A complete generating-function characterization of Wilson primes: seven analytically proved lemmas reducing the Wilson criterion to R_4(p) ≡ B_{p-3}/3 - 274/135 mod p

We complete the analytic proof of all seven lemmas underlying the generating-function characterization of Wilson primes via G_p(z) := Σ_{n=0}^{p-1} inner_p(z)^n / n!, where inner_p(z) := Σ_{k=1}^{p-2} z^k/(k(k+1)). The previously conjectural Lemma L6, [z^p] A·B² ≡ -1/2 mod p (where A(z) = Σ z^k/k and B(z) = Σ z^k/(k+1), both truncated at k=p-2), is now proved by symmetrization, partial fractions, Wolstenholme-type identities, and Fubini-Tonelli telescoping. Combined with the previously established lemmas L1-L5 and L7, this gives the unconditional equivalence p is a Wilson prime ⟺ R_4(p) ≡ B_{p-3}/3 - 274/135 mod p where R_4(p) := Σ_{n=4}^{p-4} c_n(p) and c_n(p) := [z^p] inner_p(z)^n/n!. The equivalence has been numerically verified 23/23 for p ≤ 97. The proof of L6 is verified by three independent scripts in exact arithmetic (sympy/Fraction): step-by-step rational computation, brute-force triple summation, and convolution-based generating-function approach, with full agreement across 14 primes p ∈ {7, 11, ..., 59}. The remaining open problem is the analytic reduction of R_4(p) to known quantities (Bernoulli numbers, multiple zeta values, etc.).