Reduction of the Riemann Hypothesis to a Universal Barrier for Twisted Liouville Sums

This paper proves a barrier lemma for a finite model of prime pairs. The reduction of RH is conditional on a conjecture; the paper does not prove RH.

We construct a finite-dimensional tensor model based on pairs of primes and introduce the spectral sum S_K(γ) = ∑ λ(n) n^{-iγ}.We prove a Universal Barrier Lemma for the model with two prime factors: |S_K(γ)| ≤ C√K for an absolute constant C.
Numerical experiments show that the full Liouville sum exhibits the same barrier with C ≈ 4.6 and that its local maxima coincide sharply with the imaginary parts of the non-trivial zeros of the Riemann zeta function.
The resonance is destroyed by any deformation of the phases, demonstrating an optimal rigidity.
Using the unconditional Riemann–Weil explicit formula, we show that the existence of an off-critical zero would force the spectral sum to grow and eventually violate the universal barrier.
Consequently, the Riemann Hypothesis is reduced to the single conjecture that |S_K(γ)| ≤ C√K log^A K for some absolute constants C, A.
A strategic roadmap based on Halász regularization and Dixmier trace convergence is provided in the Appendix.
All numerical data and scripts are available as supplementary material on Zenodo.