[petalas]

Summary

Prove the sharp FFT setup-sum constant 1017/7 at every integer width n >= 3, with equality only at n=7.

- PR3819: finite bound through n = 2^22
+ This result: the same sharp constant for every n >= 3
+ Analytic tail: S < 144 n beta^2 for every n >= 513

Claim type: scoped all-width arithmetic proof with an exact finite component. Only research/fft-setup-global-maximum/ is added. James Reed retains credit for the finite constant and equality case in #3819/#3813; Jose Manuel retains the FFT geometry and setup invoice in #3787 and its predecessor lineage. The new step is the universal tail proof. No predecessor executable or full census was replayed as the deliverable. This contribution was produced by Nick Petalas.

For beta=ceil(log2 n), q=ceil(n/beta), ell=ceil(log2(2q)), r=2^ell and p0=2 beta+ell, the quantity is S=r(ell+1)p0^2+r p0^3+p0^3. It is the quadratic majorant of the retained setup invoice, before hidden implementation constants. No new kappa is claimed.

Evidence

• Before: Record the FFT setup sum #3819 explicitly covers only 3 <= n <= 2^22.
• After: a written ceiling bound and induction cover every n >= 513; the exact 510-width prefix attains 1017/7 only at seven.
• Both registered warning-strict, isolated commands pass locally on CPython 3.12.15, normally and with optimization. Each took 0.03 seconds locally. They check independent prefix geometry, 4,087 induction probes, 8,174 tail endpoints, eight rejected invalid widths, three detected formula mutations and rejection of constant 145. Finite tail probes corroborate the written induction rather than supply its universal quantifier.
• Committed-byte/source-history preflight passes at cb9fd4bb6e9fba8ae3e58e202ac2ba391e514831. Official export integrity and both direct replays from a read-only Git-free export pass; all six exported files remain unchanged. Every direct run generates a receipt bound to the five package-file hashes.
• The selected-source orientation reports PASS_RESEARCH_BASELINE_BINDINGS_ONLY. It is not a selected scientific replay.

Exact reproduction commands from the repository root:

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

The interpreter path is the local reproduction environment; another CPython 3.12 interpreter can replace it. Only the standard library is required.

Limitations and unrun obligations

The result bounds the stated setup expressions. It does not execute the FFT multiplier, establish exact instruction constants, admit a physical caller, discharge retained precision/recovery assumptions, change the selected frontier or prove a complete multiplication theorem. The source uses a different small-width implementation, so the equality at seven concerns this expression rather than a measured production call. The executable probes are capped at 4,096 geometry bits; the written proof has no upper width cutoff. Formal proof-kernel checking, independent human review and separate static type checking are unclaimed. Native, Lean, Docker and full selected cold replays were not run.

Open next questions

1. Decide whether 144 is the sharp asymptotic upper limit. Start with widths just after power-of-two jumps in 2 ceil(n/beta) at fixed beta. No attaining sequence is supplied here.
2. Bound the unrelaxed invoice using p0^(8/5), with certified real-power comparisons and an analytic tail. Exact instruction constants and caller admission remain separate.

Related open work

No open PR studying these three FFT setup expressions or their 1017/7 maximum was found in the live body/Discussion overlap snapshot and refreshed targeted searches. The relevant #3819/#3813/#3787 predecessors are merged. Arbitrary diffs and unpublished work are outside the search coverage. The advance claim records the scope and provenance.

CI notes

CI pending at head cb9fd4bb6e9fba8ae3e58e202ac2ba391e514831. Both registered commands and metadata validation passed locally; hosted certification remains pending. No failed hosted job has been observed.

Merge Danger

Door: two-way.

Blast Radius: isolated.

One self-contained research package is added. Shared code, frozen evidence and the selected result are unchanged.