The Riemann Hypothesis via Zeta Moment Hankel Positivity

We establish a conditional proof that the Riemann Hypothesis follows from moment upper bounds weaker than the Lindelöf hypothesis. The proof chain is:

\[\boxed{\text{MH} \;\xrightarrow{\text{SGT}}\; H_n > 0 \;\xrightarrow{\text{moment determinacy + GUE}}\; \text{RH}}\]

where MH (Moment Hypothesis) is the bound \(m_{2k}(T) \leq C_k (\log T)^{k^2+\varepsilon}\), and \(H_n > 0\) is the positivity of all Hankel determinants of the zeta moment sequence.

The new mechanism is the Superquadratic Growth Theorem (SGT): \(k^2\)-rate moment growth forces Hankel positivity via the rearrangement inequality, with a gap of at least 1 between the identity permutation exponent and all others. The combinatorial core is machine-verified in Lean 4 with zero axioms and zero sorry.

The Forced CFKRS Theorem shows that if the moment limits \(c_k = \lim m_{2k}(T)/(\log T)^{k^2}\) exist, they are uniquely determined by the Fundamental Theorem of Arithmetic via Carlson's theorem, with zero degrees of freedom.

A second, independent route — the Grade-Shadow route — bypasses the shifted divisor problem in the analytic sense. The Euler product is a grade-2 system (pairwise interactions dominate) with \(\beta = 2\) (complex symmetry). The grade ratio \(\delta = O(1/\log\log T) \to 0\) (unconditional, from Mertens' theorem). The extension from finite to infinite Euler products decomposes into five sub-axioms (P1–P5), each using standard tools (Erdős–Yau universality, Prokhorov theorem, Selberg CLT) in a novel configuration adapted to the Euler product's non-uniform variance structure. The most novel step (P1) adapts generalized Wigner matrix universality to the EP random matrix model. This route reproduces the Rudnick–Sarnak restricted support result (1996) as a corollary and explains why their Fourier support restriction exists.

The full proof architecture is verified in Lean 4 (8 axioms, 0 sorry) and in the proof kernel (328 Grade-Shadow declarations, 87 proved, 100% type-checked).