Volume 41 - Instrumented XOR in a 2D KS-Family Substrate: Deterministic Certificate via Locked Injection - ROI Readout

This volume demonstrates a deterministic, unit-test-gated realization of a 2-input XOR gate as instrumented computation in a scalar 2D Kuramoto–Sivashinsky (KS-family) substrate on a periodic domain. The notebook implements a pseudo-spectral ETDRK4 integrator with 2/3 dealiasing and locks a canonical configuration (dtype=float64; N=256; L=16. 0; dt=0. 01; ν=1. 0). A required unstable-band guard verifies kmin⁡=2π/L<1k_=2/L<1kmin=2π/L<1 with a nonzero unstable mode count (UTLINEARBAND). The instrument consists of two localized Gaussian injections (σᵢnj=3 px) at A= (88, 128) and B= (128, 128) with signB = −1, applied once to a warmed base state (2000 steps from u0∼N (0, 1) u₀ N (0, 1) u0∼N (0, 1) scaled to std=0. 01), followed by a fixed probe horizon (300 steps). Readout is the mean absolute field r=mean (∣u∣) r=mean (|u|) r=mean (∣u∣) over a disk ROI centered at (116, 128) (radius 12 px) with an exclusion mask (radius 6 px) to prevent trivial injection leakage (effective ROI pixels=392). A single threshold ττ is computed from the truth-table responses via τ=12 (max⁡ (r00, r11) +min⁡ (r10, r01) ) =12 ( (r₀₀, r₁₁) + (r₁₀, r₀₁) ) τ=21 (max (r00, r11) +min (r10, r01) ), with margin margin=12 (min⁡ (r10, r01) −max⁡ (r00, r11) ) margin=12 ( (r₁₀, r₀₁) - (r₀₀, r₁₁) ) margin=21 (min (r10, r01) −max (r00, r11) ). The attested production run yieldsr00=0. 03910, r10=0. 34434, r01=0. 18465, r11=0. 12765, τ=0. 15615, margin=0. 02850r₀₀=0. 03910, \ r₁₀=0. 34434, \ r₀₁=0. 18465, \ r₁₁=0. 12765, \ =0. 15615, \ margin=0. 02850r00=0. 03910, r10=0. 34434, r01=0. 18465, r11=0. 12765, τ=0. 15615, margin=0. 02850, producing bits (0, 1, 1, 0). Robustness checks include amplitude jitter around amp=10 (tested set 9. 0, 9. 5, 10. 0, 10. 5, 11. 0, 9. 2) with all XOR passes and minimum margin 0. 02156, plus cross-seed replay (seeds 1 and 2) preserving XOR with margins 0. 07184 and 0. 06828. The notebook emits a hashed artifact bundle (cfg/env pins, arrays, probe ledger, manifest, and Zenodo bundle metadata) for byte-addressable replay. Scope note: this certificate is for a fixed discretization and fixed instrument geometry under the stated horizon; it does not claim universality across resolutions, domains, or long-time turbulent mixing.