TSKR-02: Quadratic Source Tangency, Velocity Reproduction Rigidity, Flux/Energy Response, Sign Fibers, and Coarse-Graining Fixed-Point Classification

Quadratic Source Tangency, Velocity Reproduction Rigidity, Flux/Energy Response, Sign Fibers, and Coarse-Graining Fixed-Point Classification

TSKR-01 reduced the surviving tangential singular obstruction to TRSK. This paper replaces an arbitrary source tensor with the actual quadratic source geometry F^act = ηu⊗u and F^mod = η(U⊗U+R), paired with Reynolds-covariance positive-definiteness. It proves a pointwise "rank-one decay rigidity" theorem: the exact tangency identity u⊗u = U⊗U+R (R positive semidefinite) forces U = θu, |θ| ≤ 1, R = (1-θ²)u⊗u — under incompressibility of u and U, θ is constant along u's streamlines, and the exact coarse flux reduces to -(1-θ²)θu·∇(|u|²/2), so flux-invisibility alone cannot force R = 0. It proves a regularized Reynolds-variance identity, so zero regularized covariance gives exact local velocity reproduction on the support of the mollifying kernel. On the exact tangential positive-covariance branch, energy-invisibility forces R = 0, reducing U to a pointwise sign fiber U = ±u. It proves a global coarse-grained fixed-point rigidity theorem: for a non-degenerate probability mollifier, a whole-space tensor pinned by a single convolution scale is zero.