Reduction of the Riemann Hypothesis to the Constancy of the Spectral Entropy of Liouville Sums (With a Possible Proof Strategy via Dixmier Measurability)

This is a conditional reduction, not a proof of the Riemann Hypothesis. The numerical evidence supports the conjecture, but the proof of constancy of entropy is open.

We reduce the Riemann Hypothesis to the statement that the local Shannon entropy of the spectral power of the twisted Liouville sum is constant for all non-trivial zeros of the Riemann zeta function.

We prove that the contribution of a single zero is a Lorentzian whose width is the real part of the zero, and that the local entropy depends only on this width. Thus, if the entropy is constant, all zeros share the same real part, which must be 1/2 by Hardy's theorem.

Extensive numerical evidence (up to K=50000, 50 zeros) shows the entropy is constant at ~0.103 ± 0.003. We also outline a possible rigorous proof of this constancy via Dixmier measurability (Ponge–Tian) and the KMS₁ state (Bost–Connes).

Version 4 is a complete rewrite: it humbly presents a conditional reduction, not a definitive proof, and invites community scrutiny. All data and code are available on Zenodo.