An Exact Symmetry-Locked Physical-Time Counterconstruction to First-Order Angular Turnover with Matched-Path and Fixed-Scale Transfer-Threshold Classifications

Abstract

M3R-NOV-02 Public v1.0 This public technical note studies an explicit smooth periodic divergence-free family for the three-dimensional incompressible Navier–Stokes equations and isolates a symmetry point at which the resolved heat-filtered strain and the deviatoric caloric covariance are constrained to opposite traceless diagonal shapes. For every fixed positive heat parameter, the preserved symmetry yields a nonzero heat-parameter-dependent physical-time interval on which the normalized favorable stress–strain alignment remains exactly saturated, $\chi=1$. Independently, a kinematic saturation lemma gives $\dot\chi=0$ at every sufficiently regular saturated state. The note distinguishes two observation protocols that must not be conflated: 1. a calorically matched path $\tau(t)=T-t$; and2. physical-time differentiation at fixed positive heat scale. For the matched path, exact finite Fourier algebra yields a family-specific transfer-threshold function with a unique interior minimizer. Exact Sturm analysis and rational enclosure certify $$0.63651056784173490941<q_*<0.63651056784173490942$$ and $$5.78812722135752519109224751394<R_{\min}<5.78812722135752519155412626161.$$ For the fixed-scale protocol, an independent exact control yields a distinct threshold that is strictly decreasing on $0<q<1$ with limiting infimum $3$. Consequently, within the declared explicit family, symmetry-locked alignment geometry, local strain growth, and favorable cross-scale transfer growth are not interchangeable dynamical statements. The package includes reproducible symbolic verification of divergence freedom, pressure-Poisson compatibility, pressure Hessian, strain derivative, heat-covariance formulas, matched-path derivative, fixed-scale derivative, exact Sturm root count, and certified threshold enclosure. ### Claim boundary This work is family-specific and local in physical time. It does not establish long-time persistence, finite-time singularity formation, global Navier–Stokes regularity, or a Millennium Prize solution. The value $R_{\min}$ is not a universal Reynolds-number threshold. No heat-scale-uniform persistence interval is claimed. The manuscript is not peer reviewed and independent external line-by-line certification is not claimed. Worldwide novelty and priority over unidentified prior art are not claimed. The earlier v0.3 Zenodo record remains the restricted historical review state. This v1.0 release is the canonical public version for mathematical inspection, independent reproduction, and defect reporting. **Prior restricted DOI:** 10.5281/zenodo.22673450 **Public v1.0 DOI:** 10.5281/zenodo.23063122 Copyright © 2026 Alexanja Senke. All rights reserved.