30 Summary and Status

30.1 Proof Status Overview

30.2 Key Results

The GIFT framework achieves:

• 0.087% mean deviation across 18 dimensionless predictions

• 455+ formally verified relations in Lean 4

• Complete Fibonacci/Lucas embeddings (\(F_3\)–\(F_{12}\), \(L_0\)–\(L_9\))

• 100% prime coverage \({\lt} 200\) via three generators

• All 9 Heegner numbers GIFT-expressible

• Monster dimension \(196883 = 47 \times 59 \times 71\) from \(b_3\)

• Joyce existence theorem for torsion-free \(G_2\) on \(K_7\)

• TCS spectral bounds: \(\lambda _1 \sim 1/L^2\) (Model Theorem)

• Literature axioms: Langlais 2024 spectral density, CGN 2024 bounds

• Physical spectral gap \(\lambda _1 = 13/99\): axiom-free from Berger classification

• Selberg bridge: trace formula connecting \(S_w(T)\) to spectral gap

• Mollified Dirichlet polynomial \(S_w(T)\): constructive core, zero axioms

• GIFT-zeta correspondences: 5 primary matches (\({\lt} 1\% \) relative error)

• Multiples of 7 pattern: \({\gt} 96\% \) match rate (2222/2300 tested)

• \(G_2\) metric non-flatness via Bieberbach bound (\(b_3= 77 {\gt} 35\))

• Spectral degeneracy \([1, 10, 9, 30]\) with topological derivations

• \(\det (g) = 65/32\): triple independent derivation (Weyl, cohomological, \(H^*\))

• \(\kappa _T^{-1} = 61 = \mathrm{dim}(F_4) + N_{\mathrm{gen}}^2\): structural decomposition

• \(38{,}231\times \) PINN compression: \(\mathrm{dim}(\mathrm{SPD}_7) = 28 = 2 \times \mathrm{dim}(G_2)\)

• TCS building block asymmetry: \(b_3(M_1) - b_3(M_2) = N_{\mathrm{gen}} = 3\)

• \(H^*(M_1) = \mathrm{dim}(F_4) = 52\): quintic block carries \(F_4\) degrees of freedom

• Matrix space: \(7^2 = 2 \cdot \mathrm{dim}(G_2) + b_2= 28 + 21\)

• Fano automorphism: \(|\mathrm{PSL}(2,7)| = 168 = \mathrm{rank}(E_8) \times b_2\)

• Kovalev involution count \(C(7,4) = C(7,3) = 35 = \mathrm{dim}(\Lambda ^3 \mathbb {R}^7)\)

• Conformal rigidity: \(28 - 27 - 1 = 0\) residual DOF (\(G_2\) holonomy + determinant)

• \(\mathrm{End}(V_7) = 1 \oplus 7 \oplus 14 \oplus 27\) (all four \(G_2\) irreps, \(49 = 7^2\))

• \(\mathrm{dim}(J_3(\mathbb {O})) = N_{\mathrm{gen}}^3 = 27\): traceless symmetric \(=\) generation cube

• Moduli gap: \(b_3- b_2= 56 = \mathrm{dim}(\mathrm{fund.}\; E_7) = \mathrm{rank}(E_8) \times \mathrm{dim}(K_7)\)

• Poincaré duality: \(\beta _{\mathrm{total}} = 198 = 2 \times H^*\) (duality doubles the spectrum)

• Structural identity: \(H^*= 1 + 2 \times \mathrm{dim}(K_7)^2\) (manifold dimension controls all)

• Holonomy chain: \(\mathrm{dim}(\mathrm{SO}(7)) = b_2= 21\), \(\mathrm{dim}(\mathrm{GL}(7)) = 7^2 = 49\)

• Codimension: \(\mathrm{dim}(\mathrm{GL}(7)) - \mathrm{dim}(G_2) = \binom {7}{3} = 35\) (3-form components)

• Torsion space: \(1 + 7 + 14 + 27 = 49 = \mathrm{dim}(K_7)^2\) (frame bundle decomposition)

• \(G_2\) branching: \(14 = 8 + 6 = \mathrm{dim}(\mathrm{SU}(3)) + 2 N_{\mathrm{gen}}\)

• Betti–torsion bridge: \(b_2+ b_3= 2 \times (1 + 7 + 14 + 27)\)