[kumagi]

Change and mathematical scope

At the unchanged Clifford tensor width f >= 4n+4, improve PR #5526's sufficient Gram interval [1/9,17/9] to [1/3,5/3]. The certified lower constant is actually 1491359/4161536 > 0.358367.

The new step counts an ordered residual-space basis together with a nonsingular residual form, fixing the form first. It gives
#{S in Sp(2n,2): rank(S-I)=k} <= 2^(2nk-k(k-1)/2).
The inherited trace-lift argument then bounds the absolute Gram row sum by sum_{k>=1} 2^(-k(k+1)/2). A short exact tail certificate proves the constant. The missing-target operator-error guarantee improves from 1/3 to at least 1/sqrt(3); the fixed Gaussian-dyadic four-width guarantee improves from M/9 to at least M/3.

Status: author-checked algebraic proof with finite corroboration, offered for independent review. This does not lower the tensor-width threshold, determine optimal spectra, audit a new caller, change paid-call counts, or improve κ. No selected-frontier or predecessor files are modified.

Task and overlap

Base: canonical main df0137efd35e11b4445734bf612bdd4c7ebc330c, unchanged at the pre-submission fetch. Priority revision 7. Picker: joint-query-controller, scope key 73442cf25caf1ed45f11, seed 1f66e16685e4ef83d568f7fabb3db680.

Scope changed: an enabling tensor-separation lemma for recursive-interface obstructions, rather than the card's complete QQ16 matched bill. The selected-baseline binding example passed; no full frontier replay or caller admission is claimed.

Direct dependency: Ali Romman's #5526, head 45f7de73b168721cdf7ec8c1769b1b594493716a; its proof is retained on the base and byte-pinned. The all-n constant is distinct from the sharp one-axis #5549/#5594 and two-axis #5675 spectra/thresholds, and from #5547 and the paid-return audits.

Canonical claim/discussion. The final fully paginated search covered 376 open PRs and 5,007 Discussion records at 05:53–05:54 UTC on 2026-10-26, supplemented by Git HEAD note/proof searches. Sparse helper omissions and the limits of overlap searching are disclosed in the README.

Validation

Passed on macOS / CPython 3.12.14, standard library only:

python3.12 -B scripts/research_package.py --package research/clifford-residual-count-bound
python3.12 -B scripts/check_research_commit.py --package research/clifford-residual-count-bound --check-source-history

• Registered normal and -O replays compare the entire symplectic sets reconstructed from signed Pauli actions versus residual forms for n=1,2. The signed Clifford groups contain 24 and 11,520 elements; the resulting symplectic rank counts are [1,3,2] and [1,15,100,300,304].
• Literal Gaussian-integer generator conjugation checks signs independently. Every Pauli fiber, trace multiplicity, exact row sum and rational tail is checked.
• Five negative controls reject a singular form, wrong transpose, lost sign, omitted lift multiplicity and unsupported width. An additional exported-input mutation is rejected by its source hash.
• Committed-byte/source-ancestry checks and deterministic-format export integrity checks pass. Direct normal/optimized replays pass in a fresh Git-free, read-only export, with unchanged files afterward.

The written proof supplies the universal quantifiers; finite enumeration is corroboration, not a proof-kernel result or independent mathematical acceptance. No new complete-caller, physical-cost or selected-frontier replay was attempted.

Credit

Prepared for kumagi with substantial OpenAI Codex assistance in derivation, code, verification and writing; Apache-2.0. Retains Ali Romman's #5526 residual-form/trace-lift/Gram/four-width framework and its classical and repository predecessor attribution. No literature-priority claim is made for residual forms or Clifford counting. Parallel finite-spectrum and application audits remain their authors' work.

Open next questions

See research/clifford-residual-count-bound/README.md and TASK.json: seek an all-n spectral estimate exploiting phases beyond absolute row sums, or audit one exact complete caller before applying the stronger error constant. Neither is discharged here.