Pure Time Theory – Chapter V - Annular Phase-Injectivity: Static RH Equivalence and Dynamic Closure

We present a two-track, self-contained framework around the nontrivial zeros of zeta. Part I (static, 2D): From the canonical bricks induced by zeros, Chebyshev/Herglotz geometry on circles and annuli, the phase-injectivity functional E, and a self-contained derivation of xi(s)=xi(1−s), we prove the purely mathematical equivalence:RH ⇔ E(Z)=0 ⇔ {E_Lambda(Z)=0 on an annulus cover}.No dynamics is assumed. Part II (dynamic, 2D+1): We give two conditional closures to RH in the sequential rotating setting: (A) a persistent co-rotation regime; or (B) a dihedral quarter-turn scheduling (IDE) together with a minimal summable nonradial tail hypothesis (HT), which yields asymptotic radialization and hence E_Lambda^dyn(Z)=0 on an annulus cover. With PTT-Ax 2 (injection fidelity) and PTT-Ax 3 (visibility), a bridge from dynamics to statics then gives E(Z)=0 and thus RH. In short: Part I provides a static characterization; Part II provides two dynamic closures. Either closure enforces E_Lambda^dyn(Z)=0 on a cover; via the bridge, this implies E(Z)=0, hence RH by Part I. Bottom line: This work identifies the only presently known conditions under which RH is proved—precisely those encoded by the sequential PTT regime. Whether these conditions describe the real world is a physical question; once physics confirms that reality operates under them, one may declare: “RH is always true in the conditions of the real.”