Anonymous Causal Decks and Cosmological Histories: Reconstruction, Identifiability and Quantitative Stability

This paper develops a unified mathematical theory connecting the reconstruction of finite causal orders with the recovery of an entire homogeneous cosmological history. It studies what can be inferred from causal relations alone when event coordinates and calibrated times are unobserved, and when the available observations may consist of an anonymous vertex-deletion deck. The central contribution is an explicit uniform upper error bound controlling both an unknown history and its first derivative throughout the full normalized time interval, including its endpoints. The finite reconstruction problem concerns independent events sampled in a four-dimensional Minkowski diamond. Their causal relations form a partially ordered set, and the deletion deck records the unlabeled orders obtained by removing each event in turn, with multiplicities retained. The paper proves that the sampled order is determined by its complete oriented deck, among all finite partially ordered sets, with probability tending to one. This conclusion holds uniformly over measurable sampling densities bounded above and below by fixed positive constants. Explicit probability estimates provide computable confidence thresholds, and the results extend to physical-volume sampling in qualifying conformal diamonds and to Poisson sprinklings. For cosmological inference, the sampling density depends only on normalized conformal time. A reduced causal observation law records the population predecessor and successor fractions of two sampled events together with their directed relation. The paper proves that this law identifies every normalized continuously differentiable history, apart from an explicit opposite de Sitter pair. A known nonnegative midpoint expansion derivative removes this ambiguity. Exact identification extends to every fixed spacetime dimension at least three. In two dimensions, an explicit family of linear histories has identical causal-order laws even after imposing the midpoint sign condition; these histories describe isometric flat diamonds in different comoving coordinates. The principal quantitative theorem applies to fixed infinite-dimensional classes of twice continuously differentiable histories whose second derivatives have a prescribed Lipschitz bound. A Wasserstein error in the reduced causal law controls the conformally aligned log-density discrepancy with exponent one over 40 and the combined uniform error in the history and its derivative with exponent one over 80. A fourth-root sampling estimate for the reduced law yields a conservative whole-history convergence rate with sample-size exponent one over 320 at fixed confidence. The estimator uses causal observations alone, and the same conclusion follows from a uniformly selected card of a complete anonymous deletion deck. All constants and small-error thresholds are explicit. The proof combines quantitative reconstruction from causal probabilities, control of local Lorentz frames in both directions, improved conditioning of degree coordinates, and quadratic extrapolation through the boundary layer. A known strictly positive lower bound on the midpoint expansion gives a sharper aligned-history exponent of two thirds, which is optimal for that comparison when the class permits curvature variation. Under this stronger assumption, the observation-law exponent improves to one over 60 and the sampling exponent to one over 240. Effective covering arguments also establish uniform consistency on fixed analytic classes, while exact polynomial calculations supply certified finite-dimensional inverses and a sharp fourth-root rate on a quadratic model. The paper also develops confidence sets whose radii can be computed from verified forward probability intervals while retaining approximation error for histories outside the finite candidate set. An executed example uses two million simulated events from a nonpolynomial history. On the declared infinite-dimensional class with zero midpoint slope, midpoint second derivative bounded in absolute value by two, and second-derivative Lipschitz bound 0.03, the resulting 95 percent simultaneous confidence band has a combined function-and-derivative error radius below 0.188, compared with a prior radius of 3.02 around the zero history. Exact rational calculations certify the forward and inverse intervals. Coverage follows from the full-class theorem; the synthetic realization illustrates the procedure. Its fixed approximation allowance prevents this single-feature example from establishing general nonparametric consistency. Lower bounds identify genuine limits to recovery. Perturbations near a temporal tip give a fifth-root statistical lower benchmark, attained when calibrated event times are supplied. A signed perturbation with exactly zero total mass sharpens the population-law obstruction: uniform inverse exponents cannot exceed one quarter for the history or three eighths for aligned density in the specified reduced-law metric. These restrictions concern population stability and do not close the finite-sample minimax gap for causal observations alone. Further results address independent event deletion, relation-entry corruption fixed before the estimator's independent random split, and nonuniform detection. The paper gives explicit error allowances and proves unavoidable error floors. Unknown detection probabilities can exactly confound the physical history; calibrated per-event detection probabilities permit corrective thinning. Joint recovery statements combine finite-order reconstruction and whole-history confidence guarantees from one complete anonymous deck without assuming independence between the two recovery events. The scope is a specified homogeneous conformal model with fixed normalization, orientation, regularity bounds and midpoint information. Quantitative history rates are established in four spacetime dimensions. Their constants are conservative, and the optimal causal-data rate, practical general-class estimation, spatially inhomogeneous history recovery and general spacetime geometries remain open. The accompanying project contains the 112-page manuscript, editable LaTeX sources, bibliography, figures, computational certificates and sixteen passing verification programs. These programs check algebra, finite combinatorial calculations and numerical certificates; they do not constitute formal verification of every analytic proof.