Two Navier–Stokes Papers, One Instrument: A comparative audit supplement for the general reader

Two papers on the three-dimensional Navier–Stokes equations were read by the same instrument on the same day. The first, Integrable Misalignment Forbids Blowup, is an eight-page paper that proves two continuation criteria from the classical base and closes the regularity question only under one disclosed premise. The second, Finite Time Blowup for Navier–Stokes, published under the author line OPENAI, is a 166-page construction claiming a smooth force that drives a viscous fluid to infinite velocity in finite time at bounded energy, which if correct resolves two of the four Millennium alternatives. This supplement compares them on identity, method, proof surface, verification state, hidden assumptions, parsimony, induction, falsifiability, physical anchoring, and provenance, and states three rulings. Neither paper is certified: both sit at the same seat, awaiting a verifier who is not the author. The axis that separates them is falsifiability. The first paper names in advance what would refute it and can be met by a fluid or a computer; the second, by its own exponents, places its singularity beyond every finite observation window, so nothing in the world can reach it and only a referee can. The second paper is also, if its towers close and to this auditor's knowledge, the first hard witness of a Navier–Stokes blowup in circulation, and the first paper's falsifier survives contact with it: extended to a smooth force, the criterion predicts that any such blowup carries a divergent misalignment integral, and a scaling of the construction's core gives exactly that divergence. The second paper also ships a Lean 4 formalization bound to a third-party formal statement of the Clay problem, which moves its certification question from prose to a build any reader can run; that build has not yet been reproduced by anyone outside the claimant, and a concurrent human-and-model program by Buckmaster and Alpöge, with Lean-verified forced blowups for Euler and two related equations and a public statement raising timing and data questions, holds the provenance door open. The unforced existence-and-smoothness problem, the one most people mean by the Millennium problem, remains open, and only the first paper addresses it. The supplement closes with a census of every infinity each paper asserts, under a strict placement law, and finds that the first paper asserts none.