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 $A_j(s)=\sum_{n\ge2}(\log n)^j n^{-s}$, one has $A_2(s)^2< A_1(s)A_3(s)<\tfrac32 A_2(s)^2$ for every $s>1$. The upper bound is equivalent to negativity of the Schwarzian derivative, whose endpoint limits are $6\gamma_1$ and $-\tfrac12(\log2)^2$. 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 $(1,\infty)$, 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 $[-4k-2,-4k]$, $k\ge5$, 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 $m\ge9$, and derive their asymptotics. For the Hurwitz family we prove a uniform numerator inequality for parameters in $(0,1]$, positive Schwarzian points for $1 MSC 2020:Primary 11M06, 37E05; Secondary 26D15, 11M35, 37B40. Version 3.0.0 (2026-09-05): Section 5.6 added — the Hurwitz family below the pole: certified trapping intervals, certified period-three orbits (hence all periods and positive topological entropy), and the numerically observed Feigenbaum cascade. New Section 6 on complex two-cycles through the inverse chart at the pole (reduction to 1-points of zeta, Laurent expansions, position neutrality; numerical maps). Section 7 records the arithmetic nature of uniqueness (Dedekind companion paper). Archive contents (zip): LaTeX source, abstract, all certificates with logs/manifests/provenance, numerical records, proof notes, change log. See README.md and CHECKSUMS.txt.