Three Conditional Clay-Problem Closures: A Shared Gaussian Observation and a Common Vocabulary

In plain words

Six Birds produced conditional closures of Navier Stokes regularity, the Riemann hypothesis and P versus NP, each resting on its own open premises. This paper does not reprove them. It compares them in two ways: one shared formula, and one shared vocabulary.

The shared formula is a family of Gaussian observations that can be evaluated on all three. For a finite window of zeta zeros mirrored across the critical line, the observation's excess is a positive multiple of how far the zeros sit from the line. On a fluid it restates the mild Navier Stokes equation. On a logic formula it gives a mass that is positive exactly when the formula is satisfiable. These identities are checked in Lean. They share a formula, not a mechanism, and move no premise from one problem to another.

The vocabulary sorts attempted proofs aimed at a target into four typed outcomes, though that these are exhaustive is assumed, not proved. Four conditional schemas, including a seven stage template, describe closures that name their load bearing source instead of deriving it. A catalogue of typed obstruction detectors ends in three named transports whose proofs are open.

What it shows

• A shared Gaussian identity across the zeta, fluid and SAT settings, verified in Lean.
• A four state classification of proof attempts, under an assumed coverage schema.
• Conditional schemas for recognition mode landings, illustrated by two of the applications.
• A catalogue of typed detectors with three open functorial gates.

What it does not claim

It adds no unconditional result about any of the problems and reproves nothing. The Gaussian identities transfer no premise between problems, and the recognition mode schemas are stated, not proved.

Cite

Tsiokos, I. (2026). Three Conditional Clay-Problem Closures: A Shared Gaussian Observation and a Common Vocabulary. Zenodo. https://doi.org/10.5281/zenodo.23086999