[petalas]

Summary

Scoped written proof with exact finite corroboration. For every finite normalized Gaussian section at either half-shift endpoint, determine its exact inverse-norm infimum over the Gaussian scale:

T[i,j] = q^((j-i)^2-epsilon(j-i)),  0<q<1, epsilon=+1 or -1
inf_q ||T^(-1)||_2 = 1/(2 sin(pi/(4N+2))).

For N>=2 every positive q gives a strictly larger norm; q tending to zero approaches the infimum. At N=1 the norm is one. The proof compares the whole positive checkerboard inverse with the triangular all-ones matrix and uses its classical singular value. This answers the scale-uniform part of Jose Manuel / jmbarrancoml's #4193 finite-cap sharpness question. The exact norm at fixed q remains open.

Only six files in research/gaussian-half-shift-envelope/ are added. The contribution was produced by Nick Petalas. Credit #4193 for the question and finite-cap framework; Vlado Vojdanovski / #1673 for the cited q-LDU factors; Jeremy Kothe and the #1758 contributors for the shifted Gaussian lineage; Cinna Davis for the later caller; and Nick Petalas #5150/#6459 for related lower and upper results. No priority is claimed for the classical triangular singular values or positivity argument. No predecessor scientific verifier is replayed.

Advance CLAIM. Canonical base e54523517d0f03e58352e647c403a5af3e525a57, priority revision 8, selected #7000 unchanged.

Evidence

The cited predecessor supplies finite-cap upper bounds and leaves sharpness open. The new written proof supplies the exact scale-uniform lower envelope and an approaching family.

Local CPython 3.12.15, standard library only:

• 80 literal rational endpoint matrices and 2,040 inverse entries; both inverse products and the new triangular domination pass. All 70 nontrivial-size cases have a strict diagonal witness.
• 32 complete Laurent-polynomial characteristic identities and eight triangular inverse pairs pass without floating-point trigonometry.
• 28 small-scale matrices and 21 strict decreases in their exact squared Frobenius errors pass. Ten invalid inputs reject and five semantic control groups pass.
• Warning-strict normal and optimized direct checks produce identical hash-bound receipts, each in under one second locally. Both registered commands pass; the complete registered replay takes under one second. These are validation durations, not speedup claims.
• In-memory source compilation, staged whitespace inspection and committed-byte/source-history checks pass at f1401fd8618394634c4f2a2171a668ba5f6915c2.

Reproduce from the repository root:

/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -I -W error -B research/gaussian-half-shift-envelope/verify.py
/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -I -O -W error -B research/gaussian-half-shift-envelope/verify.py
/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -B scripts/research_package.py --package research/gaussian-half-shift-envelope
/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -B scripts/check_research_commit.py --package research/gaussian-half-shift-envelope --check-source-history

Substitute another CPython 3.12 interpreter path elsewhere. Expected markers are PASS_GAUSSIAN_HALF_SHIFT_ENVELOPE, PASS_REGISTERED_RESEARCH_PACKAGE and PASS_RESEARCH_COMMIT_BYTES. Receipts are generated from the current six file hashes; none is handwritten or committed.

Limitations and unrun obligations

The universal statement relies on the written proof and cited q-LDU premise. Finite checks corroborate it. The result is an exact infimum over q, not the norm at a prescribed positive q. Earlier related linear-growth consequences retain their credit.

No interior-shift lower envelope, fixed-q leading coefficient, production normalization/gauges, rounded inverse, physical execution, native/compiler/stock/tape invoice, supplier admission, new kappa or complete multiplication theorem is supplied. All five selected framework assumptions remain. Lean, Docker, predecessor scientific suites, selected full cold replay and separate type/lint analysis are unrun. Independent human and formal proof-kernel review are unclaimed.

Related open work

No open PR studying this normalized constant-shift half-endpoint infimum surfaced in the current search. The cached full search covered 477 open PR bodies; fresh current PR metadata and handoff claims were checked before committing. Sparse notes, arbitrary historical diffs and unpublished work remain coverage limits.

• Add physical Gaussian row norm transfer audit #3910 bounds the physical Gaussian forward row mass and its input contract; this contribution bounds a finite Toeplitz inverse norm in a different normalization.
• Account for final-grid rounding in the Gaussian block enclosure #371 accounts for final-grid rounding in a Gaussian block enclosure; this contribution concerns exact matrix inverses and supplies no rounding policy.

Merged #4193 supplies the cited factors and upper estimate. Merged #6459 improves an upper envelope; #5150 and #6483 concern varying-phase boundary cells. No predecessor census or numerical comparison was run for this contribution.

The completion refresh examined this account's open contributions created since the CLAIM: #7178, #7179 and #7181 through #7187. They concern shell growth, Coxeter tariffs, Clifford sectors, cube-zero lifts, radical tariffs, rational headers, block signatures and samplers. None studies this constant-shift Gaussian inverse envelope or computes a conflicting shared value. No extra scientific comparison computations were run.

Open next questions

• Determine the sharp leading coefficient at fixed positive q as N grows. Start with the positive inverse in the bulk upper triangle; boundary and near-diagonal control remain missing.
• Determine the finite-size crossover as a shift approaches a half. Start with the geometrically decaying triangle under exact diagonal conjugation; justify the two-parameter limit before using it in a caller.

CI notes

At exact head f1401fd8618394634c4f2a2171a668ba5f6915c2, run 42992579989, plan job 130428771832, failed in Onboarding helper tests on exactly one of 760 tests, with 24 skipped:

test_research_current_status.CurrentCiNavigationTests.test_current_ci_section_matches_selected_record
AssertionError: '`make frontier7000-verify`' not found
FAILED (failures=1, skipped=24)

The job log shows that docs/ci-verification.md still describes selected #6030 and make frontier6030-verify. Direct inspection of canonical main e54523517d0f03e58352e647c403a5af3e525a57 confirms that its selected record instead names make frontier7000-verify. The test, document and selected record are byte-identical between that base and this PR. This six-file package did not cause the mismatch.

The same exact failing test, missing command, 760-test total, one failure and 24 skips were confirmed directly in the failed-step logs of every run below:

• Other contributors: 42905094314 and 42903313790.
• Prove sharp multiplicative-shell growth for every rank alphabet (no κ) #7178: 42992088329.
• Prove five A2 targets need five rank-three calls (no κ) #7179: 42992340683.
• Classify exact two-versus-three costs for diagonal cube-zero lifts (no κ) #7182: 42992720936.

CI did not certify this package. Its replay job was skipped because plan failed. The local normal/optimized, registered-package and committed-byte/source-history checks recorded above remain the available automated package evidence. No fix, retrigger commit, push or job rerun was made for this repository-level failure. Maintainers will repair main.

Merge danger

Reversible, isolated research package. The selected frontier and shared implementation are unchanged. Mathematical acceptance remains separate from archival intake.