Spectral Stability, Fatou Basin Contraction, and Machine-Checked Multipliers in Holomorphic Extensions of Generalized Syracuse Dynamics

Abstract

We embed the Syracuse operator $T(n)=n/2$ (even $n$), $T(n)=(3n+1)/2$ (odd $n$) and the Collatz map $C(n)$ into $\mathbb{C}$ via entire functions of exponential type $\pi$: $f(z)=\frac{1}{4}(1+4z-(1+2z)\cos (\pi z))$ and $F(z)=\frac{1}{4}(2+7z-(2+5z)\cos (\pi z))$. We prove five results: (1) every integer periodic cycle $C\subset {\mathbb{N}}^{+}$ of $T$ is a strictly attracting Fatou cycle of $f$ with multiplier ${\mu }_f(C)=3^{N_{\mathrm{odd}}}/2^N={\prod }_{n\in C\text{ odd}}\frac{3n}{3n+1}\in (0,1)$, and ${\mu }_f(O_{1,2})={\mu }_F(O_{1,4,2})=3/4<1$; (2) on the Fatou basin $B(O_{1,2})$, $g=f^{\circ 2}$ conjugates to ${\Phi }_j(g(z))=\frac{3}{4}{\Phi }_j(z)$ with Koenigs Taylor jets $c_2(1)=-\frac{13}{2}{\pi }^2$ and $d_2(2)=-\frac{9}{2}{\pi }^2$; (3) the Schwarzian derivative is strictly negative on $\mathbb{Z}$ ($S(f)(n)\le 3{\pi }^2-\frac{3}{8}{\pi }^4\approx -6.9196<0$), and a repelling Julia 2-cycle $O_{\mathrm{rep}}\subset J(f)$ ($\mu \approx 1.2148>1$) isolates $B(O_{1,2})$; (4) on $({\mathbb{Z}}_2,{\mu }_{\mathrm{Haar}})$, the 2-adic Lyapunov exponent ${\lambda }_{\mathrm{Haar}}=\frac{1}{2}\ln (3/4)\approx -0.143841<0$ equals $\frac{1}{2}\ln |{\mu }_f(O_{1,2})|$; and (5) we certify 20 theorems in Lean 4 (anc/CollatzAttractor.lean, 0 sorry). These results establish local Fatou invariants and 2-adic ergodic averages for $f(z)$, leaving open whether every $n_0\in {\mathbb{N}}^{+}$ lies in ${\bigcup }_{k\ge 0}{f}^{-k}(B(O_{1,2}))$.

arXiv comment3 pages, 3 tables; includes ancillary Lean 4 formalization (anc/CollatzAttractor.lean, 20 machine-checked theorems, 0 sorry) and Python verification suite (anc/collatz_complex_core.py, anc/collatz_mass_verifier.py). Companion replication archive: https://doi.org/10.5281/zenodo.24060394 (Concept DOI: https://doi.org/10.5281/zenodo.24060393)