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 A_j(s) = sumn>=2 (log n)^j n^(-s), one has A_2(s)^2 < A_1(s)A_3(s) < (3/2)A_2(s)^2 for every s > 1. The upper bound is equivalent to negativity of the Schwarzian derivative, whose endpoint limits are 6 gamma_1 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.