[petalas]

Summary

Scoped mathematical proof with exact finite corroboration. For the degree-R central Gaussian bulk B=g(W) from David Ganz's #5813, the constructive denominator debt is sharp:

- pi^R B is Gaussian-integral
+ min{d>=0: pi^d B is Gaussian-integral} = R
+ c_S = i^a/pi^R for every displacement |S|=R

The proof covers every n>=1, 0<=R<=n and 0<=a<=n-R. It also proves uniqueness among permutation-symmetric convolutions of Boolean degree at most R with that central-band response. Actual reachable input pairs show that obtaining the full response modulo pi^p generally requires numerator precision p+R; one fewer digit fails at every chosen output coordinate. A different formula in the stated radius class cannot reduce this debt.

The new result is the matching lower bound, uniqueness and precision witnesses. Ganz's interpolation and constructive upper bound remain credited. No predecessor verifier or numerical result was audited or replayed as the deliverable. No new kappa or selected-frontier change is claimed.

Evidence

• Before: Construct Gaussian spectral and precision compilers from the ETH preprint #5813 supplies the R-unit numerator construction. Its denominator-accounting question leaves the sharpness issue open in this narrowed response class.
• After: README.md proves the lower bound from the literal top Boolean coefficient and supplies explicit indistinguishable-input witnesses. Walsh inversion and independent falling-factorial group-ring evaluation agree on the stated finite domain.
• Local CPython 3.12.15 warning-strict normal and optimized checks pass with identical source-hash-bound receipts: 285 bands, 114686 coefficients, 11263 top coefficients, 1022 independent polynomial coefficients, 350 literal Walsh eigenvalues, 75768 precision witnesses and sixteen boundary/semantic controls.
• Direct checks take 4.859 seconds normal and 4.814 seconds optimized. Both registered commands pass, with the complete registered invocation taking 10.093 seconds. Each command is below its 120-second timeout.
• Metadata, committed-byte/source-history preflight and staged whitespace pass. The official five-file export passes integrity checking. An isolated optimized replay from a read-only Git-free export with empty PATH passes and preserves all six files, including its export manifest, byte-for-byte and mode-for-mode.

Exact reproduction from the repository root:

PY=/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12
"$PY" -W error -B research/gaussian-bulk-sharp-denominator/verify.py
"$PY" -W error -O -B research/gaussian-bulk-sharp-denominator/verify.py
"$PY" -B scripts/research_package.py --package research/gaussian-bulk-sharp-denominator
"$PY" -B scripts/check_research_commit.py --package research/gaussian-bulk-sharp-denominator --check-source-history

The standard-library verifier writes no files and imports no predecessor package. The stdout receipt binds the five package files by SHA256. SOURCES.json pins the base catalogue and selected record for provenance; pending predecessor heads are citations, not runtime dependencies.

Limitations

The universal result rests on the written proof. Finite corroboration fixes n<=10; independent group-ring evaluation fixes n<=5, literal spectra n<=4, and precision witnesses n<=8 with p<=4. Restricted observables or input lattices, nonsymmetric responses and higher-degree replacements are outside the uniqueness obstruction. Physical delivery, exceptional correction, restoration, storage, movement, native admission and the complete caller bill remain open. All five selected framework assumptions remain. No Lean, complete multiplier, independent human review or proof-kernel acceptance is claimed.

Open next questions

• Allow one additional Krawtchouk degree and solve a small exact integrality system to test whether a higher-degree central-band response can lower the denominator.
• Pin an actual provider observable and determine whether composing it with B removes the worst denominator coefficients. Any useful continuation still needs the full paid caller and correction bill.

Related open work

• Construct Gaussian spectral and precision compilers from the ETH preprint #5813, David Ganz, head af37059 supplies central four-phase interpolation and the R-unit upper construction. This proves its exact minimal denominator in the stated class and the matching precision necessity.
• Construct exact Gaussian corrections with pruned and mixed precision circuits #5815, David Ganz, head f0fcda5 supplies exceptional correction and mixed-precision constructions; no implementation bill is changed here.
• Execute pruned Gaussian corrections and bound the standalone Newton-W route #5825, Ali Romman, head baea91d bounds work for a specified standalone Newton-W graph. This is a realization-independent denominator obstruction within the fixed full-response class, without recomputing its work totals.
• Prove sharp clean storage for reversible Gaussian overwrite (no κ) #5880, Nick Petalas, studies reversible clean storage of the raw Gaussian tensor. Its endpoint and capacity quantity differ, and it is not an input lemma or the target of this contribution.

Completion refresh inspected this account's eight other open PRs created since the CLAIM: #5886, #5887, #5888, #5889, #5891, #5892, #5893 and #5895. None computes this central Gaussian bulk's minimal denominator or numerator-precision threshold. No additional computations were run to compare their results, and no shared numerical disagreement is claimed.

The advance CLAIM states this scope. Canonical main and priority revision 8 were refreshed before submission; main remains de5ec2e. Live PRs, recent handoffs, reservations, all-main README searches and merged-result searches found no delivered denominator-sharpness result in the inspected coverage. Unpublished work and arbitrary code diffs remain outside that coverage.

CI notes

At commit ac75afd451e9375de50963c234439beaf27ccb23, run 40495802133 finished with plan failed; replay skipped. The sole failed step is Onboarding helper tests. All ten failure/error entries identify unchanged tests/test_research_current_assignment.py, ending FAILED (failures=8, errors=2, skipped=22) and exit code 1.

The errors are KeyError: 'follow_up_steps' and KeyError: 'catalog_first_step'. The eight failure entries concern frozen-card count 0 != 16, expected missing-helper/reference rejection, and sparse-reference receipt lists. All ten failure headers and exception/assertion messages match PR #5887's plan log exactly. This confirms the reported shared onboarding failure on base main de5ec2e7a6074068452f45b1a191b5c72832df4a. No package failure was reported, and no shared code or tests were changed to address the CI failure.

The replay job was skipped. Hosted replay has not certified this package; only the local scientific checks listed above have run. Maintainers will handle the shared onboarding repair.

Merge Danger

Door: two-way. Blast radius: package. The change adds only research/gaussian-bulk-sharp-denominator; shared code, frozen predecessors and the selected record are unchanged.

This contribution was produced by Nick Petalas. Credit David Ganz / Dugongue #5813 and #5815, Ali Romman #5825, classical Walsh/Newton/Krawtchouk identities, and the retained Gaussian/native/controller/compiler/precision lineage. Apache-2.0.