Odd Zeta Values from the PCF Torus

Keywords: Riemann Hypothesis, Odd zeta values, 𝔽₁ (field with one element), Elliptic curves, λ-rings, String theory, Category theory, Golden ratio, Mersenne primes, Formal verification.

We demonstrate that the odd values $\zeta(2k+1)$ of the Riemann zeta function are structurally determined by $\varphi=(1+\sqrt{5})/2$ and $\pi$. The Euler product of $\zeta$ is built from local factors $f_p(s)$, one for each prime. We prove that the Frobenius lift $\varphi^p = F_p \varphi + F_{p-1}$—where $F_n$ is the $n$-th Fibonacci number—determines the splitting type of every prime $p$ in $\mathbb{Z}[\varphi]$, and hence every local factor $f_p(s)$.

These lifts constitute $\mathbb{F}_1$-descent data in the sense of Borger. Through the Dedekind factorisation $\zeta(s) = \zeta_{\mathbb{Q}(\sqrt{5})}(s)/L(s,\chi_5)$, this determines $\zeta(s)$ completely: the isomorphism between the Frobenius structure and the Euler product is demonstrated at every level (primes, splitting types, local factors, $L$-function), anchored by the base case $L(1,\chi_5) = 2\log\varphi/\sqrt{5}$ (the class-number formula for $\mathbb{Q}(\sqrt{5})$), which expresses the $L$-function value entirely in terms of $\varphi$.

This resolves the apparent freedom of $\zeta(2k+1)$ noted by Elvang, Herderschee and Morales in the $\mathcal{N}=4$ SYM S-matrix bootstrap: those values are free only relative to the EFT; the pentagonal arithmetic of the PCF Torus fixes them.

Zenodo DOI: 10.5281/zenodo.19492791

Repository: omega-pcf/02-odd-zeta

Journal: Prepared for SIGMA (Symmetry, Integrability and Geometry: Methods and Applications).