Riemann Hypothesis via Self-Dual Trace Confinement: A Conditional Closure

In plain words

Move the zeta function into a trace setting. Reflection across the critical line acts as a mirror on the zeros. Keep a weighted ledger of the nontrivial zeros.

A translation theorem shows the odd part of that ledger vanishes exactly when every zero sits on the critical line. The Self Dual Trace Confinement law then supplies conditions under which the odd part must vanish, and a named hypothesis packages those conditions.

The paper then measures what this is worth. The named hypothesis holds exactly when every represented zero sits on the line, so on the plain shell of all zeros it is equivalent to the Riemann hypothesis, and the same holds for growing mirrored windows of actual zeros. The conditional theorem is a reformulation, not evidence. On finite windows, several natural bounds one might hope to prove are again equivalent to the hypothesis. The work is formalized in Lean with Mathlib.

What it shows

• A translation between the zero ledger and the critical line statement.
• A conditional landing chain through Self Dual Trace Confinement.
• A calibration showing the named hypothesis is equivalent to the Riemann hypothesis.
• Finite window results where natural bounds are again equivalent to it.
• A Lean mechanization using Mathlib's statement of the hypothesis.

What it does not claim

It is not a proof and offers no evidence for the hypothesis, since its premise is equivalent to it. No arithmetic estimate is supplied, the intended Selberg type trace formula is not constructed, and the shell's admissibility is not established.

Cite

Tsiokos, I. (2026). Riemann Hypothesis via Self-Dual Trace Confinement: A Conditional Closure. Zenodo. https://doi.org/10.5281/zenodo.23086975