[petalas]

Summary

Scoped computer-assisted proof, produced by Nick Petalas. Every real or rational invertible 2-by-2 basis change in Cinna Davis's PR2042 paired-Walsh ten-term P112 family preserves the exact profile {8:8,9:2} and invoice 82. This closes the continuous basis orbit, extending the five-basis result without replaying its verifier. No new kappa is claimed.

56 ordered U planes x 56 ordered V planes
  -> four exceptional projective support directions + one generic stratum
  -> every added term has rank 9; every original term has rank 8
  -> 8*8^s + 2*9^s for every real exponent s

Only research/p112-walsh-gl2-strata/ is added. The exact algebraic identity, exhaustive support partition, finite rank obligations and cost boundaries are in PROOF.md. The method of projective support strata is credited to Gonzalo Mier's related PR4020; the new theorem concerns the different paired-Walsh family. Credit Cinna Davis, DreamingOfClouds, Mihir Srivastava, mightyOil and the retained P112 lineage.

Advance claim. Canonical main was fetched again before submission and remains 721c03822c421b7f7fcbb2598993b947c4d88419, priority revision 8. Exact provenance and source hashes are retained. Sparse local overlap coverage is incomplete; live open-PR/Discussion metadata and targeted merged/main searches were also checked.

Evidence

Before, PR2042 covers five basis matrices. After, the written support partition covers all GL2(R), including irrational slopes and arbitrary rational denominators. The complete finite certificate checks eight coordinate-reader spans, 280 lift/support cases with both source and reader ranks, and 25,088 corrected source-column spans. The support digest is 969fcec18477f9b7744e527669d96a51772d168b146f3e143777f40b76bd8d29.

Local CPython 3.12.15, macOS arm64, standard library only:

/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -I -W error -B research/p112-walsh-gl2-strata/verify.py
/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -I -O -W error -B research/p112-walsh-gl2-strata/verify.py
/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -B scripts/research_package.py --package research/p112-walsh-gl2-strata
/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -B scripts/check_research_commit.py --package research/p112-walsh-gl2-strata --check-source-history
/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -B scripts/export_research_package.py export --package research/p112-walsh-gl2-strata --ref HEAD --output /tmp/walsh-gl2.tar
/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -B scripts/export_research_package.py check --check-inputs /tmp/walsh-gl2.tar

Substitute another CPython 3.12 executable on other machines. Normal and optimized direct checks pass in about 1.8 seconds each, well below their 120-second registered limits. Both registered commands pass, as do committed-byte/source-history checks, official export integrity, and normal/optimized Git-free replay from read-only exported files with unchanged bytes. A private-copy corrupted geometry source is rejected before import. Literal endpoints and six domain/semantic controls also pass. The printed receipt is generated from the exact checks and bound to package hashes.

Limitations

The plane choices remain Walsh pairs and the rank-two product is fixed during each gauge. Changed planes/products, mixing the added channels with the original eight, masked endpoints and changed tariffs remain outside the theorem. Preparation, frames, source access, restoration, native delivery, hosting, precision, stock, compiler and outer costs are unpriced. This is neither a search-failure claim nor a complete multiplication theorem. No full selected replay, Lean, Docker, independent human review or formal-kernel verification is claimed. No selected-frontier change.

Open next questions

Can a different rational overlapping plane product improve the ten-term profile? First construct the complete support arrangement for one explicit changed pair and price all ten terms. A favorable example and its complete physical source/frame/restoration bill are missing. Further basis searches in the present fixed planes cannot help.

