[petalas]

Result

Scoped proof with exact finite corroboration. No new κ; the selected result, the selected suppliers and upstream/ are unchanged. This contribution was produced by Nick Petalas.

In the two-plane, four-message model of #3997, the plane-level signature of an invertible 4x4 matrix over any field depends only on two complementary 2-planes of K^4: the row spaces of its top and bottom port pairs. Sources test whether a plane lies in a coordinate hyperplane. Readers test whether a coordinate vector lies in the other plane. Three necessary conditions follow (C1 to C3). They leave exactly 439 of the 6,561 candidate signatures, and over Q every one of the 439 is realized by an explicit integer matrix with entries in {-1, 0, 1}. These 439 are exactly the signature codes that #3997 records in TERNARY-SCREEN.txt. So rational values outside {-1, 0, 1}, through exact cofactor cancellations, add no signature to the generic and ternary censuses. This answers the signature question left open in #3997's TASK.json.

Not new: the invoice consequence (minimum copy invoice 8 on all 1,540 plane pairs for every invertible basis) agrees with, and is weaker than, #4365. That merged package proves COPY mass 8 for any finite scalar factorization on two matched planes, over every field. This PR claims only the classification of signatures.

Claim type: scoped written proof (PROOF.md) with exact finite corroboration. It is not a complete multiplication theorem, a supplier, or a search failure.

Evidence

All counts come from research/two-plane-signature-classes/verify.py, which regenerates RESULT.json and WITNESSES.json and compares their bytes.

Check	Result
Candidate signatures	9^4 = 6,561
Pass C1, C2 and C3	439
Pass two of the three (without C1 / C2 / C3)	555 / 805 / 1,771
Equal to the 439 codes in #3997's TERNARY-SCREEN.txt	yes
Distinct planes spanned by two ternary vectors	362; all their complementary pairs give signatures in the 439
Integer witnesses, each checked by exact rational inversion	439
F_2: planes / complementary pairs / signatures	35 / 560 / 355
F_3: planes / complementary pairs / signatures	130 / 10,530 / 439
F_5: planes / complementary pairs / signatures	806 / 503,750 / 439
F_7 (opt-in --extended): planes / pairs / signatures	2,850 / 6,842,850 / 439
Plane rule vs direct matrix inversion	all pairs at F_2 and F_3; 8,258 at F_5 (every 61st); 6,863 at F_7 (every 997th)
Direct census of every matrix of GL_4(F_2)	20,160 matrices, the same 355 signatures
Rejection controls	11

The finite-field enumerations corroborate Lemma 1; the proof does not use them.

Reproduce

Interpreter: ~/git/integer-mult-bounds/.venv/bin/python3.12 (CPython 3.12.15). From the repository root, on head e68cb25dcc859efb03d616305f543fda2d90ece1:

PY=~/git/integer-mult-bounds/.venv/bin/python3.12
$PY -I -B research/two-plane-signature-classes/verify.py
$PY -I -O -B research/two-plane-signature-classes/verify.py
$PY -B scripts/research_package.py --package research/two-plane-signature-classes --check-only
$PY -B scripts/research_package.py --package research/two-plane-signature-classes
$PY -B scripts/check_research_commit.py --package research/two-plane-signature-classes --check-source-history
$PY -I -B research/two-plane-signature-classes/verify.py --extended   # opt-in, adds F_7

Local results: each verify.py run prints PASS_TWO_PLANE_SIGNATURE_CLASSES in about 1.4 s; the registered replay prints PASS_REGISTERED_RESEARCH_PACKAGE ... outcome=completed in about 3 s (both registered commands); the committed-bytes and source-history check prints PASS_RESEARCH_COMMIT_BYTES; the opt-in F_7 run takes about 13 s. Every registered check is far under the CI limit.

Limitations and unrun obligations

• Plane-level signatures only. The finer joint zero pattern of P and P^-1 (the cofactor strata of the original question) is not classified. It is not needed for the invoice.
• The invoice consequence relies on Research: screen generic rational joint-plane supports #3997's definition of the pricing (a function of the signature) and its recorded minimum; neither is recomputed here. The invoice floor itself is Prove the sharp two-plane COPY floor for arbitrary scalar factorizations #4365's theorem.
• Two planes of two ports and four messages. Three or more planes, overcomplete factorizations, coefficient scaling cost, physical routing and restoration are outside scope, as in Research: screen generic rational joint-plane supports #3997.
• The proof is a written argument, not machine-checked. No independent human or formal proof-kernel review is claimed.
• Over F_2 only 355 of the 439 signatures occur. "The realizable set is the 439" is a statement about Q and about the fields checked where all 439 occur (F_3, F_5, F_7); I did not determine for which other fields it holds.

Open next questions

Structured versions are in TASK.json.

1. Is the realizable signature set still the generic census for two planes of three ports (6x6)? First step: enumerate complementary 3-planes of F_2^6 by incidence data. Blocker: the generic 6x6 census has 2^36 supports.
2. Do three planes of two ports admit a signature outside the generic census? Prove the sharp two-plane COPY floor for arbitrary scalar factorizations #4365's first open question asks for a joint three-plane decoder with 49 closure-pair tariffs, which appear to be the 7 x 7 source and reader masks of three planes. First step: the same enumeration over complementary plane triples in F_2^6. Blocker: 49^6 candidates need symmetry reduction.
3. Which pairs (support of P, support of P^-1) occur? Only a port-level pricing model would see them. Blocker: no such model is written down.

Related open work

• No open PR studies the plane-level signatures of the two-plane 4x4 interface. None of this account's other open PRs opened since the claim (Link ten stored open questions to the merged packages that answer them (no κ) #6692 to Prove the eight-query mutable-source tariff is exactly 460 (no κ) #6696) studies this object.
• Research: screen generic rational joint-plane supports #3997 (merged): the source of the question; its generic and signed-ternary censuses give the 439 codes this package matches exactly.
• Research: screen joint binary bases across paired planes #3990 (merged): binary screen of the same interface; its 391 binary signatures are a subset of the 439.
• Prove the sharp two-plane COPY floor for arbitrary scalar factorizations #4365 (merged): proves the invoice floor 8 for arbitrary scalar factorizations. It agrees with the consequence stated here; this PR does not recompute its number.

CI notes

CI did not certify this head, and the cause is infrastructure, not the package. The plan job of the research-packages workflow (run 41719811223, plan job) was cancelled by the workflow's 10-minute timeout while still in its actions/checkout step. The job started 21:13:25Z, the checkout step ran 21:13:29Z to 21:23:25Z, and the job completed 21:23:27Z. The replay job was skipped. Head: e68cb25dcc859efb03d616305f543fda2d90ece1.

Other recent PRs hit the same job. Of the 19 PRs from #6676 to #6696 other than this one that I read, 11 show plan cancelled after about 10 minutes with replay skipped (#6676 to #6679, #6681 to #6685, #6689, #6692), 6 plan jobs succeeded (#6680, #6686, #6687, #6688, #6693, #6694) and 2 were still running (#6695, #6696). I confirmed the cancelled step only for this PR's job.

This result is therefore certified only by the local checks listed under Reproduce: both verify.py commands, the registered package replay, the metadata check and the committed-bytes and source-history check. I make no hosted-CI claim, and I have not re-run the workflow or pushed to retrigger it.

Merge danger

Door: two-way. One isolated new package under research/two-plane-signature-classes/; no shared code, no change to upstream/, no selected-frontier update requested.

Blast radius: package-local.