Toward a Resurgent Spectral Theory on G₂ Joyce Orbifolds: Derivation of the Electroweak Scale

A new algebraic identity is established relating the spectral invariant K_0 on the G_2 Joyce orbifold T^7/(Z_3 × I*) to the fine-structure constant: α^(-1) = K_0 + (b_3/b_2) ln(b_3/b_2), where (b_2, b_3) = (27, 451) are Betti numbers of the unique topology selected by cosmological-constant matching. This identity, combined with the cosmological relation ρ_Λ = b_3 √2 M_Pl^4 exp(-π K_0), yields three determinations of K_0 from distinct sources (spectral, electromagnetic, cosmological) that converge to within 6×10^(-6) relative precision. A resurgent transseries framework is proposed—extending Écalle's resurgence theory to spectral problems on seven-dimensional manifolds with exceptional holonomy under discrete group action—from which a closed-form expression for the electroweak vacuum expectation value is proposed, conditional on two conjectures about fractional G_2 instantons and Borel resurgence structure: v = M_Pl × exp(-9 K_0 / (7π)) × exp((1 - 7α/π) / (16π²)) Each element of this formula admits a physical interpretation: 9/(7π) = (d+2)/(dπ) with d = dim(G_2 manifold) = 7, 1/(16π²) is the universal one-loop normalization, and 7α/π is the gauge-sector contribution from eleven-dimensional Chern-Simons reduction. Numerical evaluation yields v_pred = 246.223 GeV against v_exp = 246.220 GeV, a deviation of 10^(-5). To the author's knowledge, this is the first closed-form expression for the electroweak scale in M-theory compactifications on G_2 manifolds, reducing the Standard Model's 26 free parameters to three inputs (I*, ρ_Λ, topology). The paper further constructs fractional M2-instantons via a McKay-style correspondence (Theorem B.6) and establishes the Borel singularity structure via brane classification (Theorem C.2). Seeley-DeWitt coefficients are computed via Donnelly's orbifold formula with central numerical constant SI* = 44.833 (Theorem D.6). Concrete open problems are identified with specific research programmes at the intersection of differential geometry, representation theory, and mathematical physics.