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

This work develops quantitative connections between finite causal orders, continuum geometry, gravitational action, and changes of observational resolution. Its starting data are anonymous induced orders: the observer records causal comparisons without retaining underlying event identities or a chosen spacetime embedding. The paper separates recovery of a finite order, recognition of a continuum geometry, convergence of gravitational responses, selection of a geometric ensemble, and convergence of dynamical solution maps. Results at one of these levels are not treated as automatically establishing the others. The observation theory uses the causal-profile metric, which measures disagreement between the pasts and futures of two events. A finite-law inverse theorem converts proximity of sampled-order distributions into an explicit Gromov–Hausdorff–Prokhorov bound, provided small profile balls have uniformly controlled positive mass. Ordinary event-deletion decks determine the required sampling laws, including repeated-root observations, and certify the relevant mass bounds without requiring unique reconstruction of the parent order. Further results address finite-sample confidence, degree-moment reconstruction certificates, partial-deck recovery under nonuniform Lorentzian sampling, and mesoscopic rigidity of sampled four-dimensional orders, each with its stated hypotheses. Multiscale intrinsic mass conditions give compact deterministic order limits without prescribing a target Lorentzian sampling law. Canonical joint couplings preserve both profile distances and interval observables, making interval convergence automatic for those correspondences. These limits need not be manifolds. Separate reconstruction theorems use finite interval clocks, Taylor certificates, cone representation, localization, and mass regularity to recover local charts, Lorentzian metric data, and volume. An exact rational stencil of 495 strictly timelike probes supplies fourth-jet and curvature-error estimates under regularity and nondegeneracy assumptions. Uniform small-diamond estimates and a complete constant budget give proper-time convergence on controlled temporal-slab classes, including time-dependent light cones. Counterexamples explain why sampling convergence alone does not control support, longest chains, cone shape, or curvature. An intrinsic clone-and-delete process driven by a three-chain action has a nonlinear measured-order limit and, on specified smooth classes, an evolving Lorentzian volume interpretation. Its finite action drift is certified by deletion data. The construction demonstrates an intrinsic continuum dynamics, but an explicit obstruction shows that it is not vacuum Einstein evolution. Framed and unframed realizer processes, finite path and constraint certificates, learned response generators, and positive selection processes provide additional constructions with their geometric, statistical, or constitutive inputs stated explicitly. For the established four-dimensional interval action, intrinsic thinning controls fluctuations and transfers suitable mean limits to stochastic observables. Keeping the action coefficients fixed, the paper proves relative mean-action convergence to the Einstein–Hilbert functional on regular spacelike slabs with a common boundary collar. The response limit holds for general compact tensor variations and every fixed derivative order on smooth finite parameter families, with compact uniformity on controlled smooth classes. The flat Hessian is the full Fierz–Pauli form; higher responses satisfy diffeomorphism identities. Finite stationarity tests, nondegenerate completions, and moving finite-density saddles connect these limits to Einstein stationarity and oscillatory amplitudes. These are relative action and response results: absolute boundary terms and original finite-density solution-map convergence are separate questions. Positive Einstein selection is established on supplied compact smooth metric classes, including infinite-dimensional ones. A normalized countable family of ambient tensor tests with dense linear span detects the full Einstein residual. A continuous envelope on physical pattern-law space yields an Einstein-detecting energy accessible to anonymous finite-order estimators when the pattern fibers preserve Einstein status. Gibbs concentration then follows from prior mass near the Einstein zero set. An explicit compact random-series prior allows infinitely many general tensor perturbations and has diffuse directing laws, atomless event measures, and nontrivial dependence between disjoint samples. Posterior reweighting gives exact restriction or Poisson-thinning coherence at each fixed temperature. For this prior, an explicit normalizer bound of exp[−O((log β)²)] provides a quantitative cooling and approximation budget. The example’s flat initial collar fixes flat vacuum Cauchy data; its vacuum members are flat developments. Neither this example nor the abstract selection theorem derives an unrestricted metric class, prior, or boundary data from arbitrary orders. Deletion coarse graining yields exact transformations of interval observables and a complete hierarchy of amplitude statistics. Every uniformly bounded pattern cutoff fails to close the phase on all finite orders, while subset inversion, connected overlaps, cumulants, and truncation estimates describe the additional information required. Projective positive and phase reweightings are constrained by their infinite directing laws. Coherentization of finite intrinsic sieves is distinguished from independent cardinality-by-cardinality weighting, and entropy bounds expose the limitations of specified atomic reference ensembles. The arithmetic analysis treats signed resonant sums with their counting and symmetry conventions explicit. It proves exact cancellation of the complete height-at-most-two sector, nonvanishing of the full raw partition at every cardinality, and denominator bounds yielding suppression on specified arithmetic subsequences. For each fixed number of interior vertices, the aggregate signed generating function is an entire exponential polynomial. Its connected part has an explicitly determined pole and factorial asymptotic, already in height three. The connected poles cancel in the aggregate, so these formulas do not establish the growth of the full partition or suppression of every nongeometric sector. Causal completion and evolution are examined through exact finite-density response formulas, boundary-layer obstructions, retarded constraints, and finite certificates. Homogeneous obstructions extend to regular infinite-layer kernels and bounded density-dependent families. For the actual derivative-corrected scalar operator, reflection gives an equivalent self-adjoint spectral-gap problem with compact resolvent and finite Schur-complement certificates complete at each fixed slab length. A separate theorem shows exponential cost for every positive weight in the stated absolute-kernel certificate. This closes that sufficient-certificate route without proving that the actual inverse has exponential growth. The uniform quadratic long-slab inverse estimate remains open. Modified filtered equations with supplied principal structure and constraint control do converge locally to vacuum Einstein evolution; this does not remove the filter from the original action or supply the missing nonlinear tensor inverse estimates. The central open objective is a common intrinsic law deriving general Lorentzian geometry and nonlinear Einstein evolution. Remaining obligations include original-action filter removal, uniform control of the actual long-slab inverse and nonlinear tensor solution maps, full-sequence resonant denominator growth, and suppression of all unwanted sectors. The proved obstructions apply to their stated ensembles, norms, kernels, or completion requirements; they are not universal impossibility theorems for causal-set gravity. The manuscript is organized in seven parts: Anonymous orders and continuum recognition. Intrinsic dynamics and action observables. Fixed-action responses and continuum limits. Anonymous response laws and coherent ensembles. Deletion, phases, and resonant arithmetic. Causal evolution and inverse estimates. Intrinsic and response-based selection. The package includes the main manuscript, a separate research record of attempted approaches and obstructions, complete LaTeX sources, 71 Python verification programs, three retained C++17 enumeration companions, two separately identified exploratory numerical programs, and supporting mathematical notes. Build instructions, dependencies, execution records, verification scope, and file checksums accompany the sources. Exact finite computations support algebraic and combinatorial identities; continuum, compactness, and asymptotic conclusions rely on the analytic proofs. Exploratory numerical meshes are not certified continuum inverse bounds.