16 Selberg Trace Formula Bridge
The Selberg trace formula connects the spectral side (eigenvalues of the Laplacian) to the geometric side (closed geodesics). For \(K_7\) with \(G_2\) holonomy, the Geodesic-Prime Correspondence hypothesizes that geodesic lengths \(\ell _\gamma \propto \log (p)\), linking the geometric side to the mollified sum \(S_w(T)\).
16.1 Trace Formula (Standard Result)
The multiset of lengths of closed geodesics on a compact Riemannian manifold.
For a compact Riemannian manifold \(M\) and a suitable test function \(h\):
Reference: Selberg (1956); Duistermaat-Guillemin (1975), Invent. Math. 29:39–79.
16.2 Geodesic-Prime Correspondence
For \(K_7\) constructed via TCS, primitive closed geodesic lengths satisfy \(\ell _\gamma (p) = c \cdot \log (p)\) for primes \(p\) and a geometric constant \(c {\gt} 0\). Status: Conjectural (Category E).
Under the geodesic-prime correspondence, the geometric side of the trace formula is proportional to \(S_w(T)\) from the MollifiedSum module. Status: Conjectural (Category E).
16.3 Cross-Module Identities (Proven)
The standard prime power cutoff \(K = 3\) equals the number of fermion generations \(N_{\mathrm{gen}} = 3\).
\(\mathrm{dim}(G_2) - h = 13 = \mathrm{rank}(E_8) + \text{Weyl} = \alpha _{\mathrm{sum}}\).
\(w(0) = 1\) and \(0 {\lt} 13/99 {\lt} 1\): the spectral gap falls within the compact support of the cosine-squared kernel.
\(99^2 - 50 \times 14^2 = 1\) and \(14 - 1 = 13\): the Pell equation connects bare topological constants, while the spinor correction shifts \(14 \to 13\).
Master certificate combining MollifiedSum properties, Spectral properties, and cross-module identities.
The Selberg Bridge connects three domains:
Number Theory: \(S_w(T)\) (finite sum, constructive, zero axioms)
Differential Geometry: \(\lambda _1(K_7) = 13/99\) (zero axioms for the ratio)
Trace Formula: spectral side = geometric side \(\propto S_w(T)\)