Gravity Analysis Regge Exact Flat Hessian Norm Gate4 Daudit

The normalization audit

In numerical relativity, Regge calculus approximates curved spacetime by a discrete lattice of flat simplices, much like a geodesic dome approximates a sphere with flat triangular panels. The Einstein-Hilbert action, which governs how spacetime curvature produces gravity, has both a continuum form (involving integrals over smooth geometry) and a discrete lattice form (involving sums over the lattice elements). A central consistency question is whether the discrete approximation reproduces the continuum physics in the appropriate limit, and with the correct normalization.

The module named in the question is a machine-checked audit inside the Recognition Science framework's library of formal theorems. It establishes a precise normalization fact: the discrete bookkeeping factor used in the lattice gravity calculation is exactly equal to the coefficient that appears in the continuum Einstein-Hilbert action, up to a fixed proportionality constant. The theorem states that the continuum Einstein-Hilbert discrete face evaluated at unit Frobenius norm equals the frozen preflight coefficient, which itself equals the discrete bookkeeping factor times the exact unit Frobenius trace coefficient.

In plain language, this means the discrete and continuum descriptions of gravity are not merely approximately consistent; they are exactly consistent at the level of normalization, provided one uses the correct bookkeeping factor. The audit verifies that the normalization gate passes, meaning the discrete calculation does not introduce an arbitrary scaling error relative to the continuum theory. It also confirms that the two coefficients, though proportional, remain distinct quantities: the exact unit Frobenius trace coefficient is not equal to the frozen preflight Einstein-Hilbert coefficient, they differ by the discrete bookkeeping factor.

This matters because a normalization mismatch between discrete and continuum gravity would propagate through every subsequent calculation, producing wrong predictions for gravitational phenomena. The audit closes that gap for the specific case of an exact flat Hessian norm gate, a technical condition ensuring the discrete approximation behaves well under second-order variations. Within the Recognition Science framework, this is one step in a larger program of deriving physical structure from a discrete ledger of recognition events, where gravity emerges from the bookkeeping constraints rather than being postulated independently.

theorem norm_gate_audit_package :
    NormalizationGatePass = true ∧
      exactUnitFrobeniusTTCoefficient ≠ frozenPreflightEHCoefficient ∧
        frozenPreflightEHCoefficient =
          discreteBookkeepingFactor * exactUnitFrobeniusTTCoefficient ∧
            continuumEHDiscreteFace (1 : ℝ) = frozenPreflightEHCoefficient :=
  ⟨normalizationGatePass_true, exact_unitFrobenius_ne_frozen_preflight_EH,
    frozen_EH_is_discrete_bookkeeping_times_unitF, continuumEHDiscreteFace_on_unitF⟩

theorem norm_gate_audit_package :
    NormalizationGatePass = true ∧
      exactUnitFrobeniusTTCoefficient ≠ frozenPreflightEHCoefficient ∧
        frozenPreflightEHCoefficient =
          discreteBookkeepingFactor * exactUnitFrobeniusTTCoefficient ∧
            continuumEHDiscreteFace (1 : ℝ) = frozenPreflightEHCoefficient :=
  ⟨normalizationGatePass_true, exact_unitFrobenius_ne_frozen_preflight_EH,
    frozen_EH_is_discrete_bookkeeping_times_unitF, continuumEHDiscreteFace_on_unitF⟩

What this page does not claim

This module does not derive the Einstein-Hilbert action from first principles; it audits a normalization consistency between already-defined discrete and continuum forms. The audit does not establish that Regge calculus converges to general relativity in all regimes; it verifies a specific normalization gate for the exact flat Hessian case. No claim is made about the physical interpretation of the discrete bookkeeping factor beyond its role as a proportionality constant in the normalization identity.

Verify this page

Every tagged claim above names its theorem. To check one yourself rather than trust this page, elaborate the source module with Lean 4 and audit its axiom basis:

$ lake env lean IndisputableMonolith/Gravity/Analysis/ReggeExactFlatHessianNormGate4DAudit.lean
expected axiom basis: [propext, Classical.choice, Quot.sound] (the Lean kernel's standard three; no RS-specific axioms)

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Derived articles

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