Sharp Refinements to Appendix C of OpenAI's Finite Time Blowup for Navier–Stokes: Explicit Variance Selection, Sharp Second-Order Moment Asymptotics, and Internal Exact Restoration

Abstract

This research note gives three sharp refinements to Appendix C, “Realizing the admissible stress cone,” of OpenAI’s 2026 preprint Finite Time Blowup for Navier–Stokes. First, the implicit scalar variance-selection step in Lemma C.1 is replaced by an algebraic Poisson-kernel parameterization. The associated upper admissible-cone root is also rationalized, giving an explicit algebraic lower bound once coarse bounds on the cone data are fixed. Second, the cumulative-moment estimate used in Proposition C.2 is sharpened by one full order. For the logarithmic corrugation at frequency N log X, the five cumulative radial moments satisfy ΔmN = O(N−2), rather than the recorded O(N−1) bound. At the end of a completed corrugation interval, N2ΔmN = d∞ + O(N−1), where d∞ is computed explicitly from the quadratic phase averages. The vector d∞ is nonzero at every parameter value where the completed corrugation is nontrivial, so the second-order scale is sharp. The exact five-mode restoration therefore has physical size O(N−2), with leading coefficient −B−1d∞. Third, the exact moment restoration need not be deferred to OpenAI’s first later power-law patch reserved for correction after cone realization. The corrugation interval already terminates inside a strictly admissible power-law region, and the barycentric loop equals the base shear on a full right boundary collar. The same five-dimensional transverse correction can therefore be supported inside that collar. All five moments are restored exactly before the corrugation interval ends, after which the original exterior pressure, radial velocity, integrated stress, shear, and leading stress propagate unchanged. The first dedicated post-corrugation restoration patch is therefore unnecessary for that purpose. The finite-time blowup theorem itself is not changed, and the edge analysis in Appendix C.3 remains necessary. The contribution is a more explicit, quantitatively sharp, and structurally leaner realization of the admissible-stress correction in Appendix C. The note is intended as a technical companion to the OpenAI Navier–Stokes construction and as an independently checkable treatment of the shear, moment, stress-cone, and exterior-propagation interfaces used in that part of the proof.