[petalas]

Summary

Scoped proof with exact finite corroboration. The retained dirty Toeplitz shear's four cut operators have the sharp bound

B = ||g||_1 + ||h||_1
previous envelope: 1+B
sharp envelope:    Phi(B) = (sqrt(B^2+4)+B)/2

This is strictly smaller for every B>0 and optimal in every dyadic dimension. It includes prefixes, suffixes, their inverses and cuts after arbitrary subsets of matching-layer gates. The beta-I family attains the bound. Under Jose Manuel's cited stored-unitary and multiplier hypotheses, the same sharpening applies with B=M_g+M_h+2v.

Only research/fft-dirty-shear-cuts/ is added. The contribution was produced by Nick Petalas. Credit Jose Manuel for the already dimension-independent additive shear/unitary reduction in the pinned section-5 proof, Vlado Vojdanovski for the compiler, Cinna Davis for its retained Gaussian caller, Brian for the macro-cut/forward-bound continuation, Ryanson Jonathan, unpaidmoon, James Reed and the retained FFT/phase lineage. The elementary block-shear singular-value formula is background. No predecessor executable is imported or replayed.

Advance CLAIM and explicit prior-art refinement. The latter records that dimension independence already existed; the new result is the optimal constant, not that inherited reduction.

Evidence

• Before: the retained exact-circuit envelope is 1+B. After: the written gate-cut proof gives Phi(B), with sharpness. At B=3/2 the new constant is exactly 2.
• An independent standard-library checker constructs FFT butterflies, bit reversals, scalar shears and twists exactly over Q(zeta_8). All 756 matching-subset cuts pass, 12/56/688 at m=1/2/4, including independently partial source/target twist layers. Prefix/suffix identities, unitary gates, source restoration and Toeplitz endpoints pass.
• Six semantic rejection controls and the exact sharp Gram-eigenvector/dirty-inverse control pass. Explicit checks remain enabled under optimization.
• Both registered warning-strict commands and the package runner pass locally. Direct normal and optimized checks each took 1.47 s; registered replay took 3.14 s. These are verification durations. Both commands have 240-second caps.
• Committed-byte/source-history preflight passes at head 062ff35225ed38ec601198df1a5c0dcdd7f4424a, covering all five package files and canonical base 624de9dc66858af0945e5f8a0942488d62280d43.
• Official committed export and bundle integrity pass. Optimized Git-free replay with empty PATH passes in 1.47 s and exactly matches the local source-bound receipt.
• Syntax, warning-strict execution and staged whitespace checks pass. Longest function is 35 lines. Finite domains and managed-runtime allocation limits are documented.

From the repository root:

/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -I -W error -B research/fft-dirty-shear-cuts/verify.py
/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -I -W error -O -B research/fft-dirty-shear-cuts/verify.py
/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -B scripts/research_package.py --package research/fft-dirty-shear-cuts
/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -B scripts/check_research_commit.py --package research/fft-dirty-shear-cuts --check-source-history

Any CPython 3.12 executable may replace the local path. Expected markers are PASS_FFT_DIRTY_SHEAR_CUTS, PASS_REGISTERED_RESEARCH_PACKAGE and PASS_RESEARCH_COMMIT_BYTES. Generated stdout receipts bind all five package sources; no receipt is handwritten or committed.

Limitations

The all-dimension claim rests on the written proof, with finite corroboration. Exact unitary coefficient gates and the specified dependent-layer order are hypotheses; arbitrary subsets within a matching layer are allowed. Rounded roots/data, time inside an atomic gate, all-depth caller composition, generator recursion, fixed-tape costs and complete multiplication remain unrun obligations. The stored-unitary corollary inherits its predecessor's coefficient bounds; its rounded allocation was not audited or changed. No new kappa, supplier admission or selected-frontier change. Independent human and formal proof-kernel review are unclaimed.

Open next questions

1. Compose this local bound with one literal triangular caller, retaining ancestor/sibling factors and its physical cut order.
2. Price rounded butterflies and twists using the proved local prefix/suffix envelope, including dirty-source restoration.

Both require separate caller/error ledgers; neither is supplied here.

