Faithful Terminal Descent of Autonomous Histories in Periodic Navier–Stokes Flow

Faithful Terminal Descent of Autonomous Histories in Periodic Navier–Stokes Flow Strong original-field recovery, exact proper-past energy, and registered observations This fourth paper in the periodic Navier–Stokes continuation series investigates the relationship between a hypothetical finite-time singularity of smooth, unforced three-dimensional periodic flow and the bounded ancient solution obtained through terminal magnification. Building on the continuation anatomy, nonlinear spectral constraints, and critical endpoint rigidity developed in Papers I–III, it establishes a mathematical framework for determining which observations of the original physical evolution survive in the ancient limit. The principal result is faithful terminal descent: a family-relative recovery theorem that preserves one generating history, its original pressure gauge, physical and spectral scales, and observation clocks. The analysis establishes strong compact joint (L^2) convergence of the original canonical pressure, strong compact (L^2) convergence of the original spatial gradients on the proper ancient past, exact inheritance of weighted viscous dissipation, and the full local kinetic-energy equality on fixed proper-past windows. Exact periodic-to-Euclidean Fourier correspondence and matched-filter corrections permit consistent recovery of localized pressure and signed nonlinear work. Every prescribed countable family of admissible compact observations admits a strict refinement of the same original profiles recovering all registered finite prefixes. A complementary historical-energy theorem establishes positive net donor work between the genuine rescaled initial face and a fixed ancient observation time at every sufficiently large fixed spatial radius. Together, these results characterize necessary observational and energetic relationships governing the generation of ancient Navier–Stokes profiles. The analysis combines classical PDE methods with Lean-formalized results to establish a coherent, auditable connection between original dynamics and their terminal limiting states. It provides a mathematical foundation for subsequent investigations of autonomous reachability, pressure-mediated transport, and dynamical rigidity. The conclusions are conditional on hypothetical finite-time breakdown and leave the global-regularity question open.