[MatShep]

For the full 24-class projective one-qubit Clifford alphabet, the exact normalized tensor-Gram spectrum gives sharp uniform bounds (3/8)I <= H_f <= (15/8)I for every f>=5, with Gram condition number at most 5. Both bounds and the condition number are attained at widths 5 and 6. Here H_f(g,h)=(tr(U_g* U_h)/2)^f.

This answers the one-qubit part of the best-spectral-constant question in Ali Romman's #5526 and extends MatShep's #5549, which already proved the independence threshold. A short corollary gives the sharp uniform normalized squared Hilbert-Schmidt distance 45/56 when the target class is missing from the approximation alphabet. No sharp operator-norm claim is made.

The isolated research/full-clifford-one-qubit-spectrum/ package contains the proof, exact recurrence and rational joint spectral projectors, self-contained standard-library verifier, extremizers, source provenance and complete saved result. Registered normal/optimized checks pass: independent characteristic polynomials at widths 1..10; direct spectral reconstruction at 1..32; 1,728 literal tensor Gram entries at 4/5/6; every omitted-class optimizer at 5..8; and seven semantic controls. Both direct checks also pass in a read-only export with empty PATH in about 3.3 seconds each, with all exported files unchanged. Committed bytes and export integrity pass. The optional full-history preflight is not claimed in this shallow checkout.

python -B scripts/research_package.py --package research/full-clifford-one-qubit-spectrum

Canonical base: 5e0052ca0aa3036a3afa8b0438927377c8d97d53, priority revision 7. Historical source pins are provenance only; no predecessor executable or third-party dependency is imported. Credit Ali Romman/#5526's quantitative framework, MatShep/#5549's prior rank/enumeration result, DLamdata/#5547 and josecarlosgonzalezroque-design/#5570's distinct audits, and their retained lineage. Spectral projectors, Pauli normalizers, Schur products and least squares are classical; no global literature-priority claim.

Prepared for MatShep with substantial OpenAI Codex assistance (GPT-6.1 Sol research; GPT-6 Astra brief validation and publication). This is author-checked algebra, not independent human or proof-kernel acceptance. No physical native circuit, paid supplier, selected cold replay, new kappa or selected-frontier change is claimed. The next unreserved question is a symmetry-reduced larger-qubit spectrum; no extrapolation is supplied. Apache-2.0.