Time from Anonymous Causal Records: Statistical Limits, Transport Alignment, and Physical Calibration

This working paper studies what anonymous causal records can reveal about elapsed proper time, how the loss of event identities affects inference, and which additional measurements are needed to express time in physical units. It treats three distinct observation models: repeated deletion traces of a fixed finite order, independent event samples from a continuum causal diamond, and registered field measurements under specified physical controls. For incomplete finite records, an exact stochastic reduction transfers binary deletion-trace reconstruction to anonymous event-deletion traces of dimension-two causal orders. Combined with an explicitly identified external trace-reconstruction theorem, this yields a superpolynomial worst-case sample lower bound. The result concerns repeated independent losses from arbitrary fixed parents; it does not assert the same lower bound for complete one-point deletion decks, prime parents, typical geometric samples, or duration estimation alone. For homogeneous four-dimensional causal diamonds, the paper develops quantitative duration estimation on a fixed C^{2,1} class: histories with two derivatives and a Lipschitz second derivative. With calibrated conformal event times and a fixed physical-volume normalization, the sharp estimation exponent is 2/5. From the anonymous causal order alone, a polynomial-time construction achieves uniform error O((log n/n)^(12/37)), without a supplied background density or geometric alignment. It combines nested interval counts with an estimated degree-axis frontier and a fixed degree mesh. Fixed quadratic and specified finite polynomial families admit sharp local root-n recovery under their stated assumptions. The information analysis identifies both retention and loss. At each supplied homogeneous background, pooled causal degrees asymptotically recover the local coordinate likelihood experiment for shrinking tip perturbations. A different shrinking affine experiment exhibits a strict separation between order-based and coordinate-based duration rates. A sharp aligned-Hellinger stability bound explains why lower bounds based only on deterministic aligned-coordinate comparisons cannot improve the calibrated benchmark. Explicit pooling constructions address endpoint singularities and dependence. An endpoint-corrected one-step estimator has a quadratic pilot remainder under its coordinate-observation assumptions. Growing star statistics reproduce the endpoint singularity of the duration influence at the flat background, leaving a fixed square-integrable residual. Distinct-anchor products yield exactly centered relation moments with controlled conditional variance when the true radial centering is supplied. Every fixed finite population order law admits exact duration-changing fibers, while a robust order-based tip test constrains small compensating perturbations at growing-record scales. These results preserve the distinction between fixed population laws, finite-sample fluctuations, and uniform adaptive estimation. The transport results compare reduced observation laws through canonical representations and consistent common-root couplings, with explicit kernel and regularity assumptions. Finite counterexamples show why reduced pair laws and unconstrained pair couplings need not determine the underlying causal structure. Physical calibration is developed through local massive-field correlations, classical solution jets, finite-record readouts, and bounded source pulses. Two known masses allow cancellation of common geometric terms, unknown constant detector gains, and a shared positive source coupling under the stated hypotheses. For positive conformal factors varying in space and time, local pulse protocols have a sufficient record budget of O(epsilon^(-6) log(1/(alpha epsilon))). A fixed distributed source profile gives an alternative O(epsilon^(-10) log(1/(alpha epsilon))) budget. Here epsilon is the target error and alpha the failure probability. These are sufficient bounds under explicit chart, access, regularity, noise, and repeated-record assumptions, not optimality or hardware-validation claims. Exact access and detector-drift ambiguities delimit the protocols. The optimal full-class duration rate from anonymous causal orders remains open between the established 12/37 upper-error exponent, with a logarithmic factor, and the 2/5 lower benchmark. The remaining tasks include representing the residual order influence, controlling uniform adaptation to unknown backgrounds, reconstructing the calibration chart, and extending the control model. Version 9 presents the complete mathematical content in a reorganized and polished exposition. The accompanying project includes the manuscript PDF, editable LaTeX sources, a minimal Overleaf compilation bundle, reusable code, twenty-four finite verification programs, pinned external-source records, research notes, and SHA-256 integrity manifests. The finite checks support specified identities and computations; they do not constitute complete proof-assistant verification, external peer review, or experimental validation.