Anonymous Causal Data and Quantitative Continuum Recognition Geometry, Einstein Responses, and Causal Evolution

Overview This working paper develops a quantitative framework connecting anonymous finite causal orders with continuum geometry, gravitational action responses, coherent statistical laws, and operational quantum observations. Anonymous data preserve causal comparisons while discarding event identities, coordinates, and a spacetime embedding. The analysis distinguishes three mathematical problems: recognizing a continuum geometry from those data, selecting a law on geometries, and deriving a stable constrained physical evolution. The paper proves finite-data geometric inverse theorems, intrinsic Lorentzian reconstruction results, and relative Einstein–Hilbert response limits for the fixed four-dimensional interval action. Specified entropy and stationarity tests select coherent vacuum history laws, and calibrated past observations support consistent continuation forecasts. Exact causal inverse estimates, selection obstructions, quantum instruments, and coupled matter models identify the assumptions needed to connect these constructions. The main distinction is between a proved geometric or variational limit and a proved limit of physical solutions. The original stationary constrained tensor inverse and a common microscopic derivation of geometric selection, calibrated matter, and operational records remain open. Principal contributions Explicit reconstruction bounds from finite anonymous order laws and deletion decks, with noncollapse controlling the recoverable support. Intrinsic certificates for local Lorentzian geometry, canonical globally hyperbolic interiors, and calibrated proper-time convergence. Fixed-action Einstein–Hilbert responses, coherent vacuum laws, and consistent predictions from recorded past orders under stated hypotheses. Exact obstructions to specified positive selectors, raw exponential amplitudes, scalar causal inverses, and proposed tensor-stability criteria. Operational quantum records, observable Gramian and Riccati estimates, joint response–noise models, and an exact full-reference semiclassical clock equilibrium. 1. Anonymous orders and quantitative geometric reconstruction A causal-profile metric compares events’ pasts and futures. Under noncollapse, finite sampling-law proximity gives an explicit Gromov–Hausdorff–Prokhorov bound. Ordinary deletion decks supply the required observations without unique parent reconstruction. Intrinsic mass, scalar-chart, interval-clock, Taylor, signature, comparison, and chain certificates recover metric, volume, and chronology. Exact timelike stencils control fourth jets and curvature. Finite-sample, partial-deck, degree-moment, and mesoscopic results retain their stated hypotheses. Complete coverage, global chain tests and compatible jets of every fixed order strengthen almost-everywhere reconstruction to all points of the canonical interior. Separate past and future volume bands extend the recognition to every regular smooth compact temporal slab through suitable schedules, without requiring constant difference-mark levels on its boundaries. A positive lower bound on the sum of the two masses makes their logarithmic ratio a proper continuous Cauchy time with compact levels. The resulting spacetime admits smooth compact spacelike Cauchy hypersurfaces. Each slab has its own bounds; neither one universal schedule nor a countable compact exhaustion of all smooth metrics is claimed. An exact flat four-dimensional torus example has interval volume V(T)=πT⁴/24−π(T−L)₊³(T+L)/24, which is C² but not C³ at T=L. Global cut-locus smoothness is therefore false even in vacuum; the reconstruction uses local smooth normal windows. Global causal and chronological relations agree with the reconstructed order. The certificate hierarchy excludes uncertified interior defects. Null-shell and puncture examples explain why weaker full-volume statements cannot do this; a separate curvature-nonconcentration criterion controls weak vacuum transmission when its hypotheses hold. 2. Fixed-action Einstein–Hilbert responses For the fixed four-dimensional action, relative mean responses converge to Einstein–Hilbert responses on controlled smooth slabs. General tensor variations, compact uniformity, the flat Fierz–Pauli Hessian, and higher diffeomorphism identities are included. Positive concentration on supplied smooth metric classes remains a separate construction. Finite sampled-order actions are locally constant under regular metric perturbations almost surely, so pathwise differentiation cannot replace mean responses or controlled resampling. The original-action tensor calculation includes nonzero spatial momentum. A flat-slab cross-polarization separates exactly from lapse and shift by spatial symmetries. Its finite-density Hessian includes world-function variation, outer and inner moving cones, double-cone intersections and volume-density factors. For energy-sized oscillations h12=h21=u(t)cos(ωx3)/ω, this Hessian tends to zero as ω tends to infinity at any fixed density, whereas the Einstein–Hilbert response has a nonzero negative limit. The normalized spatial-energy form discrepancy is at least one at every finite density. Frequency and density limits therefore fail to commute wherever the fixed smooth-test limit applies. This is an off-shell response theorem, not a stationary physical tensor inverse obstruction or failure for one fixed source. 