TSKR-03: Localized Quadratic Fixed Points, Harmonic Rank-One Rigidity, Nodal Sign Fibers, One-Sided Covariance Tangents, and Residual Fixed-Orbit Classification

Localized Quadratic Fixed Points, Harmonic Rank-One Rigidity, Nodal Sign Fibers, One-Sided Covariance Tangents, and Residual Fixed-Orbit Classification

After TSKR-02's global quadratic fixed-point rigidity, what remains is local: a finite-window branch might hide behind cutoff localization, low frequencies, harmonic modes, nodal sign changes, or linearized sign-changing residuals. This paper proves a localized coarse-graining commutator theorem: after multiplying by an interior cutoff, the high-frequency part of the local tangential quadratic source is controlled by the local fixed-point mismatch plus an explicit boundary/localization commutator — so once high-frequency source mismatch is small, localized approximate tangency can only hide in low-frequency/non-local boundary sectors. It shows the naive localization of the global fixed-point theorem is wrong: for radial spatially-averaged kernels, every component-wise harmonic tensor is an exact local fixed point. It proves a rank-one harmonic-matrix rigidity theorem and a dynamic harmonic-shear rigidity theorem for the projected Navier–Stokes/heat evolution. It proves analytic nodal-sign rigidity: for real-analytic u, U, u⊗u = U⊗U forces a single global sign U = ±u.