The Navier–Stokes Proof, Step by Step: What OpenAI Claimed, What the Machines Checked, and Where the Argument Breaks
A step-by-step walkthrough of the argument in OpenAI's Navier–Stokes blowup manuscript — the claim, the construction, and the closing argument of §10.3 — with a machine-verdict attached to every step where a machine can decide it.
This is the walkthrough paper of the series: the argument in OpenAI’s “Finite Time Blowup for Navier–Stokes”, step by step, with a machine-verdict attached to every step where a machine can decide it.
Each part runs two lanes: a main lane in plain language (the claim, the shape of the comparison argument, the geometry of why the flux estimate fails), and a technical lane with the actual estimates (the difference-energy identity, the pressure recovery, the two flux sums, the claimed closure, the counterexample family and its exact ε² scaling).
The walk ends where the argument breaks. The closing argument of §10.3 compares the constructed blow-up solution with any putative smooth solution and shows their difference must vanish. The load-bearing estimate is (10.19), the pressure-flux bound — and its derivation makes three support claims that are each false for every admissible cutoff, with the flux supporting them and the proposed controls living on disjoint sets. The counterexample (in the companion paper) makes the geometry precise: a family for which the flux is nonzero while the right-hand side of (10.19) is exactly zero.
What the walk settles: (8.19) is sound (formalized, kernel-accepted); the Lean code proves a different argument (the A_R-dependent bound the geometry requires — note the direction of the dependency arrow); the repair’s ingredients hold (formalized, kernel-verified); and the full closure under the corrected estimates is the one honestly-open item — a reading, not yet a verdict.
Every checkpoint in the paper carries a non-empty evidence pointer: a manuscript page and quote, a fork file/line/commit, or the signed report — the structure the test harness checks on every commit.
Companion pieces: the counterexample, the verified formalization, and the code in detail.