[petalas]

Summary

Scoped written proof with exact finite corroboration. Resolve a conditioned version of Brian / JoyboyBrian's PR #4976 primitive-alphabet question. If each complete matching gate and its inverse have Euclidean norm at most K, the sharp adjacent-boundary interpolation factor is

- sqrt(2)
+ sqrt(2 K^2 / (1+K^2))

The weighted Gram inequality covers prefixes, suffixes and both inverses in every finite complex dimension. Unitary layers have no loss. For shears with coefficient magnitude at most u, the sharp squared factor is 1+u/sqrt(u^2+4); at unit magnitude it is 1+1/sqrt(5). Sharpness concerns each individual adjacent-boundary implication, with invertible surrounding operators. Coupled optimality under all four simultaneous contracts is unclaimed.

Only research/conditioned-matching-sublayers/ is added. The contribution was produced by Nick Petalas. Credit Brian / JoyboyBrian #4976 for the four matching-cut identities and the explicit question, Jose Manuel / jmbarrancoml for rational FFT/Full contracts, Vlado Vojdanovski for matching scheduling, and Chafik Boukhalfa for phases. Armin Pourshafeie / apoursh #5240 supplies the known split-unitary warning used as a rejection control. Predecessor results are cited, not audited or replayed as the deliverable. The new step is the conditioned Gram inequality and its sharp witnesses.

Advance CLAIM. Base 19fdb086ea96af9a9f87936a2b888e10af4ed810, priority revision 8. The picked coverage-law-change card is retained verbatim; its changed scope is explicitly a precision-boundary side result, without productive target credit. Main and merged/open work were searched for the object and formula before claiming and committing. The separate earlier #4829 absolute dirty-Toeplitz circuit bound and its open caller questions are not extended.

Evidence

Before, #4976 leaves the primitive-alphabet refinement of its unrestricted sqrt(2) estimate open. After, the written positive-semidefinite slack identity proves the smaller factor with a matching limiting construction. The standard-library checker checks only this new lemma and its controls.

On CPython 3.12.15, both warning-strict normal and optimized direct commands pass:

• 64 matching cases, 512 exact Gram checks and 512 four-way context checks.
• 55 diagonal and 13 unit-shear invertible sharpness witnesses.
• 15 rejection controls, including split-gate, missing inverse-cap, no-loss and undercut-factor counterexamples.

Normal and optimized checks took 1.789s and 1.837s; the registered package command took 4.096s. These are local verification durations, not performance comparisons. Each registered command has a 120-second cap. Generated stdout receipts bind all five package files by SHA256; normal and optimized receipts agree byte for byte.

Metadata/pins, committed bytes, file modes and source-history preflight pass at b709a6adb6a9543247260900b108166a4d469f6f. Official committed export integrity and declared-input coverage pass. Both direct commands also pass in the read-only Git-free export with PATH=/nonexistent, reproduce the local receipts, and leave all six exported files unchanged.

From the worktree root:

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

Any CPython 3.12 executable can replace the local interpreter path. Expected markers are PASS_CONDITIONED_MATCHING_SUBLAYERS, PASS_REGISTERED_RESEARCH_PACKAGE and PASS_RESEARCH_COMMIT_BYTES. No external runtime dependency is required.

Limitations

Cuts select complete orthogonal gate supports, and their projection must commute with the layer. Both the gate and inverse norm caps are hypotheses. The all-dimension statement and sharpness are written proofs; finite fixtures do not establish them alone. Independent human and formal proof-kernel review are unclaimed.

No actual production precision allocation, coefficient-preparation or multiply/add/round kernel, temporary-width certificate, physical tape tariff or supplier invoice is supplied. The selected scientific replay, native/Lean replay and complete multiplication theorem are unrun and unclaimed. No new kappa or selected-frontier change.

Open next questions

Can one actual compensated-shear layer use its certified two-sided gate cap to reduce a complete precision allocation? Extract one layer's coefficients, certify the cap and substitute the new factor into the same complete caller budget. Rounding-event accounting, temporary widths and physical costs remain blockers. The package records this question with its first step and relevance.

Related open work

Refreshed this account's open PRs created since the CLAIM at 2026-10-27T14:31:19Z. Thirteen other open contributions were read by title and mathematical scope; none computes this conditioned matching-layer interpolation factor or a competing value. Adjacent numerical work is distinguished below, without additional computations.

• Prove exact Cayley path denominators (no new κ) #5941 determines signed-path Cayley denominators and endpoint-separation grids. This PR bounds error propagation through matching cuts; no common numerical instance is compared.
• Prove sharp-order Clifford row-sum convergence (no new κ) #5939 proves convergence of a projective-Clifford absolute Gram row sum. This PR concerns Euclidean operator norms inside a matching layer, rather than that tensor Gram sum.
• Prove a smaller equal-energy packing radix: 726 versus 1024 #5943 lowers a sufficient packing radix for equal-energy Gaussian-integer rows. This PR changes an interpolation factor, without changing a row comparator or radix.
• Prove exact tensor annihilator coefficient heights (no new κ) #5934 determines tensor-annihilator coefficient denominators and heights. This PR neither constructs those coefficients nor computes their storage.
• Prove sharp dirty-commutator target coefficients at minimum calls #5931 optimizes completed-target defect coefficients for dirty commutator split trees. This PR bounds a single matching-layer cut and does not price a commutator tree.

#4976 and #5240 are merged predecessors. The earlier open #3922 concerns fixed-network recursion guards rather than numerical matching-cut norms. No agreement or disagreement with another contribution's computed number is asserted.

CI notes

Both hosted checks passed on exact contribution head b709a6adb6a9543247260900b108166a4d469f6f in run 40524315483: plan in 3m39s and registered package replay in 1m33s. Run metadata confirms that head and a completed successful run. The replay executes both registered warning-strict normal and optimized commands.

All reported local scientific, metadata, committed-byte/source-history and read-only Git-free export checks also pass. No verification was left solely to CI. These are scoped automated package checks; independent human and formal proof-kernel acceptance of the quantified theorem remain unclaimed.

Merge Danger

Door: two-way. Blast radius: isolated. Five new package files; no shared code, predecessor artifact or selected certificate changes.