Finite-Window Singularity Audits and Local-to-Clean Defect Transfer for Navier-Stokes

Technical Summary: Finite-Window Singularity Audits and Local-to-Clean Defect Transfer for Navier–Stokes

1. Problem Statement
The paper addresses the local regularity problem for the three-dimensional incompressible Navier–Stokes equations. Specifically, it investigates the structural requirements for a "non-CKN branch"—a sequence of scales where a suitable weak solution fails to enter the Caffarelli–Kohn–Nirenberg (CKN) smallness regime, which would otherwise guarantee local regularity. The central question is: If a singularity persists across scales, what precise mechanisms allow it to survive finite-window observations, hide from standard observables, and reproduce from scale to scale while remaining realizable by the Navier–Stokes equation?

The paper does not claim to prove the regularity of Navier–Stokes solutions nor to construct a singular solution. Instead, it aims to reduce the problem of potential singularities to a specific, highly structured algebraic and analytic obstruction.

2. Methodology
The paper employs a two-part framework: a Finite-Window Singularity Audit and a Local-to-Clean Defect Transfer.

• The Audit (Payment and Hiding): The authors model the persistence of "badness" (critical scale quantities) as a financial ledger. They demonstrate that for a non-CKN branch to persist, it must generate either "untaxed critical supply" (energy or flux not dissipated) or "accumulated leakage" (losses due to cutoffs and localization). This is formalized through a finite-scale critical ledger inequality derived from the local energy inequality and standard interpolation estimates.
• The Transfer (Local-to-Clean): The paper constructs a mapping between "localized" defect packages (which contain artifacts like cutoffs, harmonic pressure tails, and truncation errors inherent to finite-window analysis) and "clean" defect packages (finite-dimensional models where these artifacts are removed or explicitly accounted for). The core technical tool is an algebraic transfer theorem that lifts a "clean anti-phantom gap" (a coercive estimate in the clean model) back to the localized setting, provided specific structural inputs hold.

3. Key Contributions and Innovations
The paper introduces four primary innovations:

1. Finite-Scale Payment Ledger: A quantitative alternative showing that persistent critical badness forces cumulative untaxed supply or leakage. This converts the qualitative question of "does a singularity exist?" into a quantitative accounting problem.
2. NS-Realizable Defect Audit: A rigorous filter distinguishing "formal phantoms" (algebraic kernel vectors of observation maps) from "NS-obstructions." A dangerous obstruction must satisfy coarse momentum equations, pressure compatibility, Reynolds positivity, and the energy-flux identity inherited from suitable weak solutions.
3. Local-to-Clean Transfer Theorem: An algebraic result stating that if a clean model has a quotient anti-phantom gap, and if quotient lifting, componentwise detection comparison, and residual normalization hold, then localized non-gauge defects are quantitatively detected.
4. Residual-Budget Closure Map: A systematic decomposition of all potential errors obstructing the transfer (pressure tails, cutoff leakage, truncation loss, nonlinear mismatch, reproduction drift, gauge mismatch, and profit discrepancy) into explicit finite-window ledger entries.

4. Key Results

• Theorem 3.3 (Finite-Scale Survival Alternative): Along any admissible scale-window chain, if the reservoir badness remains bounded below, the cumulative sum of untaxed supply minus taxes must be positive, or accumulated leakage must be non-negligible.
• Theorem 9.5 (Conditional Singularity-Audit Alternative): Under assumptions of dyadic defect extraction, observable depletion, and moving-window growth control, a persistent non-CKN branch implies one of five alternatives:
  • (a) Non-negligible accumulated leakage.
  • (b) Positive average untaxed critical supply.
  • (c) Ineffective moving-window observability (constants grow too fast).
  • (d) Existence of an NS-realizable, cleaned, scale-critical, combined-invisible, profitable, and reproducible defect cascade.
  • (e) A failure of vertical-duality or anti-phantom closure in one-component degeneration regimes.
• Theorem 17.4 (Algebraic Local-to-Clean Anti-Phantom Transfer): If the clean detection map satisfies a coercive gap condition ($c_{cl}^\Lambda$), and the residual budget satisfies a normalized bound ($\eta_\Lambda < c_{cl}^\Lambda(1-\epsilon_G)$), then the localized detection map satisfies a coercive estimate with a positive constant $c_{loc}^\Lambda$. This theorem is purely algebraic; its application to Navier–Stokes depends on verifying the structural inputs.
• Corollary 18.1 (Unified Alternative): Synthesizes the audit and transfer results. It concludes that if leakage is negligible, supply is taxed/depleted, constants are controlled, and NS-realizable PRV (Profitable, Reproducible, Verified) cascades are excluded, then the branch must enter a CKN scale, implying regularity.

5. Significance and Claims
The paper explicitly frames itself as a conditional reduction. Its significance lies in isolating the exact nature of the remaining obstruction to Navier–Stokes regularity.

• What is Proved: The paper proves the finite-scale ledger estimate (an unconditional PDE input) and the algebraic transfer theorem (a structural result under explicit hypotheses). It also provides a detailed bookkeeping of all residual terms that must be controlled.
• What Remains Open: The paper identifies the "genuine remaining blocker" as the normalized absorption problem. Specifically, one must prove component compatibility estimates of the form:
  $$\text{Err}_\bullet(D) \leq \eta_\bullet \text{dist}_{loc}(D, \text{Im } G_{loc}^\Lambda) + \Delta_\bullet$$
  where the sum of coefficients $\eta_\bullet$ is strictly less than the clean gap margin.
• The Final Obstruction: If regularity cannot be proved, the paper argues that any counterexample must be a highly organized object: an NS-realizable, cleaned, combined-invisible, profitable, and reproducible defect cascade. Conversely, if such a cascade can be constructed while satisfying all compatibility, positivity, and energy constraints, it would constitute a singularity.

In summary, the paper transforms the broad question of Navier–Stokes regularity into a precise structural challenge: either prove that the specific residual budgets can be absorbed into the localized quotient distance with sufficient smallness, or construct a specific type of defect cascade that evades all finite-window observability and algebraic constraints. The work does not claim to have solved the regularity problem but provides a rigorous framework to define exactly what a solution (or counterexample) must look like.