feat(spectral): add ConnesBridge module connecting Weil positivity to…#143
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gift-framework merged 2 commits intomainfrom Feb 9, 2026
Merged
feat(spectral): add ConnesBridge module connecting Weil positivity to…#143gift-framework merged 2 commits intomainfrom
gift-framework merged 2 commits intomainfrom
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… GIFT
Connes (arXiv:2602.04022, Feb 2026) shows that 6 primes {2,3,5,7,11,13}
recover the first 50 zeta zeros via Weil quadratic form minimization.
This module formalizes the algebraic bridge to GIFT's mollified sum framework.
Proven identities (zero axioms):
- |Connes primes| = 6 = h(G₂) Coxeter number
- max(primes) = 13 = physical spectral gap numerator
- All primes < 14 = dim(G₂) (natural truncation scale)
- sum(primes) - dim(G₂) = 41 - 14 = 27 = dim(J₃(O))
- 2×3×5 = 30 = h(E₈), 2×3×5×7 = 210 = dim(K₇)×h(E₈)
- Pell equation 99² - 50×14² = 1 with 14-1 = 13 ∈ Connes primes
8 axioms (Categories B/D/E) for Weil positivity, prolate operators,
and GIFT-Connes structural matching hypotheses.
Co-Authored-By: Claude Opus 4.6 <noreply@anthropic.com>
…θ(T) Formalizes the GIFT-derived adaptive cutoff θ(T) = 10/7 - (14/3)/log(T) where both parameters come from topology, not curve fitting: θ_∞ = (dim(K₇) + N_gen) / dim(K₇) = 10/7 coeff = dim(G₂) / N_gen = 14/3 23 proven theorems (zero axioms): - Topological derivation from GIFT constants - Irreducibility: gcd(10,7)=1, gcd(14,3)=1 - Bounds: 1 < 10/7 < 3/2 - Two-Weyl perspective: dim(K₇)+N_gen = 2×Weyl - Comparison with empirical: |10/7 - 1.4091| < 2% 5 Category F axioms documenting 2M-zero validation: - T5a/T5b: beats 400 random alternatives - T7: bootstrap CI contains α=1.0 - T8: no α-drift across windows Co-Authored-By: Claude Opus 4.6 <noreply@anthropic.com>
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Summary
Spectral/ConnesBridge.lean(~480 lines) connecting Connes' Weil positivity approach (arXiv:2602.04022) to GIFT's mollified sum frameworkSpectral.lean(import + 16 re-exports) andCertificate.lean(10 abbrevs + master certificate)Key proven identities
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