SAPZ Singularity Principle for Navier–Stokes (v3.7): Subcriticality Criterion, Reverse Concentration, and Closure

# Overview This record releases **v3.7** of a two-paper set developing the **SAPZ singularity principle** for the 3D incompressible Navier–Stokes equations, organized around a *renormalized trace-energy* functional \(δ(t)\) and a sharp-looking **subcriticality threshold**\[δ_c \;=\; ν^2\,y_+ .\]The companion paper packages the analytic core into a referee-facing module chain:**reverse concentration** (Theorem 19.9) \(⇒\) **\(ε\)-regularity + continuation closure** (Theorem 20.2). - **Main paper:** *SAPZ Singularity Principle for Navier–Stokes* (PDF + TeX)- **Companion:** *Aux Proof* (PDF + TeX), containing the closure modules and the detailed error ledger The guiding thesis is that **finite-time singularity is equivalent to threshold crossing** of \(δ(t)\), and that **uniform subcriticality** \(t∈(0,T)δ(t)<δ_c\) enforces regularity/continuation on \((0,T]\). # Closed results (as presented in v3.7) The companion paper provides the decisive closure chain in two steps: - **Reverse concentration principle (Theorem 19.9).** A bad cylinder scenario forces a quantitative injection of local dissipation into the SAPZ trace-energy at a matched scale, up to a fully ledgered family of error channels (pressure-local, harmonic pressure, cutoff/transport, commutator/residual, boundary normalization). - **Closure via \(ε\)-regularity and continuation (Theorem 20.2).** Reverse concentration excludes the \(ε_*\)-concentration alternative, yielding local smoothness and hence BKM/Serrin-style continuation on any \((0,T]\) under SAPZ subcriticality. # “Budget” inequality (core ledger form) A representative matched-scale injection inequality driving the closure is of the form\[δBNε_r(s_r)\;≥\;cᵢₙⱼ\; r⁻²\!\!_{Qρ_r(x_r,t_r)} |∇ u|^2\,dx\,dt\;-\;(press(r)+cut(r)+comm(r)+BN(r)),\]with explicit bookkeeping/absorption ordering (e.g. \(1/8\)-type splits) used to guarantee a strictly positive net injection at contradiction scales. # What is new in v3.7 - **Referee-facing packaging of the closure chain:** Theorem 19.9 (reverse concentration) and Section 20 (closure) are rewritten to be read as a short, explicit interface: “Section 20 consumes only the statement of Theorem 19.9.”- **Ledger transparency:** the absorption/Young bookkeeping is made explicit so that “positive remainder after absorbing all error channels” is checkable at the level of coefficients.- **Cross-reference hygiene:** Part/Section/Theorem numbering is synchronized across the main and companion papers (Part V / Sections 19–20 / Theorems 19.9 and 20.2). # Scope & non-toy status - The framework is stated for the classical 3D incompressible Navier–Stokes system with viscosity \(ν>0\).- Boundary effects are handled through an explicit boundary-normalization (BN) channel and flattening bookkeeping in the companion.- The numerical reproducibility material (if present) is separated from the proof-critical modules; the proof chain in v3.7 is organized as a purely analytic dependency graph. # Program closure and targets The set is designed as a *closure program* with a single decisive analytic bottleneck:reverse concentration \(⇒\) elimination of concentration \(⇒\) \(ε\)-regularity \(⇒\) continuation.The companion paper isolates all error channels into an explicit ledger so that each step can be independently audited. # Author Lee Byoungwoo # Recommended citation Lee Byoungwoo, *SAPZ Singularity Principle for Navier–Stokes (v3.7): Subcriticality Criterion, Reverse Concentration, and Closure*, Zenodo (2026). # Keywords Navier–Stokes equations; global regularity; blow-up criterion; local energy inequality; \(ε\)-regularity; reverse concentration; pressure decomposition; Calderón–Zygmund; BKM–Serrin continuation; boundary flattening.