The Cumulant Bridge: Reducing the Moment Hypothesis to a Single Distributional Condition

We reformulate the Moment Hypothesis for the Riemann zeta function as a cumulant boundedness condition and show that it reduces to a single distributional hypothesis about the Dirichlet polynomial approximation at \(\sigma = 1/2\). Working in the log-domain with \(X(t) = \log|\zeta(1/2+it)|^2\), the multiplicative structure of \(\zeta\) becomes additive, and the Euler product over primes gives an exact cumulant decomposition. The approximate functional equation provides a second decomposition into modulus and phase components. We prove three unconditional results: (1) the per-prime cumulant sum \(\sum_p \kappa_m(2\operatorname{Re}(X_p))\) converges for \(m \geq 3\) (Theorem 3); (2) a reflection symmetry forces the cross-cumulant \(\kappa_{2,1}(\operatorname{Re}(X_p), \operatorname{Im}(X_p)) = 0\) exactly for every prime (Theorem 2); (3) the AFE phase correction has bounded cumulants by equidistribution (Theorem 5). The full Moment Hypothesis then follows from a single hypothesis (H1): that the \(L^m\) norms of the correction \(\varepsilon = \log D - \sum_p X_p\) remain bounded. Hypothesis (H1) is strictly weaker than the Moment Hypothesis itself — it concerns only the non-multiplicative truncation error of the Dirichlet polynomial, not the full zeta function moments. Numerical evidence at \(T\) up to \(20{,}000\) supports all conditions. The algebraic core of the reduction chain is machine-verified (10 proved theorems, 14 Mathlib references, 0 novel axioms, 0 type errors).