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

This paper develops a unified mathematical theory connecting two inverse problems: reconstructing a finite causal order from anonymous event-deletion observations, and inferring a homogeneous cosmological history from the statistics of that order. The common setting is a fixed causal diamond in three spatial dimensions and one time dimension. The analysis distinguishes recovery of a sampled combinatorial structure, identification of its generating history, and quantitative control of reconstruction error. The finite reconstruction problem concerns the complete multiset of orders obtained by deleting one event at a time. Event labels are not shared between observations, but causal orientation and repeated-card multiplicities are retained. For independent samples drawn from any measurable density between fixed positive bounds, the paper proves that the sampled order is determined by this anonymous deletion deck with probability tending to one, uniformly over the density class. Uniqueness holds among all finite posets, without requiring competing reconstructions to be geometrically realizable. The reconstruction procedure does not require knowledge of the sampling density. The deterministic mechanism combines recovery of bidegree information with a sparse reinsertion certificate. Geometric probability estimates show that suitable certificates become typical in the sampling model. Supporting results establish isolated-event and finite-degree boundary laws, identify the role of the equatorial boundary region, and provide separation estimates for bulk causal profiles. Explicit finite-sample failure bounds yield computable confidence thresholds. The bounded-density conclusions extend to physical-volume sampling in qualifying conformal diamonds, including specified spatially flat cosmological and de Sitter examples. These thresholds are conservative sufficient guarantees rather than predictions of practical sample requirements. For history inference, the sampling density depends only on conformal time, with fixed interval and volume normalization. The principal identification theorem uses a reduced causal law: the joint distribution of two sampled events’ population past and future masses together with their directed causal relation. Radial degree coordinates and rigidity of the conditional causal kernel show that this law determines every continuously differentiable normalized history, apart from an explicitly classified pair of opposite de Sitter histories. This ambiguity concerns expanding and contracting descriptions of the same de Sitter diamond. A nonnegative midpoint expansion condition selects a unique representative; it is additional model information. Two quantitative estimates address distinct parts of the inverse problem. A construction using independent witnesses and disjoint root pairs estimates the reduced causal law from a finite causal order, with an explicit uniform Wasserstein error bound decreasing at a fourth-root rate at fixed confidence. The construction also applies to a uniformly selected deletion card. Separately, after alignment by a diamond-preserving conformal transformation, an explicit square-root stability bound controls the entire history and its first derivative. This exponent is sharp on admissible classes containing the linear mode. The first estimate concerns an observable probability law, while the second concerns aligned log densities. Effective compactness and rational approximation provide uniform consistency on fixed analytic classes and on nonanalytic classes with a Lipschitz second derivative. The constructions include internal approximation nets, computable forward probability enclosures, and terminating finite-pattern separation searches. They establish recovery guarantees for an entire unknown function within a declared class, without assuming that the true history has finitely many coefficients. Related conclusions control Hubble histories and proper-time quantities under the stated assumptions. Computability is distinguished from computational efficiency: these results do not assert a practical full-class search or a useful closed-form statistical rate. The finite-dimensional theory supplies explicit inverse theorems and exact certificates. A quadratic history model admits a certified nonlinear inverse and a sharp fourth-root statistical rate. General polynomial calculations characterize the common first-order kernel, give sufficient finite event counts and constructive local inverses at each fixed degree, and establish second-order inversion with any fixed even polynomial background. Additional results quantify the effects of deletion, relation errors and controlled model discrepancy on selected statistics. Lower bounds identify limitations that persist beyond a particular reconstruction algorithm. A perturbation concentrated near a temporal tip gives a fifth-root minimax lower bound on the full finite-regularity class when the Lipschitz allowance is positive, even if event coordinates are observed. An explicit estimator attains this exponent when calibrated conformal event times are supplied, establishing an optimal benchmark for that stronger observation model. A joint recovery theorem combines finite-order reconstruction and uniform history consistency from one anonymous deck, using common density bounds and a union bound without assuming independence between the two recovery events. The remaining quantitative problem is stated explicitly. A useful upper error bound for the entire unknown history from causal observations alone requires a controlled estimate converting reduced-law error into conformally aligned log-density error. Exact identification and compactness ensure that some modulus exists, but do not supply the needed useful formula. The endpoint construction also restricts the possible strength of a power-law estimate. Neither the calibrated-time benchmark nor the aligned-density stability theorem is presented as closing this gap. The paper likewise does not solve unrestricted finite-poset reconstruction or recover arbitrary inhomogeneous spacetime geometries. The accompanying project contains the 70-page manuscript, editable LaTeX sources, bibliography, figure source, six appendices with full supporting arguments, both computational code bases, and a common verification entry point. Eleven verification programs pass, checking exact algebraic identities, finite combinatorial cases, inverse certificates and conservative probability bounds. These computations support specified proof components rather than formally verifying every analytic argument. The study is theoretical and uses no observational cosmological dataset.