The Parity-Shifted Cyclotomic Series and the Exceptional 5-Adic Resonance: Complement Lifts, Mixed Socles, and a Five-Block Filtration

For k = 4n−2 = 2m with m odd and ζ a primitive 2m-th root of unity, the parity-shifted value S2m = ∑r<m ζr²/(ζ;−ζ)r2 is proved equal to the denominator-free complement lift V−m(ξ) at ξ = −ζ. The block multiplier is −4, so characteristic 5 is the unique resonant prime for this alternating corner, while the companion corner V+ resonates at 3; a single parameter d = (ℓ−1)/2 controls every exponent in the theory. Proved: prime-divisor descent and the identification of the resonant prime; a resonant Hasse–Frobenius identity; the exact product coordinate at prime-power levels (μ3 = −1, μ5 = 1); an exact five-block decomposition; the first mixed-layer socle congruence; and a local-to-norm bridge converting initial forms into exact norm valuations. The all-level 5-adic tower is reduced to a three-derivative filtration, verified exactly through N = 125. Two explicit gates remain: the partial-block derivative filtration and residual nonvanishing. Complete splitting is a third, logically independent conjecture. Corrections to the source record (doi:10.5281/zenodo.20481749), verified in both directions: the mixed socle congruence is false modulo Φse and true modulo Φse−d+1 (the coefficient was right, the modulus was not); and the proposed monomial closed form does not exist. The package contains the LaTeX source and PDF, a Lean 4 formalisation of the proved algebraic core containing no sorry, no project axiom and no use of the native evaluator, and a self-contained exact-arithmetic verification suite that regenerates every numerical claim using only the Python 3 standard library.