Spectral approximation and smeared Sorkin–Johnston convergence for two-dimensional sprinkled causal sets

We study the massless Pauli–Jordan operator sampled uniformly in a flat two-dimensional causal diamond. Exact permutation and twisted-Fourier reductions, dependent-permutation concentration, and finite-dimensional spectral comparisons give growing-index relative eigenvalue fidelity through j_N=o(N/log N), together with a global absolute approximation bound. A general bounded-Hermitian-kernel moment argument proves almost-sure convergence of smeared Sorkin–Johnston observables for each fixed pair of bounded measurable tests, simultaneously for continuous spectral functions. Vanishing cutoffs preserve this smeared limit. A spectral-index sandwich counts the positive modes above the standard truncation threshold sqrt(N)/(4*pi) on iDelta_N. Rank-coordinate eigenvector and window-projector comparisons are presented as numerical evidence. No Hadamard regularity, entropy convergence, universal spectral knee or physical cutoff prescription is established. The Cauchy bulk law remains conjectural and is not used. AI assistance: ChatGPT, Claude and Codex assisted with planning, exploration, literature searches, drafting, numerical implementation, related formalization work and adversarial review. The author checked and revised their outputs and takes responsibility for the content; the systems are not authors.