Note 22. Metric reconstruction from causal structure: FLRW pilot, de Sitter H2 scaling, and sign flip in Anti-de Sitter

This note records the results of the GR-DYN-03a/b/c experimental series on metricreconstruction from causal structure and measure (C,≺, μ). The goal is not to derive theEinstein equations and not to measure curvature independently, but to establish that observablesbuilt from causal-set data and evaluated at externally prescribed H and externalproper time τext reproduce the known volume laws of the causal diamond in controlledstatic geometries. Three blocks are presented.FLRW pilot (dynamical geometry, coordinate reconstruction): the functional formsC2 ∼ a3/2 and FLRW ∼ a2 are reproduced, but with a control-test C1 drift of about 9%;a clean separation between density and geometry is not achieved. Status: preliminary.de Sitter, magnitude and H2 scaling: the full function FdS(Hτext) is reproducedat five values of H (0.10–0.35) with mean q = 0.994 and no significant trend of q inH. The effective slope of the linear approximation in the large-H range (0.25–0.35) is−0.166 ± 0.020 and is statistically consistent with the leading coefficient −1/6.Anti-de Sitter, sign flip: the positive volume correction (opposite sign to de Sitter)is confirmed quantitatively for all five H ∈ [0.20, 0.60] with the exact containment formula(q = 0.9996–1.0070). The deficit at H ≥ 0.50 observed in the previous version is fullyeliminated by the exact formula for the extent of a non-central diamond.Key limitations: flat bias depends on the coordinate system (AdS-spherical +0.6%, dSCartesian+1.3%, opposite signs, both statistically indistinguishable from zero at 1.2σ);not universal. At H = 0.60 the confirmation refers to 11% of pairs (fully containeddiamonds); the rest are excluded correctly because of the box size. H and τext are inputparameters, not measured.