3. Coherent laws and intrinsic vacuum selection Finite rational Ricci tests provide one vacuum selector. A second uses the original interval action: certified mean-action secants on synthetic local extensions imply stationarity against every compact tensor variation, hence Ricci-flatness, without imposing explicit Ricci residuals. Probe collars are computational constructions, not prescribed physical boundaries. Shifted Kasner torus slabs establish simultaneous feasibility with an additional rational lower bound on the Kretschmann invariant. That extra predicate makes every selected directing interior nonflat. A bounded finite-marginal entropy objective has a unique coherent exchangeable maximizer, obtained as the full-sequence limit of finite optimizers. Geometric certificates, probe stationarity, the nonflatness predicate, and entropy weights remain additional selection principles; uniqueness does not mean a unique spacetime. 4. Predicting vacuum continuation from past observations A measurable realization in standard Borel ADM history codes preserves every selected order marginal, including finitely many connected Cauchy components. The marked history law is fixed throughout the prediction construction; its marking and acquisition slab remain specified. Calibrated order laws of a smooth compact past slab determine its terminal Cauchy metric and second fundamental form up to spatial isometry. Four past-anchor interval clocks extend the reconstructed isometry smoothly to that boundary. Physical volume fixes the otherwise invisible homothety. No future samples or whole-history future-dependent marks enter the reconstruction. A finite anonymous past order, its count and known observation exposure are sufficient for an explicit posterior likelihood, even when arrival identities and earlier nested cards were retained. Increasing exposure recovers physical past volume and every finite past-order probability. Posterior forecasts of identifiable geometric observables converge almost surely to their true values; forecasts of future Poisson orders converge in total variation to their true distributions. A compact-class confidence theorem supplies an explicit concentration inequality with a class-dependent inverse modulus and stated effective-computation requirements. An adaptive continuation target removes the need for a fixed future window contained in every selected history. For each dyadic proper-time duration, the relation between smooth vacuum data and normal-collar volume and order statistics is analytic; vacuum uniqueness makes its nonempty sections singletons. Choosing the first admissible duration gives a positive invariant horizon. A law-relative Borel version makes the target available for the entire unchanged prior and gives consistent finite-past forecasts. This is a continuation inside the unique vacuum development and may exceed the recorded history's cutoff. It does not predict that cutoff, choose a maximal-spacetime code, supply an effective posterior algorithm, or identify exposure with physical time. The selectors and the marked lift remain supplied, and no finite-density physical-time solution-map limit follows. 5. Effective selection, computability, and finite prediction Computability has a sharp criterion. Under an effective rational hierarchy and effective entropy tail, the selected coherent law is computable if and only if its optimum value is computable. Equivalent information is a convergent sequence of certified entropy lower bounds or a marginal convergence modulus. A rational chain–antichain hierarchy has explicit computable finite maximizers but a noncomputable two-event limit encoded by the halting set. A known feasible law is therefore insufficient. This does not prove the manuscript's particular vacuum optimum noncomputable. The minimum dyadic continuation horizon is itself noncomputable on a smoothly parametrized contracting Kasner family. An explicit computable mixture of an atom at the discontinuity and a uniform parameter distribution has no almost-sure computable version of that exact index. Smaller safe horizons remain possible. On effectively compact classes with computable forward statistics and an identifiable continuous target, a terminating finite-net search computes an inverse modulus and a certified finite-exposure anonymous-past predictor without any prior. A computable prior additionally gives finite positive-evidence posterior probabilities with an explicit normalization error budget. Effective continuation classes, rather than merely measurable lifts, are the requ