Identical Local Data and Spectra, Different Navier–Stokes Dynamics: Constructive non-identifiability and sharp pressure-recovery bounds

AI-generated manuscript: not independently verified ChatGPT generated the research content, mathematical arguments and proofs, and manuscript text. SONTAEYOUNG proposed the topic and uploaded the manuscript; SONTAEYOUNG did not conduct the research or independently verify the mathematical claims or proofs. The exact ChatGPT model/version and generation transcript are not supplied in this deposit. This document is shared for critical examination. Theorems and proofs in the text are AI-generated claims, not independently validated results. No external peer review or proof-assistant formalization is reported. Correctness, novelty, and the adequacy of citations require independent assessment; no publication-priority claim is made. Depositor and citation SONTAEYOUNG is the human depositor/distributor. Appearance of this name in creator metadata or an automatic citation identifies the depositor of the record and does not assert human authorship of the research. Please describe this item when citing it as an AI-generated, unverified manuscript deposited by SONTAEYOUNG. Data and code availability This deposit contains the disclosed PDF manuscript only. No manuscript source, verification scripts, dependency specification, or execution outputs are supplied. Earlier statements that code and execution outputs accompany the manuscript have been corrected. No independently reproduced computational results are claimed. This disclosure revision is not a mathematical validation. Reuse CC0 1.0 Universal applies to any copyright or related rights the depositor holds and can waive in this document. This notice does not assert copyright over uncopyrightable AI output and does not waive third-party rights. Abstract (from the manuscript; unverified claims) We construct periodic, real-analytic, mean-zero, divergence-free velocity fields that agree in every prescribed finite-order spatial jet at a point, in all Fourier coefficients below a prescribed finite cutoff, and in their complete modewise velocity spectral tensors, yet have opposite nonzero values of a pressure-Hessian component at that point. The fields can be normalized to have equal energy. On an explicit finite-parameter family, the pressure observable is exactly linear in the coefficients, allowing a sharp minimax recovery formula even when the full spectral tensors are supplied. Additional linear observations with bounded deterministic noise lead to matching primal and dual expressions for the optimal error. For the actual unforced Navier–Stokes solutions at any fixed positive viscosity, the corresponding small-time velocity-gradient prediction error equals time times the static recovery error, up to a uniform quadratic remainder. A normalized example with 20 signed initial Fourier modes matches all spatial derivatives through order five; one additional spatial observation resolves its one-dimensional ambiguity. The results concern deterministic information limits under specified observations, not average prediction errors for natural turbulence or nonuniqueness from identical full initial data.