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_{\min}=2\pi/L<1kmin=2π/L<1 with a nonzero unstable mode count (UT_LINEAR_BAND). The instrument consists of two localized Gaussian injections (σ_inj=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)u_0\sim\mathcal 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=\mathrm{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 τ\tauτ is computed from the truth-table responses via τ=12(max⁡(r00,r11)+min⁡(r10,r01))\tau=\tfrac12(\max(r_{00},r_{11})+\min(r_{10},r_{01}))τ=21(max(r00,r11)+min(r10,r01)), with margin margin=12(min⁡(r10,r01)−max⁡(r00,r11))\mathrm{margin}=\tfrac12(\min(r_{10},r_{01})-\max(r_{00},r_{11}))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=0.02850r_{00}=0.03910,\ r_{10}=0.34434,\ r_{01}=0.18465,\ r_{11}=0.12765,\ \tau=0.15615,\ \mathrm{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.