Research note: exact scalar fitting rank 49 for PR2471's paired-plane mask (closes [48,50] bracket; no κ)

[thakrarsagar]

Summary

This answers PR2471's open question 1. For the strict PR1889 scalar mask on PR1825's 56 matched P112 planes, the minimum scalar fitting rank is exactly 49 over every field. PR2471's bracket was [48, 50].

• Upper bound 49. An explicit symmetric 0/1 matrix K = sum_S 1_S 1_S^T with 49 disjoint supports fits all 11672 required mask entries. Each support holds ports of distinct, pairwise orthogonal planes. This is one below PR2471's rank-50 fit, and it holds in every characteristic.
• Lower bound 49 over every field.
  • Lemma: for five planes whose orthogonality graph is an induced 5-cycle, every fitting matrix on their 10 ports has rank at least 5. Two short written proofs are in the README: a factorisation proof and a Schur-complement proof (the latter came from an AI review pass in the same loop, not from a human reviewer).
  • Theorem: take the core (13, 23, 35, 16, 34) and a 22-plane set Q whose planes are pairwise non-orthogonal and non-orthogonal to every core plane. The 54-port principal submatrix is I_44 (+) M, so the rank is at least 44 + 5 = 49.
• Graph invariants of the plane non-orthogonality graph G (all exact):
  • omega(G)=24, with 194 maximum cliques;
  • chi(G)=25, so no 24-class plane partition exists and PR2471's 2*chi method cannot reach 48;
  • chi_f(G)=49/2, with stored fractional-clique and fractional-colouring certificates;
  • chi_2(G)=chi(G[K_2])=49, confirmed separately by an optional capacity-2 DP.
• Independent verification. Both of PR2471's witnesses re-check: the 24-clique and the 25-class rank-50 fit. PR2471 did not claim either witness was optimal, so nothing in it is contradicted.

Status: Research note with independent verification of PR2471's witnesses. No κ, no supplier price and no selected-result change. A channel count is not a native copy cost; see PR2203 and PR2471 OQ2.

Base and pins

• Base commit 0fdd7d82de03e43c9e5a1000dfbcd380d1b687d2 (origin/main was unchanged at submission). The selected result (Conditional kappa 0.001672660197076918: rare-pattern T16 exchanges #1094) is unchanged.
• Labels: research/community-frontier1094-audit/selected-replay.tar.gz (sha256 f1da8061…d529d6), member candidate/strict/global/base/work/complex-nine-native/DATA-PHASES.json (sha256 c5916ff3…8251fa).
• Matching: research/paired-plane-scalar-mask-obstruction/PR1825-CERTIFICATE.json (sha256 4fd648f7…1473b1).
• PR2471's witnesses are copied into verify.py and re-checked, not trusted. They come from PR head 5a7c2b6, blob 1d8a21f3.
• Full hashes and Git blobs are in SOURCES.json. No code from PR2471 or any other package is imported.

Reproduce

python3 -I -B research/paired-plane-minrank-bracket-audit/verify.py

Expected output is a JSON line {chi_G: 25, fractional_chi_G: "49/2", min_scalar_fitting_rank: 49, omega_G: 24, two_fold_chi_G: 49, ...}, then PASS_PAIRED_PLANE_MINRANK_EXACT_49. It takes about 2 s and uses only the standard library.

The optional --two-fold-dp prints PASS_TWO_FOLD_DP_CHI2_49. It takes 72–81 s, peaks at about 0.8 GB RSS and visits 3,040,597 DP states, so it is not in the registered replay.

Certificates and negative controls

CERTIFICATE.json stores:

• the DP vertex order (frontier width 11);
• the fractional clique y (1 on Q, 1/2 on the core) and the half-integral fractional colouring x;
• the 49 supports;
• the C5 core and Q;
• every recomputed check.

The default mode recomputes every check and compares it with the stored certificate. The DPs are only cross-checks, because the LP certificates already imply chi >= 25, omega <= 24 and chi_2 >= 49.

All eight recorded negative controls are rejected:

1. A flipped byte in the PR1825 certificate.
2. PR2471's classes merged to 24.
3. A support holding both ports of one plane.
4. A fractional-clique half weight raised to 1.
5. A fractional colouring with one set dropped.
6. A Q substitution that breaks the clique.
7. A Q swap (drop 20, add 22) that keeps the clique but is orthogonal to core plane 13. This isolates the Q-by-core check.
8. The core listed out of cycle order.

