From Ergodic Attention to Einstein's Equations: Explicit Computational Verification of the Spectral Gap–Area Law–Gravity Chain

The Algorithmic Theory of Reality (ATR) derives the Einstein Field Equations from three information-theoretic axioms via a four-link proof chain: (i) ergodic attention produces Lindblad operators that generate the full matrix algebra; by the Frigerio-Verri theorem, this guarantees an irreducible generator with a unique steady state and strictly positive spectral gap Δ > 0; (ii) the spectral gap implies exponential correlation decay at length ξ ≤ vLR/Δ; (iii) exponential decay yields the mutual information area law I(Σ:Σc) ≤ η0 · A(∂Σ), which produces the variational area law δS = η0δA; and (iv) the variational area law, combined with the first law of entanglement thermodynamics and the Raychaudhuri equation, produces the Einstein equations Gμν + Λgμν = (8πG/c4)Tμν via the Jacobson argument. Each link has been proven rigorously in the parent framework. What has not been done is an explicit, end-to-end computational verification on a concrete system. This paper provides that verification. We construct explicit Lindblad generators on a periodic one-dimensional lattice of n = 4 to n = 8 qubits (Hilbert space dimension d = 16 to d = 256; superoperator dimension up to 65,536 × 65,536) with spatially local, attention-parameterized POVM elements and energy-dependent transition rates, and subject each link to 12 independent numerical falsification tests. Key Findings & Computational Results Comprehensive Falsification Suite: All 12 checks pass for n = 5 to 8. At n = 4, Checks 6 and 7 (exponential correlation decay) fail because a 4-site periodic chain provides only two distance points—a trivial finite-size limitation. Robust Spectral Gap: The spectral gap ratio Δ/(γ/d2) grows from 14x to over 2,400x the universal worst-case bound, confirming that the gap remains well above the exponential floor at all system sizes tested. Information-Theoretic Prerequisites: No geometric or gravitational inputs are assumed—the area-law scaling and the emergent Newton's constant arise entirely from the Lindblad dynamics. This explicitly verifies the pre-geometric mechanisms that feed into the Jacobson derivation. Code Availability The complete computational suite, including the matrix-free Arnoldi and dense eigendecomposition solvers used to generate these results, is open source and available at: GitHub Repository.