, 34 s wall time) under CPython 3.12.15 in a fresh venv with numpy 2.2.6. The commit check passed (PASS_RESEARCH_COMMIT_BYTES, commit 2e29f06). The direct verifier also passed under numpy 2.5.4. All of these ran on macOS (Apple silicon).

Limitations

• The optimum of the clique programme is not computed. Prove a global all-field COPY floor 140 for the 56-plane port decoder (no new κ) #5560's uncertified pricing estimate is about 167. Pricing here is a local search, so no upper bound is certified.
• The 224 conjecture over Q is untouched, and [141, 224] stays open in characteristic 0.
• Native realization, hosting, routing, the coverage law and concave pricing are outside the contract, as in Prove the sharp two-plane COPY floor for arbitrary scalar factorizations #4365 and Prove a global all-field COPY floor 140 for the 56-plane port decoder (no new κ) #5560.
• No Lean or other formal proof-kernel check of this certificate was run. Prove a global all-field COPY floor 140 for the 56-plane port decoder (no new κ) #5560's Lean lemmas cover only the abstract Lemmas 1-2. Independent human review is unclaimed.

Open next questions

1. What is the optimum of the clique programme, and is it above 164? The first step is to raise STOP in generate/colgen.py and add an exact MILP pricer, which would also certify an upper bound. Local-search pricing found fewer than ten new columns per round near 140.
2. Can non-clique columns (sharp C5 covers from c5-residual-weighted-cover, three- and four-plane rank duals) beat the clique programme? Each exhaustive separation pass takes about 15 seconds.
3. Does every decoder cost at least 224 over Q (Prove a global all-field COPY floor 140 for the 56-plane port decoder (no new κ) #5560's conjecture)? The first step is the 2-adic induction in Prove a global all-field COPY floor 140 for the 56-plane port decoder (no new κ) #5560 THEORY.md E2.

Related open work

• Prove plane-port clique savings need an odd cycle in both term graphs (no new κ) #5977 studies the same strict plane-port decoder from Prove a global all-field COPY floor 140 for the 56-plane port decoder (no new κ) #5560. It proves a necessary condition (an odd cycle in both term graphs) for a local clique saving. It computes no global floor, so there is no shared number to compare. The other PRs this account opened after the CLAIM (Spend the power-variation root budget on outside-domain roots: 391 → 2987 certified (no new κ) #5976, Prove four-term address-twist syzygies survive at m≥3 with residue rank ≤2 (no new κ) #5978, Prove exact Pauli paid-history minimum for arbitrary targets: X/Y/Z squares need 2 returns, not 3 (no new κ) #5979, Prove three Gaussian blocks need five internally restoring stages (no κ) #5981-Prove a fused two-root readout reaches 72 connector calls only at 30 fused reads (no new κ) #5990) study different objects.
• Construct rank-135 fitting matrix on 336 alternating-plane ports (no κ) #5950 studies a different port set: a 336-port alternating-plane fitting matrix. It is not the same object, and there is no shared number.

CI notes

Both hosted checks passed on commit 2e29f06 in run https://github.com/CrocSwap/integer-mult-bounds/actions/runs/40567317254: plan (5m41s) and replay (research/plane-port-clique-floor-141) (3m13s). The replay job runs the registered command, which is the full exhaustive check of all 39,087^2 ordered pairs, not a sample.

Credit

The clique programme, Lemmas 1-3 and the 140 certificate are Mark (Matarener)'s (#5560). The contract and two-plane theorem are CubeJerry's (#4365). The rank-dual functional is pinterbanget's (#4421). The 2-integral 224 bound is Philippe Van Velzen's (Discussion 1587, #2532). The budgets are josie's (#1808), and the labels and matching are Cinna Davis's (PR1825/2126). The new certificate, generator and verifier were produced by Nick Petalas.