A holonomy action for U(1) gauge fields on causal sets, I: the exact Maxwell mean and L² convergence in 1+1 dimensions

Version 3 (October 2026) — Paper I of two. The record is split: this version is Paper I (the 1+1-dimensional theory, 20 pages); the 3+1-dimensional material is Paper II, 10.5281/zenodo.23374678. Changes over Version 2, after two external audits, an internal red/green/verification cycle and a further independent referee-style audit: the smooth-field theorem is stated with an additive remainder and its admissible centre–volume–boost domain and boundary-strip estimates are written out in full (Lemmas 4.3, 4.6); the L² corollary is stated in absolute form on bounded C¹ classes, the relative form appearing only under a lower bound on ∫F² (sufficient, not necessary); the 1+1 theorems are stated for every finite-support family with m₂ = 0 (limit −m₂^log/12), the normalisation is stated for ρE[S], the raw-scheme formula carries its exponential tail and the exact truncated-moment formula is displayed; the U(1) object is described as the real line-integral lift (non-compact quadratic flux functional); a product-order remark explains why the 1+1 closed forms are dominance-order combinatorics; and the novelty paragraph is narrowed: the chain-pair loop and its squared circulation already appear in Sverdlov's 2008 construction, and what is claimed is the cardinality-weighted all-pairs observable, its moment system and the convergence theorems. Nothing in Version 2's 1+1 results is withdrawn. The Version 2 status notice remains on Version 2.

Abstract. We construct a gauge-invariant action for a background U(1) connection on causal sets obtained by Poisson sprinkling of Minkowski spacetime: an Alexandrov-cardinality-weighted sum of squared real-lifted holonomies of four-edge chain-pair loops (chain pairs) with finite-support coefficients C_n. In 1+1 dimensions, on a causal diamond, the sprinkling mean is computed exactly at every density for every coefficient family and constant field strength; for a nonzero constant field the normalized mean ρE[S] has a finite continuum limit iff one moment of the pair-count kernel vanishes (m₂ = 0), a condition no nontrivial family of nonnegative coefficients satisfies. For every C¹ field strength and every finite-support family with m₂ = 0 the mean is the Maxwell action, E[S] = −(m₂^log/12ρ) ∫_D F² + R_ρ(F), with an additive remainder bounded explicitly in ‖F‖_∞, ‖∂F‖_∞ and the diamond size; the proof writes out the admissible centre–volume–boost domain of the pair measure and shows that the moment condition killing the boost-logarithm also annihilates the boost-channel dependence on the field profile to leading order. A Poincaré-inequality bound on the add-one-point cost gives Var[S] ≤ c‖F‖_∞⁴ V_D ln²(ρV_D)/ρ³, hence L² convergence of ρS to the Maxwell action at rate ln(ρV_D)/√(ρV_D), uniformly in the absolute sense on bounded C¹ classes. Sprinkling simulations reproduce the exact finite-density means. The 1+1 closed forms are dominance-order combinatorics: the causal order is a product of two chains in null coordinates. Higher dimensions, where the loop bivector is not boost invariant, are treated in Paper II.

Files. The manuscript (PDF, 20 pages) and its LaTeX source; a zip with the 1+1 verification scripts and their outputs (the mean/variance script with fixed seeds; the chart Monte Carlo and Table 1 reproduction) and the two figures.