Audit of OpenAI's Navier-Stokes Formalization: Weak Bochner vs. Clay Statement C

An audit of OpenAI’s Lean 4 formalization for 3D forced Navier-Stokes identifies internal structural contradictions and fatal specification mismatches that break the claimed C∞ blowup construction. The formalization proves a bounded-forcing lemma yet injects an explicit forcing term into the momentum evolution, creating a direct logical mismatch between assumptions and the evolution equation. A time-dependent frame transformation Q(t) used in the code can produce coordinate-artifact divergence: blowup occurs in transformed coordinates while the physical Eulerian velocity and forcing remain smooth. Local stage lemmas bound each packet’s forcing by stage-dependent constants F_k and V_k, but the global solution sums infinitely many stages; if F_k or V_k grow without uniform control the global force violates the bounded-forcing hypothesis. Evolution is stated using weak Bochner derivatives in TimeLp T L2 rather than pointwise classical derivatives, so the formal proof addresses a weak solution class that does not satisfy the Clay Problem’s Statement C. Structure parameter guards can also render hypothesis domains vacuous, making Blowup provable only from impossibility.

Concrete numeric verification of the paper’s recurrence relations shows mutually exclusive parameter constraints and catastrophic scaling failures. Using the paper’s formulas produces a stage-J ratio log(P_J)/log(k_J) ≈ 3.68×10^6 versus the required ≤10^-8, and asymptotics would need j>89,000, which in turn forces other parameters to explode. Arbitrary-precision tests compute a pressure Hessian increment at j=3 of order 10^698, destroying coercivity for linear solvers; forcing x0 large enough to suppress that spike makes high-moment H^m norms diverge, so the initial datum cannot be C∞. Included scripts (parsers, a Bochner-vs-C∞ type checker, and ph_bound_verification.jl) reproduce these failures and demonstrate that the parameter tuple required by the inductive construction does not exist for finite choices, invalidating the claimed C∞ blowup.