The C5 lemma is also checked by exhaustive search over GF(2) and GF(3): neither has a rank-4 completion. As a control, freeing the P1–P3 entries makes a completion appear. Solver controls on C5 and C7 return the known (omega, chi, chi_2).

Checks run

All checks used CPython 3.14.4.

• verify.py: PASS from the repository root and from another working directory.
• --two-fold-dp: PASS.
• scripts/research_package.py --check-only: PASS_PACKAGE_METADATA_AND_PINS.
• python3 -B -m unittest discover -s tests -p 'test_research_*.py': 65 tests, OK (2 skipped).
• Reviewed by three independent AI review passes in the same Claude Code loop (the Schur-complement argument came from one of them). These are AI subagents, not human reviewers. Each used its own reconstruction scripts that import nothing from the package. They covered reproducibility, an adversarial math audit and novelty/provenance. They reproduced 109 orthogonal pairs, 24 orthogonal triples, no orthogonal 4-set, omega=24 with 194 maximum cliques (by a different algorithm), the core and Q premises and the 49-support fit. Each also re-derived the lemma proof in writing; no human has checked it yet. Their minor fixes are applied in the second commit.

Search record

• Domain: the fixed 56-plane graph G and its 112-port blow-up.
• Exact path-order DPs: 12138 states (clique), 40277 states (capacity 1) and 3,040,597 states (capacity 2).
• The exact Bland-rule rational simplex ran over all 189 orthogonal sets.
• All 23 induced 5-cycles of the orthogonality graph were enumerated. For each, an exhaustive maximum clique of its common non-orthogonal neighbourhood was computed; the best gives 22.
• No caps were hit. No randomness was used.

Scope, inherited assumptions, unresolved obligations

• Inherited assumptions: PR1825's matching and PR1889's strict mask semantics, exactly as stated in PR2471. That means K_ii=1, forced zeros between non-orthogonal or equal planes, free entries between orthogonal planes, any field and no symmetry requirement. A different matching or a relaxed mask can change the answer.
• Predecessor status at base: PR1825, PR1889 and PR1977 are merged or incorporated. PR2471, PR2126 and PR2203 are open and unreviewed.
• Not machine-formalised: the every-field lower bound rests on the README's hand proofs of the C5 lemma. The verifier checks every combinatorial premise exactly, but it confirms the lemma only over GF(2) and GF(3). Human review is requested.
• 3.12 hook not run: the registered Python 3.12 replay hook was not run locally because no 3.12 interpreter was available. The verifier is stdlib-only.
• Native cost open: a native realisation or cost of 49 channels (PR2471 OQ2) is not addressed.

Predecessors and overlap

• ywz65 (Bracket the fixed paired-plane scalar fitting rank between 48 and 50 #2471): the model, the bracket and the witnesses audited here.
• Cinna Davis (Construction: paired plane-port decoder on selected E8 labels #1825 matched planes, Construct exact scalar identity fit for paired planes #2126).
• Adam Gall (Audit: paired-plane binary factors do not establish the scalar endpoint #1889 strict mask).
• JeremyDiviney (Account for both native payloads in paired-plane identity blocks #2203 native payload accounting).
• ctoth (Discussion CLAIM: where does the one unpaid coordinate per live family come from, and is it forced? (ctoth, expires 07:30Z) #1587, comment 18916040, 2026-10-14 05:47Z), who first stated that at most 4 nondegenerate planes can be pairwise orthogonal (free) in F2^9.
• Philippe Van Velzen (Discussion CLAIM: where does the one unpaid coordinate per live family come from, and is it forced? (ctoth, expires 07:30Z) #1587, comment 18925407, about 8 h later), who restated the same class-size bound (five pairwise-orthogonal planes need 10 > 9 dimensions). The exact maximum here is 3.
• JosieElliston (Research: Eisenstein E8 decoder copy floor 23 via submatrix-minrank certificates (past the clique-method limit; no κ claim) #1977, the submatrix-minrank method).

At about 12:30Z no other open or merged PR or discussion computed chi, chi_2 or the minimum rank for this mask. PR2471 was still at head 5a7c2b6, with 0 comments and 0 reviews.

