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Abstract:We prove a finite-scale supply--tax reduction for suitable weak solutions of the three-dimensional incompressible Navier--Stokes equations near the local regularity threshold. Along an admissible chain of parabolic windows, let B_k = A_k + C_k + D_k denote the scale-critical reservoir badness. The main PDE theorem says that if B_k stays outside a decay basin for many consecutive scales, then either cutoff/window leakage accumulates or the solution generates a positive cumulative amount of untaxed critical supply. The supply terms come from nonlinear flux, pressure transport, interpolation amplification, and pressure regeneration; the tax terms come from viscous dissipation, expected decay of old reservoir coordinates, and harmonic pressure decay. The theorem is deliberately finite-scale. It is not a proof of global regularity and it is not a construction of a singular solution. Its role is to turn persistent scale-critical badness into an explicit accounting alternative: badness can survive only by paying through untaxed supply or through non-negligible leakage. The second part of the paper interprets the ledger coordinates as a PDE-realizable finite-scale defect package. This leads to a dynamic obstruction language in which a dangerous mechanism must be profitable, reproducible, Navier--Stokes-realizable, and visible or invisible through specified pressure, flux, energy, and trace channels. The final part records finite-window quotient tests, a clean positive-cone anti-phantom theorem, and a conditional localized transfer framework. These results are finite-window or conditional tools built around the main ledger theorem, not an unconditional Navier--Stokes regularity theorem.

Submission history

From: Runlong Yu [view email]
[v1] Thu, 11 Jun 2026 20:24:00 UTC (53 KB)