23 Summary and Status
23.1 Proof Status Overview
|
Module |
Theorems |
Status |
|
E8 Lattice |
15 |
|
|
G2 Cross Product |
10 |
|
|
Betti Numbers |
8 |
|
|
SO(16) Decomposition |
11 |
|
|
Joyce Theorem |
10 |
|
|
Physical Relations |
50+ |
|
|
TCS Spectral Bounds |
15 |
|
|
Literature Axioms (Langlais/CGN) |
8 |
|
|
Physical Spectral Gap (13/99) |
28 |
(zero axioms) |
|
\(G_2\) Metric Properties |
25 |
|
|
TCS Piecewise Metric |
30 |
|
|
Conformal Rigidity |
37 |
|
|
Spectral Scaling |
35 |
|
|
Poincaré Duality |
41 |
|
|
\(G_2\) Differential Geometry |
18 |
(zero axioms) |
|
Dimensional Hierarchy |
25 |
|
|
Extended Observables |
50+ |
|
|
Mass Gap Ratio & Universal Law |
20 |
|
|
Selection Principle & Yang–Mills |
30 |
|
|
Total |
460+ |
– |
Mathematical infrastructure: E\(_8\) lattice (240 roots), G\(_2\) cross product, \(G_2\) differential geometry (axiom-free), octonion bridge, K\(_7\) Betti numbers, Joyce existence, conformal rigidity, Poincaré duality, G\(_2\) metric properties, TCS piecewise metric.
Published predictions: 33+ dimensionless derivations, 50+ observables, hierarchy solution, exceptional chain, SO(16) decomposition.
Spectral gap programme: bare algebraic ratio \(\mathrm{dim}(G_2)/H^*= 14/99\), physical ratio \(({\mathrm{dim}(G_2) - h})/H^*= 13/99\), analytical mass gap \(\lambda _1 = 6\pi ^2/475\), TCS spectral bounds, selection principle, Cheeger inequality, KK spectral bridge.
Complete GIFT framework: 460+ certified relations spanning E\(_8\) lattice, \(G_2\) holonomy, \(G_2\) differential geometry (axiom-free), dimensional hierarchy, 50+ physical observables, spectral theory, bare ratio \(14/99\), physical ratio \(13/99\), selection principle, KK spectral bridge, Cheeger inequality, \(G_2\) metric properties, TCS piecewise metric, conformal rigidity, and Poincaré duality.
Structure: three domain-organized pillars — Foundations.statement \(\wedge \) Predictions.statement \(\wedge \) Spectral.statement.
23.2 Key Results
The GIFT framework achieves:
0.24% mean deviation across 32 well-measured observables (PDG 2024 / NuFIT 6.0)
460+ formally verified relations in Lean 4; 4 logical axioms on the main prediction chain, plus 11 interval-arithmetic certificates for the K3 block of \(g^*\)
Joyce existence theorem for torsion-free \(G_2\) on \(K_7\)
TCS spectral bounds: \(\lambda _1 \sim 1/L^2\) (Model Theorem)
Physical spectral gap \(\lambda _1 = 13/99\): axiom-free from Berger classification
\(G_2\) 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: \(7^2 = 2 \cdot \mathrm{dim}(G_2) + b_2= 28 + 21\)
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, \(49 = 7^2\))
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: \(\mathrm{dim}(\mathrm{SO}(7)) = b_2= 21\), \(\mathrm{dim}(\mathrm{GL}(7)) = 7^2 = 49\)
Betti–torsion bridge: \(b_2+ b_3= 2 \times (1 + 7 + 14 + 27)\)