A holonomy action for U(1) gauge fields on causal sets: an exact Maxwell mean in 1+1 dimensions and the structure of the higher-dimensional limit

We construct a gauge-invariant action for a background U(1) connection on causal sets obtained by Poisson sprinkling of Minkowski spacetime, built from holonomies of minimal closed causal loops weighted by Alexandrov-interval cardinality. In 1+1 dimensions we prove the sprinkling mean converges to the Maxwell action for every C¹ field strength, with L² convergence — an unconditional theorem; the canonical Benincasa–Dowker-type coefficient alternation does not cure this action, and the required moment-kill condition is derived. In 2+1 dimensions we derive the kill conditions and predicted coefficient with a numerical pilot consistent at the few-percent level. In 3+1 dimensions we solve the kill system and prove the Maxwell limit for constant field strength on a causal diamond via a Lorentzian covariogram identity, with E[S] → (3/20π)∫F² and all constants in closed form; a parity lemma shows the kill system and continuum constant are region-independent on the integer-power covariogram class. To our knowledge this is the first gauge-field action on a causal set whose sprinkling mean is proven to converge to the Maxwell action. The smooth-field and variance analyses in 3+1 dimensions remain open. A plain-language companion note and reproducibility scripts are included.