A holonomy action for U(1) gauge fields on causal sets, II: the 3+1-dimensional constant-field mean, the variance obstruction, and the capped L² window

Abstract

Paper II of two. Paper I (10.5281/zenodo.23010937, Version 3) defines the action and proves the 1+1-dimensional theorems. This paper treats d ≥ 3. Provenance: its content is the higher-dimensional part of the earlier single manuscript (Versions 1–2 of the Paper I record), revised after two external audits, an internal verification cycle, and a further independent referee-style audit (October 2026): the general-region characterization lemma was corrected (shadow condition) and its perimeter statement restated with a counterexample to a universal second-order remainder; the capped-variance bound, previously a proof sketch, is proved (Appendix B) for smooth exact curvatures of bounded norm and was confirmed by an independent audit; the 2+1-dimensional mean is an exact truncated-moment series (Appendix D), replacing earlier quadrature values; the diagonal-variance theorem's inputs are closed forms (Appendix C); the intrinsic-weight proposition is restricted to scale-free, exchange-symmetric weights; the compact-action discussion is corrected (boundedness of 2(1−cos Φ) does not remove the diagonal mechanism; a corollary gives the same leading asymptote); region independence beyond the diamond is stated as conjectural under regularity that condition (M) alone does not supply; and a limitations section gathers the physical costs, with the plane-wave pilot reported as a finite-window numerical result. Abstract. Paper I defined a gauge-invariant action for a background U(1) connection on sprinkled causal sets — a cardinality-weighted sum of squared real-lifted chain-pair holonomies with finite-support coefficients — and proved its exact Maxwell mean and L² convergence in 1+1 dimensions. Here we treat d ≥ 3, where the loop bivector is not boost invariant and the boost divergences are exponential in the rapidity (polynomial in e^χ). For constant field strength on a causal diamond in 3+1 dimensions, an exact endpoint-axis bivector identity and a Lorentzian covariogram identity for Alexandrov intervals collapse the ensemble mean to an exact closed form at every density; with the kill ray (14, −16, 9, −2) it converges to (3/20π) ∫ F_{μν}F^{μν}, with the ln ρ · ρ^{−1/2} correction series explicit, and the finite-density formula is sprinkling-tested in the raw channel at the 0.2–0.4% level. The 2+1 mean is an exact truncated-moment sum (a finite series of complete moments up to exponentially small tails). The diagonal class of the 3+1 second moment is exactly computable, diverges as ρ^{1/2} relative to the squared mean, and cannot be cancelled coefficientwise, nor by scale-free intrinsic geometry weights; numerics indicate it dominates the total variance over the tested range, while analytic divergence of the total variance remains open. A mesoscale cap in the region frame preserves the asymptotic Maxwell coefficient for every fixed cap, with the leading finite-density difference and its remainder explicit, and Var[S_{ℓ_m}] ≤ c Λ⁴ V_D (√ρ ℓ_m⁶ + ρ ℓ_m⁸) for smooth exact curvatures F = dA of bounded norm, giving per-configuration L² convergence for constant F along ℓ_m/T = (ρV_D)^{−a}, 1/8 < a < 1/4, though not at fixed physical ℓ_m. Beyond the diamond, region independence of the constant is conjectured under suitable weighted null-edge and joint-corner regularity; a null-displacement condition (M) is isolated and verified for three region families, but (M) as stated does not yet establish the required implication. The 3+1 smooth-field limit and the uncapped off-diagonal variance classes are open. Files. The manuscript (PDF, 46 pages) and its LaTeX source; a zip with the verification scripts and their outputs (3+1 variance, repair and close stages; the 2+1 exact series; the flux-bound Monte Carlo; the plane-wave pilot cited in the limitations section) and the figure.