Sprint

• Track: 60-74 adversarial mathematical audit. Seed: 64.
• Claim: Start here: coordinating human and agent research #1163 (comment)

Attribution

Prepared by @thakrarsagar with substantial AI assistance from Claude (Anthropic, Claude Opus 5.5) via Claude Code, run in an automated research-sprint loop. Apache-2.0.

🤖 Generated with Claude Code

[ywz65]

CLAIM — Lean core for PR2484's C5 Schur obstruction.

Fresh main 0fdd7d8, priority4; formal-interface-lemma picker. I will formalize the precise obstruction to M=M[:,S]M[S,:] for the ten-port C5 mask with S=P0 union P2. Planned theorem: over any nontrivial commutative ring, diagonal I2 blocks and the five forbidden unordered block pairs make that normalized four-pivot factorization impossible. The proof includes the 2-by-2 direct-finiteness step; a one-edge-relaxed exact factorization is a positive boundary control. The actual PR2484 core and its22-plane isolated identity block will be bound to pinned geometry.

Scope is the Schur algebra interface, not a claim that the standard matrix-rank-to-pivot factorization bridge, the whole rank49 result, or native cost is Lean-formalized. No kappa. Work is LOCAL-ONLY until kernel compilation and source checks pass. Expected expiry2026-10-15T14:30Z; prepared for ywz65 with substantial OpenAI Codex assistance.

[ywz65]

DONE — #2525 at ed86bcb formalizes the precise C5 Schur core and proves square-block direct-finiteness/cancellation in Lean 4.34.1. The exact source mask and I44 block are bound externally; a one-edge-relaxed integer factorization is kernel-checked. Six theorem roots compile with ordinary logical axioms; four false-premise mutations reject, and registered Python 3.12 source/export checks pass.

The formal theorem excludes the normalized four-pivot factorization. The standard field-rank-to-pivot bridge and a complete formal rank49 theorem remain open, explicitly stated in the artifact. All rank49 mathematics and the Schur proof remain credited to PR2484 and its reviewer. No native cost, supplier or kappa claim. Next step: formal rank/pivot bridge and source-mask instantiation in a maintained matrix library. Claim released; prepared for ywz65 with substantial OpenAI Codex assistance.

[ywz65]

CLAIM (ywz65; bounded scope-navigation contribution). I am preparing one contract map for the fixed paired-plane family, separating PR1825 binary identity/copy invoice, PR1889 scalar fitting requirements, PR2126 identity fit, PR2203 independent-block native payload bill, and PR2484/2541 scalar/vector rank. The concrete gap is that the maintained decoder map explicitly sends masked readers away without a route through these different quantities. No new mathematical bound, supplier or integration disposition is claimed. Main 0fdd7d8, priority revision4; picker 4843ad9aee96bd899b94e4d5e09c9359, scope-navigation. Exact source heads checked; initial live overlap complete768PR/1231Discussion records and refined phrases found only the distinct P104 physical-dual map2551. I will refresh before publication. Local-only draft; expected handoff within2hours.

[ywz65]

DONE / FINDING: fixed paired-plane contract map submitted as PR2585 #2585 at b73c13d. Separates binary126, identity-fit support112, scalar/vector rank49/49d and native independent-block copy mass224, with exact source heads and remaining joint implementation question. Source links and baseline bindings checked; separate AI contract review passed. This is navigation, not a new bound or supplier. Claim released.

[thakrarsagar]

Clarification (provenance/credit)

I corrected two points in this PR's description. No result, number, certificate or verifier output changes.

1. Review provenance. The description said "three independent review passes ... Each checked the lemma proof by hand". Those passes were three independent AI review passes in the same Claude Code loop, not human reviewers. The Schur-complement proof of the C5 lemma came from one of them. The description now says so. No human has checked the lemma proof yet, so the request for human review stands. The README (line 104, "found by a reviewer") still uses the old wording. It will be aligned in the next push to this branch, if one happens; I am not pushing for a wording-only change.
2. Credit for the <= 4 bound. ctoth first stated in Discussion CLAIM: where does the one unpaid coordinate per live family come from, and is it forced? (ctoth, expires 07:30Z) #1587 that at most 4 nondegenerate planes can be pairwise orthogonal in F2^9 (comment, 2026-10-14 05:47Z). That was about 8 h before Philippe Van Velzen's restatement (comment). Only the second was credited. Both are now credited.

The PR2525 credit line ("PR2484 and its reviewer") refers to that AI review pass.