[petalas]

Summary

Scoped algebraic proof with exact finite corroboration. This determines the exact least common denominator of every signed-path Cayley matrix and sharpens Ali Romman's #5339 endpoint-separation bound. For positive coprime a,b,

D_0=1, D_1=b, D_n=b D_(n-1)+a^2 D_(n-2)
den((bI-aK)^(-1)(bI+aK)) = D_n/gcd(D_n,2b).

Signed edges preserve the result, inverses have the same denominator, and disconnected label runs combine by lcm. A far-corner cofactor proves minimality. Substituting exact per-call denominators into the existing funded error implication yields a smaller sufficient grid. In the specified nine-dimensional, eight-call alternating-label example:

- 318 fractional bits from the coarse determinant bound
+ 108 fractional bits from exact label denominators

The reduction is exactly 210 bits for that example. It is not an optimal-precision or runtime claim. No new kappa or selected-frontier change.

The contribution was produced by Nick Petalas. Credit Ali Romman #5339 for the separation/error contract and optimization question, #5315 for rational orthogonal witnesses, Krzysztof Wojt #5262 for the inverse compiler/shortest-word theorem, zxv #5235 and the retained dirty-echo/tensor/phase and rational-series lineage. Cayley, continuant, adjugate and integer-lattice methods are classical; global literature priority is unclaimed. Predecessor scientific results are cited as premises, not audited or replayed.

Advance CLAIM. Canonical base and refreshed main: 19fdb086ea96af9a9f87936a2b888e10af4ed810, priority revision 8. Source #5339 at 26fe40cbca6972a0e392e9248a9cb8a84368530b matches both pinned source documents byte for byte. The original card and explicit change to an arithmetic precision side result are retained in TASK.json.

Evidence

Before: #5339 uses C(3B)^(dq) as a numeric denominator upper bound. After: H=C product_j Q_(label_j) uses the exact matrix denominators and never exceeds that coarse bound. The complete register-lattice induction preserves every dirty role, rational coefficient and old-value alias.

Local CPython 3.12.15 standard-library checks pass:

• 540 connected matrices, with whole-matrix and far-corner reduced denominators checked by dense rational inversion.
• All 1,092 signed-support matrices in the declared domains, including independent exact inverse/transpose checks.
• 42 register programs on all 420 independent input columns, including linear assignments, true inverses and aliased queries.
• Both dyadic grids in the strict example, with every returned coordinate inside the same error allowance, and 12 rejection/boundary controls.

Both warning-strict normal and optimized registered commands pass. Direct checks took 0.70s and 0.66s locally, and the registered pair took 1.48s. These are verification durations, not performance comparisons. Each registered command has a 120-second cap. Generated stdout receipts bind all seven package files.

Committed bytes, executable modes and source-history preflight pass at 82b99d923adbef5a0f3b03d48130fb6a2c4ff2f5. Official export integrity and both declared input checks pass. Both read-only Git-free exported replays with empty PATH pass, in 0.55s each. An actual changed-source control rejects before mathematical execution.

Exact reproduction from the worktree root:

/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -I -W error -B research/cayley-path-exact-denominator/verify.py
/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -I -O -W error -B research/cayley-path-exact-denominator/verify.py
/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -B scripts/research_package.py --package research/cayley-path-exact-denominator
/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -B scripts/check_research_commit.py --package research/cayley-path-exact-denominator --check-source-history --check-modes

Any CPython 3.12 path can replace the local interpreter. Expected markers: PASS_CAYLEY_PATH_EXACT_DENOMINATOR, PASS_REGISTERED_RESEARCH_PACKAGE, and PASS_RESEARCH_COMMIT_BYTES.

Limitations

The all-path result is a written proof; finite fixtures do not prove generality by themselves. Matrix denominator minimality does not imply that the composed-word product H or the resulting grid is optimal. The general error implication remains a credited #5339 premise. Independent human and formal proof-kernel review are unclaimed.

No predecessor scientific census, actual native-compatible witness, general optimized detector, physical cost, coefficient producer, selected cold replay, compiler/ownership/tape/stock/recovery admission, Lean or Docker check was run. No selected assumption or complete multiplication theorem is discharged. Only the isolated new package changes.

Open next questions

• Can a specified composed register word use cancellation to reduce H further? Begin with a literal inverse-pair word and retain all scalar factors and dirty returns.
• Can an actual native-compatible witness use this refinement in a fully paid precision contract? Pin its operator relations and caller before applying the bound; physical execution and costs remain open.

Related open work

The closeout refresh lists 14 open account PRs created since the CLAIM at 2026-10-27T14:36:05Z, including this contribution. The 13 other titles and stated mathematical scopes supply no competing value for this skew-path Cayley matrix family or the #5339 precision bound. No additional mathematical computations were run for comparison. The adjacent denominator and precision work is distinguished below. Earlier sparse-local coverage limitations remain recorded in TASK.json.

• Prove exact tensor annihilator coefficient heights (no new κ) #5934 studies exact Gaussian tensor-annihilator coefficient denominators and heights. This result concerns rational path-matrix denominators; no common value is computed or compared.
• Prove sharp conditioned matching-layer interpolation (no new κ) #5940 studies conditioned matching-layer norm interpolation. That changes an error-propagation factor; this result changes the endpoint-separation lattice. No common numerical fixture is compared.

CI notes

Both hosted checks passed at exact contribution head 82b99d923adbef5a0f3b03d48130fb6a2c4ff2f5 in run 40524880909: plan in 4m32s and registered package replay in 1m26s. Run metadata confirms this head and a completed successful run. The replay executes both registered warning-strict normal and optimized commands.

All reported mathematical, metadata, committed-byte/source-history and read-only Git-free export checks also passed locally. No required check was left solely to CI. Hosted replay certifies the bounded registered checks; independent human and formal proof-kernel review of the written theorem remain unclaimed.

Merge danger

Door: two-way. Blast radius: isolated. Seven new research-package files with opt-in replay; shared code, predecessor artifacts and selected evidence are unchanged.