The arcs of Re zeta(s) = 0 next to the critical line: partners, depth and side

Through each simple zero rho = 1/2 + i gamma of zeta(s) passes a curve on which Re zeta(s) = 0. Followed into sigma > 1/2, it returns to the critical line at a point where zeta(1/2 + it) is purely imaginary (a half-Gram point, Conrey's Gram-minus point). Writing zeta(1/2 + x + it) = exp(-i theta(w)) Z(w) with w = t - ix, the paper proves a local theorem with explicit hypotheses on the sector between the zero and a half-Gram point t_L: bounds on Z'/Z and Z''/Z there, and Stirling bounds on theta''. Under them, the arc of rho is a single analytic arc that stays in sigma > 1/2 and returns to the line exactly at t_L; its depth is delta^2 / (4 theta'(gamma)) (1 + E), with E bounded explicitly and of order delta, where delta is the phase advance of theta along the arc. RH is not assumed. Arb computations verify the hypotheses for 197 of 256 arcs (zeros 100-300 and three windows near t = 2000, 3000 and 7000), and a separate rectangle-chain certification covers the other 59, including the Lehmer pair, so all 256 arcs are certified: their side, their partner and an enclosure of their depth. Measurements show the arc always lands on a half-Gram point, the phase-nearest one except at 15 zeros next to failures of Gram's law, and a first-order depth coefficient of -1/4. The pairing is related to an unpublished theorem of Conrey; the global two-crossing statement is equivalent to the Riemann Hypothesis under Conrey's hypothesis on zeta', discussed in a remark only. Written jointly with Claude (Anthropic); the human author is responsible for the content.