Spectral structure of noise thresholds for contextuality: Walsh modes, a heralded-CHSH invariant, and model dependence

Noise thresholds of contextuality witnesses depend both on the quantum behaviour and on the noise model used as a yardstick. We make this dependence explicit for products of odd-cycle (KCBS) scenarios. Mixing-type noise models act diagonally in the Walsh basis of each context, with eigenvalues ρ^N(A) counting the blocks touched by a mode; the threshold of a fixed witness is therefore a root of a polynomial in ρ whose coefficients are mode weights of the behaviour. This framework is standard in essence; we use it to separate two kinds of statements. For the CHSH window of a face of C_m ⊗ C_n we show that in several noise models (local reset L_bits, bit-flip BF, local reset to an admissible outcome A_wr, reset of the joint context J_bits, and every product spectral noise) contextuality of the window is governed by a single condition on the visibility of a heralded CHSH sub-behaviour, v_eff(η) = (1+S)^(−1/2), and that the contextual fraction of the window is given in closed form and never exceeds √(1+S) − 1. This window law follows from standard ingredients (Fine's theorem, Landau's identity for the CHSH operator, substitution of an effective visibility); we record it with explicit threshold values. The window threshold is not the face threshold in general: under bit-flip noise a certificate (exact noncontextual bound, rigorous interval evaluation) shows that the face of C_5 ⊗ C_5 stays contextual at η = 0.1609, above the window value 0.1531. By contrast, laws previously established for one noise model — the max-rule for product states and the equality of the face threshold with the window threshold — fail under other models; we give exact certificates over Q(√5) (new values obtained by standard linear-programming methods). Finally, by a standard expansion with rigorous tail bounds we give the expansion of the CH-branch constant κ*_n up to O(n^−3) with an explicit remainder bound, by residue class of n mod 3.