[petalas]

Summary

For every odd axis count n>=3 and every tensor width f>=4n+4, improve the sufficient lower constant for the full projective Gaussian-dyadic Clifford tensor Gram matrix:

- Gram_f >= (1491359/4161536) I
+ Gram_f >= (14601049/29130752) I > (1/2) I

The improvement is exactly 1/7 in this domain. Claim type: scoped written mathematical proof with exact finite corroboration. The exact spectrum and optimal constant remain open; no new kappa or physical supplier is claimed.

The new step uses seven commuting Pauli quarter turns whose product is I. Their regular-action adjacency sum is at least -10 I. Averaging over Clifford conjugation gives a 5/7 bound on the rank-one contribution, and the cited higher-rank envelope supplies the remainder. Entrywise Gram multiplication propagates the result beyond the initial width. The even-axis initial width has the opposite rank-one sign and remains outside this improvement.

This answers the odd-axis part of Open next question 2 in Javier Vera's #5714. Credit Hiroki KUMAZAKI #5694 for the displacement-rank count and rational tail certificate, and Ali Romman #5526 for the projective trace/Gram framework. Their classical and native/kernel attribution remains with the pinned sources. The predecessor results are cited, without running their executables or repeating their censuses.

Canonical base: de5ec2e, priority revision 8. The six new files are confined to research/clifford-seven-rotation-gap/. Advance claim.

Evidence

• Before: the absolute-row certificate gives lower constant 1491359/4161536, approximately 0.358367. Audit Clifford Gram row method: asymptotic sharpness and Sp(6,2) census #5714 proves its asymptotic limitation and leaves spectral strengthening open.
• After: the written proof gives 14601049/29130752, approximately 0.501224582187. The verifier prints PASS_CLIFFORD_SEVEN_ROTATION_GAP.
• New finite evidence enumerates 135 isotropic three-spaces with uniform point incidence, 1,080 literal endpoint columns, all 4,096 admissible fourth-root tuples, and 512 diagonal-subgroup characters. Forty-nine word mutations, two semantic controls, five scope controls and 120 width fixtures pass.
• Warning-strict normal/optimized registered runs agree exactly. Each was timed locally, approximately 0.065/0.087 seconds under its 120-second limit. Environment: CPython 3.12.15, macOS 27.0.1 arm64, standard library only.
• Package metadata, committed-byte/source-history preflight, official export integrity and both read-only Git-free export replays pass. All ten exported files, including the export manifest, retain their bytes and modes.

Exact reproduction from the repository root:

/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -I -W error -B research/clifford-seven-rotation-gap/verify.py
/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -I -O -W error -B research/clifford-seven-rotation-gap/verify.py
/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -B scripts/research_package.py --package research/clifford-seven-rotation-gap
/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -B scripts/check_research_commit.py --package research/clifford-seven-rotation-gap --check-source-history

Another CPython 3.12 path can replace the absolute interpreter path. The verifier emits a generated SHA-256-bound receipt to stdout and writes no files. SOURCES.json pins the three cited proof/question inputs on the base commit.

The written argument supplies the universal quantifiers; the finite checks corroborate its new local ingredients. Independent human and formal proof-kernel review are unclaimed. No full three-axis Clifford census, selected cold replay, complete caller, paid/native invoice, ownership/restoration, compiler, precision/tape, stock or outer admission was run or discharged.

Limitations and unrun proof obligations

• The improvement requires odd n>=3 and f>=4n+4. The even-axis initial-width problem, the exact full-group spectrum and the optimal lower constant remain open.
• The alphabet is the full projective Gaussian-dyadic Clifford group with exact arithmetic. Scalar-phase copies must be combined before using a projective Gram matrix. Arbitrary non-Clifford operators, nonunitary scales, nonlinear leaves and rounded equalities are outside scope.
• The universal statement is a written proof using the cited count and trace inputs. Its finite checks corroborate the new local ingredients. Independent human and formal proof-kernel review are unclaimed; no full three-axis Clifford census or proof-kernel check was run.
• No complete physical caller, paid/native invoice, ownership/restoration proof, finite compiler, precision/tape, stock or outer admission is supplied. The selected cold replay and the five inherited framework assumptions were not discharged. No new kappa or complete multiplication theorem is claimed.

Related open work

• Prove the selected #5599 scalar instructions lift at every column mul… #5837 proves scalar-chart lifting for selected Raise conditional kappa to 0.00310045 with independent helper splits #5599 instructions at every column multiplicity. This result supplies a quantitative full-Clifford Gram bound and makes no selected-caller admission claim.

• Prove sharp all-Clifford helper-detour bounds and correction budgets #5779 studies all-Clifford helper-detour tariffs and actual source budgets. Its quantity is a path invoice; this contribution bounds an operator Gram spectrum.

• Prove identity scratch does not unlock full monomial two-FULL routing. #5843 studies monomial two-FULL routing with restored identity scratch. Its algebraic endpoint obstruction has a different target from this spectral lower bound.

• Prove exact tariffs for consecutive queries sharing a detour #5884 extends Prove sharp all-Clifford helper-detour bounds and correction budgets #5779 to consecutive queries sharing a detour and evaluates a path invoice on Clifford charts. It does not compute this full-alphabet Gram lower constant.

• Prove exact additive cost for two anticommuting query types (no new κ) #5877 proves an additive tariff for two anticommuting involution query types over a faithful coefficient algebra. Its call-count target differs from this full-projective-group spectral bound.

• Construct a 22-channel monotone delivery counterexample (no new κ) #5881 constructs a many-channel immutable-payload response under a monotone tariff. It addresses a temporal delivery quotient and supplies no competing Clifford Gram constant.

The completion refresh read this account's nine other open PRs created since the CLAIM at 2026-10-27T10:39:47Z: #5876, #5877, #5878, #5879, #5880, #5881, #5882, #5884 and #5885. None supplies this same odd-axis constant or a conflicting same-instance number. No extra computations were run to check another contribution.

Open next questions

1. Improve the even-axis bound at f=4n+4 with an upper adjacency or joint-rank argument that handles the negative rank-one coefficient and trivial character.
2. Determine the best odd-axis constant by relating higher displacement classes to the same local subgroups. Local sharpness of the seven-root inequality does not establish full-group spectral sharpness.

Merge danger

Door: two-way. Blast radius: package. This adds an isolated proof and explicit replay hook. The selected frontier and predecessor files are unchanged.

CI notes

At commit cf069a8b7fbaaf9927f66cf6915f57bf4b02feba, run 40491695048 finished with plan failed; replay skipped.

The only failed step was Onboarding helper tests. All ten failure/error entries point to unchanged tests/test_research_current_assignment.py, ending FAILED (failures=8, errors=2, skipped=22), exit code 1. The two errors are KeyError: 'follow_up_steps' and KeyError: 'catalog_first_step'. The eight assertion failures are 0 != 16, five missing-helper/reference controls reporting ValueError not raised, and two empty-versus-expected sparse omission-list comparisons.

All ten failure/error headers match PR #5876's plan log, checked directly. This confirms the reported shared onboarding failure on main de5ec2e7a6074068452f45b1a191b5c72832df4a. Shared tests and implementations were not changed or rerun to address it; maintainer repair remains outstanding.

The replay job was skipped. Hosted replay has not certified the package; only local scientific checks have run. The local checks and their limitations are recorded above. The package and mathematical result remain unchanged after publication.