We introduce a capacity–defect method for the kissing number problem that applies uniformly across dimensions. The framework consists of: (1) an exact Gegenbauer capacity identity converting the problem to a constrained triangle-free graph; (2) sharp d-ray energy caps c(d) = (d+1)(d+2)/(8d); (3) an odd–even corridor forcing antipodality at equality; (4) elementary local reconstruction of the optimal configuration. Only the Gegenbauer coefficients change between dimensions — the architecture is identical. Paper 1 demonstrates the method on K₄ = 24: the same formulas that Musin's 32-page modified LP argument required are here reduced to an 8-page self-contained proof of D₄ equality rigidity, infinitesimal jamming, and saturation. Paper 2 applies the identical framework to prove K₇ = 126, resolving a problem open since the Bachoc–Vallentin bound of 2008. The proof combines the capacity identity, exact finite sector certificates, a full-space compression to a single five-cycle Gram problem, and an exact PSD closure with rational contradiction gap 187/6720. The archive contains all computational certificates in exact integer/rational arithmetic. No floating-point comparison enters any theorem decision. 52C17 (Packing and covering in n dimensions)52C25 (Rigidity and flexibility of structures) 05C35 (Extremal problems in graph theory)68V20 (Formalization of mathematics in connection with theorem provers)