Related open work

These study adjacent Toeplitz conditioning or macro-prefix questions. This contribution concerns internal dirty-shear Euclidean cut norms; it does not recompute their reported numbers.

• Sharpen leading-section bounds for all completed downsets #4788 bounds completed downsets; unfinished FFT fragments remain outside that theorem.

• Record one four-entry n=8 parity prefix at 3/2 (no new κ) #4782 records a four-entry triangular macro-prefix fixture.

• Record one signed four-entry n=8 parity prefix at 3/2 (no new κ) #4796 records a signed four-entry triangular macro-prefix fixture.

• Record one five-entry n=8 parity prefix at 3/2 (no new κ) #4798 records a five-entry triangular macro-prefix fixture.

• Record one signed five-entry n=8 parity prefix at 3/2 (no new κ) #4799 records a signed five-entry triangular macro-prefix fixture.

• Record one six-entry n=8 parity prefix at 3/2 (no new κ) #4800 records a six-entry triangular macro-prefix fixture.

• Record one signed six-entry n=8 parity prefix at 3/2 (no new κ) #4802 records a signed six-entry triangular macro-prefix fixture.

• Record one full-support n=8 parity prefix at 23/16 (no new κ) #4811 records a full-support triangular macro-prefix fixture.

• Record one signed full-support n=8 parity prefix at 23/16 (no new κ) #4812 records a signed full-support triangular macro-prefix fixture.

• Record one degree-6 sign n=8 parity prefix at 23/16 (no new κ) #4815 records the degree-6 sign triangular macro-prefix fixture.

• Record one degree-5 sign n=8 parity prefix at 23/16 (no new κ) #4816 records the degree-5 sign triangular macro-prefix fixture.

• Record one degree-4 sign n=8 parity prefix at 23/16 (no new κ) #4820 records the degree-4 sign triangular macro-prefix fixture.

• Record one degree-3 sign n=8 parity prefix at 23/16 (no new κ) #4821 records the degree-3 sign triangular macro-prefix fixture.

• Record one degree-2 sign n=8 parity prefix at 23/16 (no new κ) #4822 records the degree-2 sign triangular macro-prefix fixture.

• Record one leading-quarter n=8 parity prefix at 25/16 (no new κ) #4823 records a leading-quarter triangular macro-prefix fixture.

• Certify all 128 signed n8 prefix maxima at 23/16 (no new κ) #4824 certifies all 128 signed n8 macro-prefix maxima; its claim and finite quantity are distinct from this gate-cut norm.

• Prove exponential parity-prefix growth above tail one (no κ) #4827 studies exponential macro-prefix growth above tail one; this local theorem remains conditional on its own split-symbol budget.

• Prove sharp conditioning of the complete complex port (no new κ) #4828 proves sharp Euclidean endpoint conditioning for a different complete two-bank complex port. Both use the classical block-shear formula; its coupling and complete interface differ from this internal Toeplitz FFT circuit. No extra numerical comparison was run.

• Prove exact all-depth binomial suffix norms (no new κ) #4831 proves exact composed binomial forward-suffix infinity norms at completed macro cuts; this theorem bounds Euclidean cuts inside one dirty shear. No common numerical instance is compared.

This account's open PRs created since the CLAIM at 2026-10-22 21:10:34 UTC were refreshed before DONE. Related #4824 and #4827 were already listed; #4828 and #4831 have been read and added above. The other new account contributions study address-algebra kernels, connector or projector tariffs, partial-channel rank or integer job splitting, without computing this Toeplitz gate-cut norm.

CI notes

Both hosted checks passed on contribution head 062ff35225ed38ec601198df1a5c0dcdd7f4424a in run 39856141612: plan in 3m6s and the registered package replay in 1m18s. Run metadata confirms this exact head and a successful completed run. The replay executes both registered warning-strict normal and optimized commands.

All registered local checks, committed-byte/source-history preflight and isolated export replay also pass. No check was left solely to CI; no CI failure was observed or diagnosed. These are scoped automated package checks, not independent human or formal proof-kernel acceptance of the all-dimension theorem.

Merge Danger

Door: two-way. Blast radius: isolated. Five new package files, no shared implementation or selected record changes.