[GosuBoy]

Change and mathematical scope

This contribution provides a scoped independent audit and complementary mathematical result for PR #5694:

#5694

For N = 2n and 1 <= k <= n, we establish the lower bound:

product(i=0 to k-1) [2^(N-i) - 2^i] <= c_(N,k)

Here, c_(N,k) counts symplectic transformations S in Sp(N,2) satisfying rank(S-I) = k.

Together with #5694's upper bound, this proves that the full projective-Clifford absolute Gram row sum at width f = 4n+4 converges to:

lim(n -> infinity) R_(2n) = sum(k >= 1) 2^[-k(k+1)/2]

Numerically, the limit is approximately 0.641632560655154.

Consequently, #5694's certified rational Gram lower constant, alpha = 1491359/4161536, is within 1.48e-11 of the best uniform lower constant obtainable using the absolute off-diagonal row-sum method alone.

Scope: This establishes asymptotic sharpness of the counting-based diagonal-dominance method, not spectral optimality. It does not rule out phase-aware improvements, validate a physical multiplication caller, change paid-call counts, or improve the selected exponent saving (kappa).

The written proof and independent verifier are included in research/clifford-row-sharpness/.

Task selection and overlap

This contribution addresses an independent audit of the live #5694 handoff rather than a randomly drawn task. Its scope is refined to the symplectic displacement-rank argument, a complementary lower bound, and its asymptotic implication.

• Canonical base: df0137efd35e11b4445734bf612bdd4c7ebc330c
• Priority revision: 7
• Selected conditional kappa: 0.00310045, unchanged
• Direct subject: kumagi's PR Strengthen uniform Clifford tensor Gram bound from 1/9 to 1/3 #5694, head aa01976e51aa1731433a9e65443303a7c2f869ca
• Predecessor: Ali Romman's PR Prove Clifford tensor faithfulness and a 104-return paid-repair floor #5526, head 45f7de73b168721cdf7ec8c1769b1b594493716a

This work complements #5694's nonsingular residual-form upper bound and #5526's Clifford trace-lift and Gram framework. Existing independent audits and parallel finite-spectrum investigations retain their attribution.

Overlap checks were targeted, not an exhaustive novelty certification.

No upstream, frozen evidence, historical source, or selected-frontier file has been modified.

Validation

Reproduce from the repository root using Python 3.10+ and the standard library:

python3 -B research/clifford-row-sharpness/verify.py

python3 -O -B research/clifford-row-sharpness/verify.py

Both normal and optimized verification passed on Linux x86-64 / CPython 3.13.5.

The verifier uses independent ordered symplectic-frame enumeration rather than #5694's signed-Clifford generator and residual-form enumeration.

Verification evidence:

• Exhaustive enumeration of all 1,451,520 elements of Sp(6,2).
• Complete displacement-rank histogram: [1, 63, 1932, 28980, 202944, 608832, 608768].
• Independent Sp(4,2) verification through all 65,536 binary 4x4 matrices.
• Exact rational Gram row sums for n = 1, 2, 3.
• Verification of symplectic group orders, rank-one transvections, and finite upper/lower rank-count inequalities.
• Exact rational-tail and asymptotic-gap certificate.
• Negative control rejecting an invalid faster tail ratio.

Finite enumeration corroborates the written argument but does not establish its universal quantifiers. The all-dimension result relies on the mathematical proof in README.md.

No formal-kernel verification, complete caller audit, registered source-bound replay, or selected-frontier cold replay is claimed.

The initial GitHub Actions workflow returned action_required without executing test jobs. This is neither a verification failure nor a CI pass.

Credit

This contribution preserves the following predecessor work:

• kumagi, PR Strengthen uniform Clifford tensor Gram bound from 1/9 to 1/3 #5694: Nonsingular residual-form upper bound and improved Clifford Gram certificate.
• Ali Romman, PR Prove Clifford tensor faithfulness and a 104-return paid-repair floor #5526: Clifford trace-lift, residual-form and Gram/four-width framework.
• Earlier independent audit contributors: Retained Clifford pointwise-faithfulness audit coverage.
• Classical mathematical literature: Symplectic residual forms, Clifford counting, and related character-orthogonality methods.

The new scoped contributions are the complementary lower bound, asymptotic row-sum sharpness observation, and independent three-axis symplectic-frame census.

Substantial OpenAI assistance was used in the mathematical derivation, verification implementation, and documentation.

License: Apache-2.0.

No literature-priority claim is made beyond the stated comparison with the repository's retained results.

Open next questions

1. Independent mathematical scrutiny: Verify the fixed-nonsingular-form lower count, reconstruction correspondence, and all-dimension limiting argument.

2. Spectral strengthening: Investigate whether phase cancellation provides a stronger uniform minimum-eigenvalue bound than absolute-row-sum diagonal dominance.

3. Physical caller application: Independently verify the exact operator alphabet, retained dirty state, ownership and restoration requirements, paid-call budget, and inherited framework assumptions.

The present contribution is a scoped mathematical result with independent finite corroboration. It does not request a selected-frontier update or claim a complete multiplication theorem.

Workflow feedback

Publication required an authorized personal fork because direct upstream writes through the GitHub integration returned HTTP 403.

The resulting branch and PR preserve the original research files and attribution. The description identifies the exact mathematical scope, reproducible evidence, and remaining obligations without implying maintainer approval.