[petalas]

Summary

Scoped written mathematical proof with exact finite corroboration. Under the historical Gaussian caller's unchanged assumptions ||H||_2, ||H^-1||_2 <= p^10 and physical bandwidth W<p, replace its Neumann locality witness by an explicit Chebyshev polynomial:

-residual degree p^42; conservative halo p^48; error <= p^10 exp(-p^2)
+residual degree p^22; conservative halo p^24; error <= 2p^10 exp(-2p^2)

The new error is strictly below the old majorant for every integer p>=2. The proof gives exact principal-section row agreement beyond (2p^22-1)W, the ordered error identity for nonnormal matrices, and an integer recurrence with polynomial coefficient bit lengths. Every inverted section retains its original conditioning hypothesis.

This contribution was produced by Nick Petalas. Credit Cinna Davis's #4100 caller, Nick Petalas's #4217 locality distinction, Mihir Srivastava's #4501 nonnormal/section contract, and their retained Gaussian/framework predecessors. Chebyshev approximation and approximation-based inverse decay are classical; DLMF 18.5.1 and Demko, Moss and Smith (1984) are cited. The new contribution is the explicit parameter replacement at this interface. No new general inverse-decay theorem or global literature priority is claimed.

Evidence

• Before: the pinned HALO witness uses residual degree p^42 and halo p^48 under these norm assumptions.
• After: the written proof gives degree p^22 and halo p^24. Exact normal and optimized checks each pass 3,696 scalar samples, 45 nonnormal operator fixtures, 33 full section rows, six too-small-halo rejections, six invalid-input rejections and 127 parameter substitutions.
• Local CPython 3.12.15 direct checks take 0.34 and 0.33 seconds respectively. Both registered VERIFY commands pass, each far below its 120-second bound. These are check durations, not runtime speedup claims.
• Metadata and committed-byte/source-history preflight pass, including nine package/declared-input files. The exact stdout receipt binds all six package files by SHA256. No predecessor scientific suite was replayed.

From the repository root:

/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -W error -X dev -I -B research/chebyshev-inverse-locality/verify.py
/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -W error -X dev -I -O -B research/chebyshev-inverse-locality/verify.py
/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -B scripts/research_package.py --package research/chebyshev-inverse-locality
/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -B scripts/check_research_commit.py --package research/chebyshev-inverse-locality --check-source-history

Another CPython 3.12 interpreter may replace the shown path. Direct replay prints PASS_CHEBYSHEV_INVERSE_LOCALITY; registered replay prints PASS_REGISTERED_RESEARCH_PACKAGE. The package is standard-library-only and its verifier writes no files.

Limitations

This is a component improvement under retained conditioning premises, not their proof. It supplies no optimized core geometry, complete physical coefficient-page cost, rounding/stability analysis, native supplier admission, new kappa, rounded multiplier or complete multiplication theorem. All five selected assumptions remain. The finite matrix fixtures use reduced parameters; the universal quantifiers rest on the written proof. Independent human and formal proof-kernel review are unclaimed. Selected cold replay, Lean and Docker are unrun. Ruff is unavailable locally; warning-strict normal and optimized execution passes.

Open next questions

Can a complete coefficient-page producer realize this polynomial with a favorable rounding and physical setup bill? Start with one exact page at a named caller section, charging coefficient widths, recurrence intermediates and output error. Missing rounding, page implementation and consumer costs block physical adoption. No further search is included.

Related open work

• Conditional κ = 0.001679745508802299: gauge-free (fractional-shift) interiors, structured boundary solves and radix-distributed sparse-union schedule on PR1673's capped phase-cell fusion outer, pivot-566+554 T74 supplier #1758 studies historical gauge-free Gaussian interiors and structured boundary solves. This adds a conditional inverse-locality witness only; it neither retunes nor reviews that construction.
• Prove three Gaussian blocks need five internally restoring stages (no κ) #5981 studies internally restored moving-Gaussian stages. Its stage-count endpoint is separate from this numerical inverse polynomial and halo radius.
• Prove sharp 7TP table travel for skew-radix cached chirps (no new κ) #5987 studies skew-radix cached-chirp table travel. Its layout and head-movement tariff do not compute this inverse-error or locality quantity.
• Construct a corrected two-guard reciprocal (no new κ) #5989 constructs a rounded scalar reciprocal with two guard bits. Its scalar Newton schedule and rounding residual differ from this exact matrix Chebyshev polynomial; no numerical comparison is claimed.

The account's open PRs created since the advance CLAIM were refreshed: #5981, #5982, #5983, #5984, #5985, #5987, #5988, #5989, #5990, #5992. The remaining entries concern Walsh tape travel, G20 helper splitting and carries, the outer ledger, rank-one call floors, fused connector readouts, or pooled carrier moment ceilings. None computes the same inverse-locality polynomial or the same norm/error contract. No additional mathematical computation was run to compare their results. Main and merged-work searches were also checked; sparse cached local-note coverage remains incomplete.

Related merged work: #5145 optimizes the fixed scalar Neumann tail 2(1-g)^J/g < 2^(-16p) with g=1/(16p) for #3851's normalized Gaussian core. This PR instead constructs a Chebyshev polynomial on H*H under #4100's ||H||_2, ||H^-1||_2 <= p^10 assumptions and targets the operator error p^10 exp(-p^2). It neither optimizes nor extends #5145's scalar tail or caller. The hypotheses and error budgets differ, so no numerical comparison of truncation lengths is claimed.

CI notes

Both hosted checks passed at exact head 31bbc7b22ea697437660a0af4c006e114b90710b in run 40570704729: plan and replay (research/chebyshev-inverse-locality). The check list was read once and the completed run's head matches the published PR. Hosted replay certified the registered normal and optimized bounded checks. All package commands and the committed-byte/source-history preflight also passed locally. This does not establish independent acceptance of the written proof, conditioning premises or physical caller.

Advance claim: #1166 (comment)

Merge Danger

Door: two-way. This adds one isolated research package and its opt-in replay manifest.

Blast Radius: package. Shared implementations, frozen evidence and the selected record are unchanged.