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

STATUS NOTICE (October 2026) — external adversarial audits: core confirmed, two results downgraded pending revision. Two independent external audits (October 2026) have reviewed this version. Independently reconfirmed: the 1+1-dimensional exact mean, its kill ray C₃ = −C₂/5, and the 1+1-dimensional L² theorem; the 2+1-dimensional cancellations (40, −25, 6); the 3+1-dimensional bivector constant b₄ = 2/(5π); the causal-diamond covariogram identity; the kill ray (14, −16, 9, −2) with m₀ = m½ = m₁ = 0 and mˡᵒᵍ₁ = −1/2; and (provisionally) the constant-field Maxwell mean 3/(20π) on the causal diamond. Downgraded pending a revised version: (1) Lemma 8.5(i) is false as stated — the two-1-Lipschitz-graph criterion does not characterise compact causally convex regions (a rectangle satisfies the criterion but is not causally convex); the general-region covariogram formula and the region-independence package are therefore NOT established at theorem grade. The causal-diamond results are unaffected, having separate proofs. (2) Proposition 9.8 is a proof sketch, not a complete proof: the four-dimensional sharing-class enumeration is asserted by analogy to the 1+1-dimensional appendix; accordingly the "unconditional" capped L² convergence on the window 1/8 < a < 1/4 (Corollary 9.9, and the corresponding sentence of the Version 2 note below) is downgraded to conditional pending the full enumeration. (3) Lemma 4.2’s Taylor remainder requires more than the stated C¹ regularity (repairable via the null-strip argument). (4) The bound |Φ| ≤ 2Λ_F·V of Section 2 holds in 1+1 dimensions only. The revised version is published: Version 3 of this record is Paper I (the 1+1-dimensional theory, 10.5281/zenodo.23374774) and the 3+1-dimensional material, with all four corrections incorporated, is Paper II (10.5281/zenodo.23374678). Nothing in the 1+1-dimensional results or in the 3+1-dimensional constant-field diamond mean is withdrawn.

Version 2 (August 2026). This version folds in three closures over v1. (1) The 3+1-dimensional variance of the action is shown to diverge for this action class (relative variance ~ 42.2 sqrt(rho V_D)), so the constant-field Maxwell result is an ensemble-mean theorem; (2) the divergence is repaired by a mesoscale cap in the region's rest frame, and the capped L2 convergence is now an unconditional theorem on the window 1/8 < a < 1/4, with the explicit variance bound Var[S] <= c Lambda_F^4 V_D (sqrt(rho) l_m^6 + rho l_m^8) and the co-scaled mean error computed in closed form; (3) the class of regions on which the Maxwell constant (3/20pi)Vol(R) holds is characterized: compact causally convex regions are exactly slabs between 1-Lipschitz graphs, their covariogram obeys a master formula whose first-order term is the Lorentzian analogue of Matheron's perimeter term, and null boundary segments do not break the class. Four ancillary scripts, 170 machine-checked asserts. A plain-language companion note is included.

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.