TSKR-01: Tangent Source Geometry, Complementary Channel Recovery, Sign-Changing Stress Kernels, Adjoint Synchronization Compatibility, and Residual Rigidity

Tangent Source Geometry, Complementary Channel Recovery, Sign-Changing Stress Kernels, Adjoint Synchronization Compatibility, and Residual Rigidity

IDRP Cycle IX reduced the surviving impulse obstruction to a "tangential singular-impulse phantom" (TSIP), with failure mode TANG∨ADJ∨SKER∨AMP. This paper attacks the geometry of tangential source bursts and the singular residual kernel directly. Core question: if a source burst sits inside clean/model source geometry, and is therefore not itself a source defect, must its dynamics still become visible through some other native channel? It proves an operator-level NO-GO: there exists a smooth, Fourier-localized, symmetric source tensor whose pressure–source symbol is zero, yet whose Leray-projected divergence is nonzero — so active pressure alone cannot universally recover the forcing-relevant source direction. It proves a stronger forcing-pairing NO-GO: a large causal forcing pairing can come entirely from the clean/model source subspace. It defines the "tangential-reproduction principle": a tangential source is not a defect, and must be tested through coordinates such as state reproduction, flux, energy, trace, strain/model-cone, and increments. It proves an exact/perturbative adjoint synchronization theorem for the causal and PFET finite-window dualities under shared terminal data.