Halves of the Riemann zeta function: exact pair counting, Möbius nesting and phase locking

We study the Riemann zeta function and its Riemann–Siegel truncations through "halves": functions F such that Λ(s) = F(s) + χ(s)·conj F(1 − s̄), where χ is the factor of the functional equation. Examples are the main sum P_n(s) = Σ_{k≤n} k^(−s), Siegel's auxiliary function ℛ, the main sums of Dirichlet L-functions and the halves of self-reciprocal polynomials. Writing s = 1/x + it, the functional equation becomes the Hölder conjugation x ↔ x/(x−1), and zeros off the critical line appear as Hölder pairs. We prove an exact counting identity for halves: the number of Hölder pairs equals the number of zeros of the half to the right of the critical line minus the number of backward crossings of its phase. Equivalently N_line = N_tot − 2Z⁺ + 2R, an exact form of Siegel's lower bound. Each zero of the half to the right of the line creates a well in the phase velocity. The zeros of Λ are fixed points of nesting maps, and a Möbius (dipole) nesting attracts every backward crossing exactly when a separation condition holds. Numerically, with Siegel's ℛ, every well contains exactly one zero of ζ in seven complete Riemann–Siegel windows up to t ≈ 6.3·10^8 (1.63 million zeros), and the certificate Z⁺ = R holds exactly in these windows. In random-matrix analogues this local balance depends on eigenvalue repulsion: unbalanced wells decay like e^(−5.5β) in circular β-ensembles. Random halves with logarithmic frequencies, with independent or multiplicative coefficients, have wells statistically similar to those of ζ, yet only 65% and 76% of their zeros lie on the line. For them a well captures a zero with probability δ/π, where δ is its phase drop; for ζ and Dirichlet L-functions the capture probability is close to 1 at every depth. We call this phase locking; locally it is equivalent to the Riemann hypothesis. The paper separates proved results, numerical observations and conjectures. It does not claim a proof of the Riemann hypothesis.