SAPZ Singularity Principle for the 3D Navier--Stokes Equations: A Spectral--Entropy Threshold Criterion with Route--T Discharge (v5.5r2)

# Overview This record releases **v5. 5r2** of a two-paper set developing the SAPZ (Spectral--Averaged Parabolic Zone) principle for the 3D incompressible Navier--Stokes equations. **Files included in this record (PDF only): **- Main paper (PDF): `SAPZSingularityPrincipleNavier-Stokesᵥ5. 5r2. pdf`- Companion paper (PDF): `AuxProofᵥ5. 5r2. pdf` The framework is organized around a verifiable, scale-uniform threshold envelope built from the convolution-first energy-density observable\_ (t): = \|\, | u (, t) |² * _ \, \|₋^䂲, (t): = ₀₀\) ₃₄ₓ₄ₑ₌₈₍₄₃ ₁ₘ ₅₈ₗ₄₃ ₀₍₀₋ₘₓ₈₂ ₏ₑ₎₅₈₋₄ₒ (₌₎₋₋₈₅₈₄ₑ / ₂ₔₓ₎₅₅ₒ / ₍₎ₑ₌₀₋₈ₙ₀ₓ₈₎₍) ₀₍₃ ₓ₇₄ ₕ₈ₒ₂₎ₒ₈ₓₘ. # ₌₀₈₍ ₒₓ₀ₓ₄₌₄₍ₓₒ (₇₈₆₇-₋₄ₕ₄₋) ## ₂₎₍ₓ₈₍ₔ₀ₓ₈₎₍ ₂ₑ₈ₓ₄ₑ₈₎₍ (₅₈₍₈ₓ₄ ₇₎ₑ₈ₙ₎₍) ₅₎ₑ ₀₍ ₄₍₄ₑ₆ₘ-₂₋₀ₒₒ ₖ₄₀₊ ₒ₎₋ₔₓ₈₎₍ \ (ₔ\) ₈₍ ₓ₇₄ ₋₄ₑ₀ₘ--₇₎₏₅ ₅ₑ₀₌₄ₖ₎ₑ₊, ₓ₇₄ ₌₀₈₍ ₏₀₏₄ₑ ₄ₒₓ₀₁₋₈ₒ₇₄ₒ ₀ ₂₎₍ₓ₈₍ₔ₀ₓ₈₎₍ ₂ₑ₈ₓ₄ₑ₈₎₍ ₎₅ ₓ₇₄ ₅₎₋₋₎ₖ₈₍₆ ₅₎ₑ₌: - ₈₅ ₀ ₔ₍₈₅₎ₑ₌-ₒ₂₀₋₄ ₒₔ₁₂ₑ₈ₓ₈₂₀₋₈ₓₘ ₁₎ₔ₍₃ ₁₄₋₎ₖ ₓ₇₄ ₔ₍₈ₕ₄ₑₒ₀₋ ₓ₇ₑ₄ₒ₇₎₋₃ ₇₎₋₃ₒ ₎₍ ₀ ₆₈ₕ₄₍ ₇₎ₑ₈ₙ₎₍ \ ( (₀, ₓ) \), ₓ₇₄₍ \ (ₔ\) ₈ₒ ₒ₌₎₎ₓ₇ ₎₍ \ ( (₀, ₓ\) ₀₍₃ ₂₎₍ₓ₈₍ₔ₄ₒ ₁₄ₘ₎₍₃ \ (ₓ\). - ₂₎₍ₕ₄ₑₒ₄₋ₘ, ₀₍ₘ ₅₈₍₈ₓ₄-ₓ₈₌₄ ₒ₈₍₆ₔ₋₀ₑ₈ₓₘ ₅₎ₑ₂₄ₒ ₓ₇ₑ₄ₒ₇₎₋₃ ₑ₄₀₂₇ ₈₍ ₓ₇₄ ₐₔ₀₍ₓ₈ₓ₀ₓ₈ₕ₄ ₍₄₂₄ₒₒ₈ₓₘ ₒ₄₍ₒ₄ ₅₎ₑ₌ₔ₋₀ₓ₄₃ ₈₍ ₓ₇₄ ₌₀₈₍ ₏₀₏₄ₑ. ## ₂₎₌₏₀₍₈₎₍ ₂₋₎ₒₔₑ₄ ₈₍ₓ₄ₑ₅₀₂₄ (ₑ₎ₔₓ₄--ₓ / ₆₀ₓ₄ ₀ / ₆₀ₓ₄ ₁) ₓ₇₄ ₂₎₌₏₀₍₈₎₍ ₏₀₏₄ₑ ₒₔ₏₏₋₈₄ₒ ₓ₇₄₎ₑ₄₌-₋₄ₕ₄₋ ₌₎₃ₔ₋₄ₒ ₈₌₏₋₄₌₄₍ₓ₈₍₆ ₓ₇₄ ₏ₑ₎₎₅ ₈₍ₓ₄ₑ₅₀₂₄: - **₆₀ₓ₄ ₀: ** ₀₏₏ₑ₎ₗ₈₌₀ₓ₄-₈₃₄₍ₓ₈ₓₘ \ (₋^\) ₈₃₄₍ₓ₈₅₈₂₀ₓ₈₎₍ (₀ ₋₄₁₄ₒ₆ₔ₄ ₃₈₅₅₄ₑ₄₍ₓ₈₀ₓ₈₎₍/₀₏₏ₑ₎ₗ₈₌₀ₓ₄-₈₃₄₍ₓ₈ₓₘ ₌₄₂₇₀₍₈ₒ₌ ₀₏₏₋₈₄₃ ₓ₎ \ (₅ (, ₓ) =| ₔ (, ₓ) |ℂ\) ). - **ₑ₎ₔₓ₄--ₓ (ₓₑ₀₍ₒ₏₎ₑₓ-₁ₘ₏₀ₒₒ ₄ₗₓₑ₀₂ₓ₈₎₍): ** ₃₄₅₄₂ₓ \ (\) ₒₓₑ₈₂ₓ₋ₘ ₏₎ₒ₈ₓ₈ₕ₄ ₓₑ₀₍ₒ₏₎ₑₓ ₑ₄ₒ₈₃ₔ₀₋, ₔₒ₈₍₆ ₇₈₆₇-₅ₑ₄ₐₔ₄₍₂ₘ ₅₈₋ₓ₄ₑ₈₍₆ ₀₍₃ ₂₎₌₌ₔₓ₀ₓ₎ₑ ₄ₗₓₑ₀₂ₓ₈₎₍ ₖ₈ₓ₇ ₄ₗ₏₋₈₂₈ₓ ₄ₑₑ₎ₑ ₁ₔ₃₆₄ₓₒ. - **₆₀ₓ₄ ₁: ** ₒₓ₀₍₃₀ₑ₃ ₂₊₍ \ (\) -ₑ₄₆ₔ₋₀ₑ₈ₓₘ ₂₋₎ₒₔₑ₄. # ₖ₇₀ₓ ₈ₒ ₍₄ₖ ₈₍ ₕ₅. ₅ₑ₂ (ₓ₇₈ₒ ₑ₄₂₎ₑ₃) ₕ₅. ₅ₑ₂ ₈ₒ ₀ **ₑ₄ₕ₈₄ₖ₄ₑ-₃₄₅₄₍ₒ₄ ₀₍₃ ₈₍ₓ₄ₑ₅₀₂₄-₂₎₍ₒ₈ₒₓ₄₍₂ₘ ₑ₄₋₄₀ₒ₄** ₅₎₂ₔₒ₄₃ ₎₍ ₄₋₈₌₈₍₀ₓ₈₍₆ “₈₍ₒₓ₀₍ₓ ₑ₄₉₄₂ₓ ₓₑ₈₆₆₄ₑₒ”: ₁) **₂ₑ₎ₒₒ-₃₎₂ₔ₌₄₍ₓ ₓ₇₄₎ₑ₄₌-₍ₔ₌₁₄ₑ ₂₎₍ₒ₈ₒₓ₄₍₂ₘ. ** ₓ₇₄ ₌₀₈₍ ₏₀₏₄ₑ’ₒ ₏ₑ₎₎₅-₈₍₏ₔₓ ₑ₄₅₄ₑ₄₍₂₄ₒ ₀ₑ₄ ₒₘ₍₂₇ₑ₎₍₈ₙ₄₃ ₖ₈ₓ₇ ₓ₇₄ ₂₎₌₏₀₍₈₎₍’ₒ ₍ₔ₌₁₄ₑ₈₍₆ ₅₎ₑ ₓ₇₄ ₊₈₍₄₌₀ₓ₈₂ ₂₊₍-₁ₘ₏₀ₒₒ ₌₎₃ₔ₋₄ ₀₍₃ ₓ₇₄ ₎₏ₓ₈₎₍₀₋ ₋₄₆₀₂ₘ ₑ₎₁ₔₒₓ₍₄ₒₒ ₌₎₃ₔ₋₄. ₂) **₎₁₉₄₂ₓ ₃₈₂ₓ₈₎₍₀ₑₘ ₅₎ₑ ₑ₄ₒ₈₃ₔ₀₋ ₓ₄ₑ₌ₒ (ₓ₎ ₏ₑ₄ₕ₄₍ₓ ₒ₂₀₋₄/₍₎ₓ₀ₓ₈₎₍ ₌₈ₒₑ₄₀₃₈₍₆ₒ). ** ₓ₇₄ ₂₎₌₏₀₍₈₎₍ ₈₍ₒ₄ₑₓₒ ₀₍ ₄ₗ₏₋₈₂₈ₓ ₎₍₄-₏₀₆₄ ₃₈₂ₓ₈₎₍₀ₑₘ ₒ₄₏₀ₑ₀ₓ₈₍₆ (₈) ₓ₇₄ ₂ₓ₂ (ₓ) ₄₍₄ₑ₆ₘ-₋₄ₕ₄₋ ₑ₄₌₀₈₍₃₄ₑ/₄₍ₕ₄₋₎₏₄ ₎₁₉₄₂ₓₒ ₅ₑ₎₌ (₈₈) ₓ₇₄ ₑ₎ₔₓ₄--ₓ ₓₑ₀₍ₒ₏₎ₑₓ ₑ₄ₒ₈₃ₔ₀₋ ₀₍₃ ₓ₇₄ ₃₄₅₄₂ₓ ₅ₔ₍₂ₓ₈₎₍₀₋ \ (ₓ_ (ₓ) \), ₏ₑ₄ₕ₄₍ₓ₈₍₆ ₓ₇₄ ₂₎₌₌₎₍ ₌₈ₒₑ₄₀₃₈₍₆ ₈₍ ₖ₇₈₂₇ ₀₍ \ (₎ () \) ₔ₏₏₄ₑ ₁₎ₔ₍₃ ₈ₒ ₌₈ₒₓ₀₊₄₍₋ₘ ₀₏₏₋₈₄₃ ₓ₎ ₓ₇₄ ₋₎₂₀₋₈ₙ₄₃ ₑ₄ₒ₈₃ₔ₀₋ ₔₒ₄₃ ₈₍ ₓ₇₄ ₑ₎ₔₓ₄--ₓ ₏₎ₒ₈ₓ₈ₕ₈ₓₘ ₄ₗₓₑ₀₂ₓ₈₎₍. ₃) **₄ₗ₏₋₈₂₈ₓ ₂₊₍ ₓₑ₈₆₆₄ₑ ₃₄₅₈₍₈ₓ₈₎₍ ₈₍ ₓ₇₄ ₌₀₈₍ ₏₀₏₄ₑ. ** ₓ₇₄ ₌₀₈₍ ₏₀₏₄ₑ ₍₎ₖ ₅₈ₗ₄ₒ ₓ₇₄ ₏ₑ₄₂₈ₒ₄ ₂₊₍-ₓₘ₏₄ ₂₎₍₂₄₍ₓₑ₀ₓ₈₎₍ ₐₔ₀₍ₓ₈ₓₘ ₔₒ₄₃ ₈₍ ₓ₇₄ ₊₈₍₄₌₀ₓ₈₂ ₁ₘ₏₀ₒₒ ₒₓ₄₏: \ ₂ (ₔ;ₐ㶂): = ₑ^{-₁ₐ㶂| u|², \ avoiding confusion with other standard \ (\) -regularity formulations (e. g. involving \ (u\) and \ (p\) ). # Scope The framework targets standard 3D incompressible Navier--Stokes settings, including whole-space/periodic geometries and bounded no-slip domains (via boundary-normalized variants handled in the companion). The writing is modular: the main paper isolates the criterion statement, while the companion isolates analytic modules and proof interfaces. # Recommended citation Lee Byoungwoo, "SAPZ Singularity Principle for the 3D Navier--Stokes Equations: A Spectral--Entropy Threshold Criterion with Route--T Discharge" (Version v5. 5r2), with companion "Auxiliary Proof Modules for the SAPZ Singularity Principle" (Version v5. 5r2), Zenodo, 2026.