Gravity Analysis Regge Exact Flat Hessian Norm Gate4 Daudit Norm Gate Audit Pack

The audit gate

In numerical relativity, the Regge calculus approximates curved spacetime by a discrete lattice of flat simplices, and the Einstein-Hilbert action becomes a sum over deficit angles. The continuum limit is delicate: the discrete Regge action converges to the continuum action only under specific conditions on the lattice and the test functions. A common failure mode is that the discrete coefficient and the continuum coefficient drift apart as the lattice refines, so the discrete theory does not reproduce general relativity in the limit.

The framework's machine-checked library of formal theorems contains a declaration, norm_gate_audit_package, that pins down one precise relationship at unit scale. It proves four facts together: the normalization gate passes (NormalizationGatePass = true); the exact unit Frobenius time-time coefficient is not equal to the frozen preflight Einstein-Hilbert coefficient; the frozen preflight coefficient equals the discrete bookkeeping factor times the exact unit Frobenius coefficient; and the continuum Einstein-Hilbert discrete face at unit argument equals the frozen preflight coefficient. In plain terms, the theorem establishes a scaling relation between the discrete bookkeeping factor and the continuum coefficient, and it certifies that the two are genuinely different objects, not accidentally equal.

The theorem is a formal statement in the framework's library, not a numerical simulation or a measurement. It is a logical implication: if the four antecedent facts hold, the conjunction follows. The declaration is a theorem, so it is proved in the machine-checked library with no unproved assumptions beyond the ambient type theory. It does not claim that the Regge action converges to the Einstein-Hilbert action in general, nor that the discrete bookkeeping factor has a particular numerical value, nor that the normalization gate pass implies physical correctness of any specific lattice.

What the theorem does give is a sharp, checkable identity that anchors the discrete-to-continuum comparison at unit scale. It tells a reader that the framework's discrete gravity analysis has a built-in audit: the gate passes, the coefficients are distinct, and their ratio is exactly the discrete bookkeeping factor. That is a precise statement a future derivation can build on, and it is the kind of exact relation that numerical checks alone cannot supply.

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

The theorem does not assert that the Regge action converges to the Einstein-Hilbert action for general lattices or test functions. The theorem does not assign a specific numerical value to the discrete bookkeeping factor. The theorem does not imply that a passing normalization gate makes any particular lattice physically correct.

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)

A page whose claims cannot be reproduced this way does not ship. In production, every anchor links to the exact declaration in the public source release, and this block carries the build receipt for the page itself.

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