[petalas]

Result

Scoped proof. No new κ. This sharpens spectral condition E3 of Mark (Matarener)'s merged PR #5560 (research/plane-port-global-rank-dual) for the strict plane-port decoder. Take a clique S of pairwise non-orthogonal planes. The cross-term sum N on S equals I minus the diagonal-term sum. Its off-diagonal block support therefore lies in both term graphs:

supp N ⊆ G_cross ∩ H_diag

E3's signature argument then gives the following in every characteristic other than 2. If the diagonal-term plane graph is bipartite, the clique costs at least 4|S|, whatever the cross terms and coefficients are. A saving therefore needs an odd cycle realized by both term types. On a 3-clique, diagonal terms must reach all three plane pairs.

Claim type: written scoped proof of a necessary condition, with exact finite corroboration. It is not a finite certificate for any class, a search result or a multiplication theorem.

Evidence

• research/plane-port-clique-odd-cycle/PROOF.md: the lemma and its proof. The bipartite step is E3's. The inclusion into the diagonal graph is new.
• verify.py, standard library and exact Fraction arithmetic:
  • authenticates the E3 text by SHA256;
  • builds 400 seeded decompositions of I_6 on three planes, with random multi-plane terms plus an exact rank factorization of the remainder;
  • checks supp N ⊆ G ∩ H and at least 6 - m diagonal terms on every decomposition;
  • on the 384 decompositions with bipartite supp N, checks dim ker(I+N) = m and 2m <= rank N;
  • checks a triangle control: three single-source cross terms with N = I - J on one coordinate per plane give rank N = 3 and m = 2. E3's inequality alone therefore admits a triangle, and the lemma forces diagonal terms on all six directed pairs.

The verifier uses explicit checks, not assert, so the -O run is meaningful. No sampled decomposition had 2m above its cross-term count, so that branch was only exercised by the explicit triangle control.

Reproduction, from the repository root (local interpreter ~/git/integer-mult-bounds/.venv/bin/python3.12, CPython 3.12):

python3.12 -B scripts/research_package.py --package research/plane-port-clique-odd-cycle
python3.12 -B scripts/check_research_commit.py --package research/plane-port-clique-odd-cycle --check-source-history
python3.12 -B research/plane-port-clique-odd-cycle/verify.py
python3.12 -O -B research/plane-port-clique-odd-cycle/verify.py

Local results: PASS_REGISTERED_RESEARCH_PACKAGE ... outcome=completed (under 1 s), PASS_RESEARCH_COMMIT_BYTES, and PASS_PLANE_PORT_CLIQUE_ODD_CYCLE from both direct runs.

Limitations

• This is a necessary condition only. It does not decide the 224 conjecture, the global floor 140, or any open three- or four-plane class.
• It does not price the multi-plane diagonal terms that a triangle forces.
• Native realization, hosting and κ are untouched.
• The random decompositions corroborate the proof; they are not a census.
• The proof is written and has not been checked in a proof kernel. Independent human review is unclaimed.

Open next questions

1. Price the forced diagonal terms on the open three-plane classes of three-plane-rank-dual, and compare their tariff excess with 2m - Σ t_j. First step: tabulate the cheapest diagonal support reaching each plane pair in each open class.
2. Directed refinement. Does a saving need a directed odd cycle?
3. Global use. Can the bipartite-diagonal exclusion act as a structural cut in a clique programme, rather than another LP column?

Related open work

• Raise the all-field plane-port decoder floor from 140 to 141 (no new κ) #5980 raises the all-field plane-port decoder floor from 140 to 141 with a stronger clique programme (Prove a global all-field COPY floor 140 for the 56-plane port decoder (no new κ) #5560's open question 2). Same decoder and contract, but a different question: it certifies a global numerical floor, while this package gives a structural necessary condition for a saving on one clique. This package computes no global floor, so no numbers are compared. The diagonal-graph exclusion could act as a structural cut in such a programme (open question 3 above); neither package depends on the other.

Credit

The contract is CubeJerry's (#4365). The floor 140, spectral condition E3 and the 224 conjecture come from Mark (Matarener)'s PR #5560. The three-plane classes come from Ryanson Jonathan's three-plane-rank-dual (#4421).

CI notes

All checks passed on head 7c49d1da1af141de374ec4c2267192b4d381a601: plan and replay (research/plane-port-clique-odd-cycle), run https://github.com/CrocSwap/integer-mult-bounds/actions/runs/40565760253.