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Lemma 4 and Theorem 5 — whole-matrix noise: the sign of Ξ decides

The other demos noise only the cross-block. Here every entry of the covariance is perturbed: rank-one exchangeable displacements of amplitude e_α on the A block, e_δ on the D block, and e_ζ on the cross-block, with independent signs (8 equally likely states, averaged exactly). Theorem 5 says interiority is decided by the sign of Ξ = (9963/320)e_ζ² + (1674/125)e_α² − (189/500)e_δ², with 1 − γ* ≈ Ξ/((d−a)²c²K) at small amplitudes. Noise confined to the D block makes Ξ < 0 and full coupling stays optimal: the support of the noise does not decide the question, the sign of Ξ does.

What to look for: with any noise on the cross or A channels the marked minimizer is interior and the measured 1 − γ* tracks the Ξ prediction at small amplitudes. Click "δ only": Ξ goes negative and the minimizer parks at γ = 1 exactly, at every amplitude.

The widget runs the recursion of the paper's equation (4) with the naive fitness (5), ported directly from the reference implementations, live in your browser. Nothing is pre-baked.