Odd Zeta Values from the PCF Torus: φ and π as Arithmetic-Geometric Sources

Abstract

We demonstrate that the odd values ζ(2k+1) of the Riemann zeta function are structurally determined by φ=(1+√5)/2 and π. The Euler product of ζ is built from local factors fp(s), one for each prime. We prove that the Frobenius lift φp = Fpφ + Fp-1—where Fn is the n-th Fibonacci number—determines the splitting type of every prime p in ℤ[φ], and hence every local factor fp(s). These lifts constitute 𝔽₁-descent data in the sense of Borger. Through the Dedekind factorisation ζ(s) = ζℚ(√5)(s)/L(s,χ5), this determines ζ(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,χ5) = 2logφ/√5 (the class-number formula for ℚ(√5)), which expresses the L-function value entirely in terms of φ. This resolves the apparent freedom of ζ(2k+1) noted by Elvang, Herderschee and Morales in the N=4 SYM S-matrix bootstrap: those values are free only relative to the EFT; the pentagonal arithmetic fixes them.