A Schwarzian inequality for the Riemann zeta function and its real dynamics

We prove a two-sided moment inequality for the Riemann zeta function: for Aⱼ (s) = sum₍>=₂ (log n) ʲ n^ (-s), one has A₂ (s) ² 1. The upper bound is equivalent to negativity of the Schwarzian derivative, whose endpoint limits are 6 gamma₁ and - (log 2) ²/2. Our proof separates three discrete atoms from a truncated gamma integral and bounds the Euler-Maclaurin remainder explicitly; all finite inequalities used in the proof are certified by ball arithmetic. A convexity argument then shows that zeta composed with itself has exactly one fixed point on (1, infinity), and every other orbit alternates toward 1 and infinity. This resolves Conjecture (i) of OEIS A344428. On the negative axis, we construct expanding two-branch horseshoes in every band -4k-2, -4k, k >= 5, and finite horseshoes with arbitrarily large symbolic alphabets. Consequently the supremum of entropy over compact negative-real invariant sets is infinite. We also enumerate exactly the two-cycles meeting (0, 1), one per trivial zero -2m for m >= 9, and derive their asymptotics. For the Hurwitz family, proved shape and endpoint identities and a conditional multiplier asymptotic are separated from unproved numerical bifurcation observations. We distinguish the resulting Cantor repellers from the unrestricted bounded-orbit set, which contains an interval. MSC 2020: Primary 11M06, 37E05; Secondary 26D15, 11M35, 37B40. Archive contents (zip): LaTeX source, abstract, verification scripts with recorded logs (python-flint 0. 9. 0 / Arb ball arithmetic; mpmath 1. 3. 0), and expository proof notes. See README. md and CHECKSUMS. txt inside the zip.