Hölder Pairs of the Symmetrized Riemann–Siegel Main Sum: an Exact Counting Theorem and the Lehmer Mechanism

Let P_n(s) = Σ_{k≤n} k^{-s} and A_n(s) = P_n(s) + χ(s)P_n(1−s), its symmetrization by the functional equation of ζ; on the critical line, in the Riemann–Siegel regime n = ⌊√(t/2π)⌋, A_n is the Riemann–Siegel main sum. Writing s = 1/x + it, every zero of A_n either lies on the line x = 2 or belongs to a pair of Hölder conjugate exponents (x, x’), 1/x + 1/x’ = 1. We prove an exact Riemann–von Mangoldt formula for A_n, with no error term, and an exact integer identity for the number of Hölder pairs in any height range: it equals the number of zeros of the unsymmetrized sum P_n to the right of the critical line, minus the number of zeros of the main sum at which its total phase is decreasing, plus an integer winding defect that vanishes at generic heights. The key is the exact relation D = 2φ' between the derivative of the modulus equation across the line and the derivative of the phase along it. Every Hölder pair is shown to be born at the edge of a “phase retreat” produced by a single zero of P_n hugging the critical line; exact computations up to height 5.7·10^5 confirm the identity to the last integer, trace each pair to a Lehmer pair of close zeros of ζ, and show that the first Riemann–Siegel correction returns every pair to the line, while a parameter-free GUE model reproduces the number of pairs per window. Since the counting theorem holds for any splitting ζ = Q + χQ̃, it reformulates the Riemann Hypothesis in a range as the statement that no right-half zero of the half-function Q is left uncompensated by a backward crossing of its phase.