A Discrete Ledger Theory of Gravity: Weak-Field Einstein–Hilbert and Constraint-Algebra Recovery

Abstract

We present a discrete theory of gravity whose microscopic primitive is a conserved ledger whose dynamical step is a single one-coordinate, one-bit posting, and we prove exactly where such a theory must import geometry and how much of general relativity it recovers once it does. Three unconditional theorems carry the new positive mathematics. A counting obstruction: no positive rational per-class weighting of the substrate reproduces the Euclidean moment tensor of the mesh, even up to an overall real factor (by linear independence of {1, √2, √3} over the rationals); a companion shows every positive real weighting is anisotropic, while an explicit real counterexample shows the stronger “rational or real” claim is false. The geometric import is therefore irreducible within the rational counting class, and the substrate matches the geometry sector by sector up to two exact constants. A rigidity theorem: on the stated n = 2 point-split class with a kinetic intensivity normalization, the Hojman–Kuchař–Teitelboim closure conditions force the ADM Hamiltonian shape with c_mom = 4 c_kin c_grad, four counterexamples showing every stronger unconditioned statement false; beside it, the scaled lattice bracket converges to the continuum hypersurface-deformation form of the H–H bracket for the one-dimensional model. And the weak-field limit: the discrete action restricted to transverse-traceless modes converges to the Einstein–Hilbert quadratic form on a four-dimensional periodic lattice family, with Regge’s normalization constant ρ = 1/2 derived twice independently rather than adopted. Two further contributions are derivation routes, each conditional on a named postulate stated in full before use: a path-sum measure in which label erasure produces the factor n_V! n_E! n_T!/|Aut K| and the uniquely normalized relabeling-invariant labeled weight removes the factorial, yielding μ(K) = 1/|Aut K| under a labeled-carrier postulate, with the rival weights excluded by theorems; and a ledger-to-geometry bridge in which the recognition ratio is computed, in closed form, as the strain of a unique stationarity minimizer of the recognition cost against a postulated constitutive source; the logarithm of the ratio matches the supplied deficit to cubic accuracy, and the odd part of the ratio, the recognition phase, matches it exactly. An exactly solved negative result sharpens the frontier: the tilted path sum factorizes over independent edges, elementary algebra on that factorization yields the closed-form threshold t_c = −ln(2^(1/3) − 1), and because the tilt weight is a function of the edge-class census alone, the construction carries class-count information into the limit and nothing else: observables separating equal-census configurations are excluded at any tilt strength. The Lorentzian continuation of the causal Regge action holds on every finite causal triangulation with the two-length causal metric, the Kuhn 4-torus among them, with closed-interval continuity through the cut refuted. Further results (an axis-model resonance family, the identification layer, and the pinning of the four otherwise-free constants by two unit conventions plus two proved relations) are conditional on seven physical postulates; for two of the postulates the natural derivation routes are proved impossible, so they are genuinely independent inputs. The theory is not empirically confirmed: it supplies no discriminating prediction and no external test, and we separate those empirical obligations from the mathematics established here. The principal results are machine-verified in Lean 4, with the continuum threshold obtained by elementary algebra from a verified factorization; the complete formal development is publicly available, and a single appendix maps each numbered statement to its formal source.