The Architecture of the Riemann Hypothesis: Thermal Dynamics, Noncommutative Geometry, and Integrable Systems

This paper provides a unified view of three major contemporary approaches to the Riemann Hypothesis (RH): the thermal evolution of De Bruijn–Newman, the integrable dynamics of the Toda lattice, and Connes' noncommutative geometry. We show how the RH translates into positivity of the Hankel determinants associated with the Xi function and how this positivity is linked to Jacobi matrices and a Lax flow. We identify three central barriers that remain open: the absence of a well-defined convergent invariant, the impossibility of propagating positivity without assuming the conclusion, and the lack of a proof of total positivity (PF_infinity) for the kernel Phi(u).

For the first barrier, we propose a regularized relative invariant and test it numerically with arbitrary-precision arithmetic (150 decimal digits, Jacobi coefficients up to n=20, thermal parameter t in [0,5]). Our computations confirm the strict positivity of all Hankel determinants and Jacobi coefficients at t=0, consistent with the RH. However, the proposed relative invariant is not conserved along the flow: a quantitative fit of the boundary terms yields c_N(5)-c_N(0) ≈ 4.3×10⁻³ + 3.9×10⁻³/N, confirming a nonzero limit as N tends to infinity.

We further test the Fredholm determinant det(I + K_t), where K_t is the difference of truncated Jacobi matrices. This quantity is also not conserved: its logarithm grows linearly with the truncation size N, with a per-site contribution of approximately 6.1×10⁻³ matching the boundary term variation. This demonstrates that the non-conservation is an extensive (bulk) phenomenon, ruling out both the relative regularization and the simplest Fredholm approach.

The Python code for all numerical experiments is available as supplementary material.