19 Summary and Status
19.1 Proof Status Overview
|
Module |
Theorems |
Status |
|
E8 Lattice |
15 |
|
|
G2 Cross Product |
10 |
Octonion structure, G2 characterizations pending |
|
Betti Numbers |
8 |
|
|
SO(16) Decomposition |
11 |
|
|
Fibonacci/Lucas |
20 |
|
|
Prime Atlas |
20 |
|
|
Heegner Numbers |
10 |
|
|
Monster Group |
15 |
|
|
McKay Correspondence |
12 |
|
|
Joyce Theorem |
10 |
|
|
Physical Relations |
50+ |
|
|
TCS Spectral Bounds |
15 |
(v3.3.13) |
|
Literature Axioms (Langlais/CGN) |
8 |
(v3.3.13) |
|
Physical Spectral Gap (13/99) |
28 |
(v3.3.16, zero axioms) |
|
Selberg Bridge |
8 |
(v3.3.16, algebraic identities) |
|
Mollified Dirichlet Polynomial |
9 |
(v3.3.16) |
|
GIFT-Zeta Correspondences |
20+ |
(v3.3.16) |
|
Total |
290+ |
– |
Complete GIFT framework: 290+ certified relations spanning E\(_8\) lattice, G\(_2\) holonomy, physical predictions, spectral theory, number-theoretic correspondences, and the Selberg bridge.
19.2 Key Results
The GIFT framework achieves:
0.087% mean deviation across 18 dimensionless predictions
250+ 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)