Related open work

Related open work was refreshed after CI by reading this account's open PRs created since the advance claim. No newly opened PR studies this P112 Walsh-plane basis orbit or computes its invoice. The previously listed broader P112 work remains separate:

• Checkpoint symmetric two-star P112 supports and inconclusive COPY63 search #5088 studies a two-star masked support dictionary and bounded search; this theorem keeps the exact all-pairs Walsh product.
• Checkpoint eight masked-decoder basis searches (exploratory; no new K) #5064 changes off-mask completions and nine-factor bases; this theorem fixes a ten-term all-pairs family.
• Screen P112 tetrad coefficient fusion under relaxed rank ceilings #3604 studies tetrad coefficient co-delivery and its consumer costs; this theorem has no physical consumer.
• One-sided padding lemma: P112 to P120 at cost exactly +8 (L120 suffices for Delta >= n_nf) #3074 transfers one-sided decoder costs by padding; this theorem determines one fixed overcomplete orbit.
• Copied-information floor of the selected centre path #3120 studies copied-information demand in a retained centre consumer; this theorem prices its own support factors.
• Add selected centre decoder demand floor audit #3918 binds a retained provider's demand rank; this theorem has no claim about that provider's implementation.
• Add exact dirty-input coefficient audit for P112 tetrad block #3916 checks a tetrad dirty-input coefficient contract; the present rank certificate supplies no dirty-state realization.
• C(P112) >= 64: complete reduction to stated conjectures #3154 reduces general P112 decoder floors to conjectures; this theorem is a narrower exact family classification.
• Research: one-sided residue closures — Circuit Lemma, relative root counting and plane lemma close 480 of the 603 d>=4 residues (116 open; no kappa claim) #2985 studies one-sided residue closures; the present result changes neither its residue obligations nor its claimed counts.
• Two-sided deficiency budget: Theorem BX for b_Q <= 4 (conditional frame, exact residue, no κ claim) #3029 studies two-sided deficiency budgets; the present fixed-family theorem does not settle those general budgets.
• Bind P112 tetrad decoder to selected ports; no kappa change #3566 maps a tetrad decoder to selected ports; the present Walsh factorization has a different interface.
• Non-dense fitting B: every E8 term with x·y ≠ 0 has rank ≥ 9; square root-code floors E8 ≥ 72, P112 ≥ 64, P104 ≥ 56, D4 ≥ 12 for any fitting B (no κ claim) #3045 studies square root-code floors for non-dense fitting tables; this theorem fixes the dense pairing and ten terms.
• Decoder floors: universality, projection transfer and batching reductions; C(P112) ≥ 64 reduced to named conjectures, C(P120) ≥ 72 restated on N₀ (no κ claim) #2832 studies general decoder-floor reductions; this theorem does not resolve its general conjectures.
• perf(research): compile the P112 ten-deletion census with popcnt (2.69x, byte-identical receipts) #4090 speeds a P112 deletion census; this theorem does not change that census or its mathematical coverage.
• Tooling: 3.09× faster production frame-pair enumeration via eight-bit neighborhoods with identical receipts #2375 speeds production frame-pair enumeration; this package uses a separate small rank certificate and leaves production unchanged.

CI notes

Uncertified by CI at 017c9d467c8d3878be320373b8cf992b33c52ee1. In run 40724607013, plan was cancelled by the job's timeout-minutes: 10 limit. It ran 16:18:09–16:28:25 UTC and was still in "Onboarding helper tests" when cancelled. The package replay job was skipped; the package did not run in CI. These outcomes do not certify or refute the local mathematical checks.

Separately, even without that timeout, the unchanged main base 721c03822c421b7f7fcbb2598993b947c4d88419 would fail two onboarding helper tests. The PR6030 admission changed the selected record without the matching documentation updates:

• CurrentStatusTests.test_current_section_matches_selected_record expects **0.003125804** in docs/research/current-status.md.
• CurrentCiNavigationTests.test_current_ci_section_matches_selected_record expects make frontier6030-verify in docs/ci.md.

The failed logs for PR6150 run 40724809789 and another contributor's PR6120 run 40691776027 confirm exactly those two failures. PR6050 targets the shared navigation repair. No documentation fix, push or CI rerun is included here.

The local scientific, committed-byte/source-history and portable-export checks above passed. This PR needs a maintainer rerun after main is fixed before hosted package replay can be certified.

Merge Danger

Door: two-way. The contribution is an isolated research package.

Blast Radius: package. No shared code, source archive or selected result changes.