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NS-TSKR

Navier–Stokes Tangent Singular Kernel Rigidity Program · Cycle X · 4/4 built · global regularity OPEN

NS-TSKR continues from NS-IDRP Cycle IX: IDRP reduced the surviving impulse obstruction to a "tangential singular-impulse phantom" (TSIP), whose failure modes split into TANG (tangential classification), ADJ (adjoint synchronization), SKER (singular-kernel classification), and AMP (amplitude). Core question: if a source burst sits inside clean/model geometry, and is therefore not itself a defect, must its dynamics still become visible through some other native channel? Four rounds carry this forward: TSKR-01 proves pressure alone cannot universally recover the forcing-relevant source direction; TSKR-02 uses the actual quadratic source geometry to prove rank-one decay rigidity and global coarse-grained fixed-point rigidity; TSKR-03 handles the localized version and finds a genuine harmonic branch; TSKR-04 completes the Cycle X final audit.

TSKR-04 concludes that energy- and flux-invisibility compress the two-sided mismatch stress to an at-most-one-dimensional hyperbolic mismatch fiber w ∥ u×Su; second-order zero-base degeneracy re-enters the rank-one decay theorem and is largely excluded; harmonic-pressure tail recurrence is routed into the existing harmonic-tail ledger. Cycle X thereby reduces TRSK to a "two-sided residual phantom" (TSRP), formally handing off to "NS-RKAP: Navier–Stokes Residual Kernel and Amplitude Program." The confirmed relay so far: NS-CSP (II) → NS-DRC (III) → NS-ANP (IV) → NS-CFOP (V) → NS-FCBP (VI) → NS-MORP (VII) → NS-DCRP (VIII) → NS-IDRP (IX) → NS-TSKR (X) → NS-RKAP (XI).