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Lemma 4 and Theorem 5: under 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. An interior minimizer under δ-only noise would refute Theorem 5(ii).

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 precomputed.