Weyl curvature from the Hasse diagram: a parameter-free bridge formula for causal sets

We construct a statistical estimator (CJ) on Poisson-sprinkled causal sets in four spacetime dimensions and derive a parameter-free formula connecting it to the electric part of the Weyl tensor. On a vacuum causal diamond of proper time T with N sprinkled elements:

⟨CJ⟩ = (32π²)/(3 · 9! · 45) · N^{8/9} · E_{ij}E^{ij} · T⁴

The coefficient C₀ ≈ 6.45 × 10⁻⁶ decomposes into five factors of distinct geometric origin: the two-leg structure of the link score (4 = 2²), the squared Benincasa–Dowker normalisation (8/3), the nine-simplex beta overlap (1/9!), the angular average of the squared tidal deformation (8π/15), and the four-dimensional diamond volume (π/24). No continuous free parameters enter the formula.

The formula is verified against Monte Carlo data on exact pp-wave causal diamonds for N = 500–15,000: the ratio of measured to predicted CJ is 1.016 ± 0.015. Seven diagnostic tests are reported, including a de Sitter null test (CJ = 0 exactly), Kottler cross-term, and polarisation independence (cross/plus = 1.042 ± 0.053, p = 0.43, M = 50 paired seeds). All rational coefficient identities are formally verified in Lean 4 (105 sorry-free theorems). Five failed approaches and thirty closed spectral routes are documented.

The derivation rests on two explicitly stated conditions concerning the continuum limit of the stratified estimator. The N^{8/9} exponent is consistent with the theoretical value 8/9 at the 1σ level (α = 0.915 ± 0.023). To our knowledge, this is the first discrete causal-set observable shown to respond to Weyl curvature.

35 pages, 8 figures, 105 Lean 4 theorems. Full source code, Monte Carlo data, and formal proofs available in the accompanying repository.