Gravity Analysis Regge Exact Flat Hessian Bloch Symbol4 D Gate Passes With Discr

The discrete bookkeeping gate

The declaration gate_passes_with_discrete_bookkeeping is a theorem in the framework's machine-checked library of formal theorems. It establishes a precise numerical agreement. The theorem states that a certain discrete bookkeeping factor, when multiplied by a unit-normalized coefficient, equals a frozen preflight coefficient. It also states that a continuum expression, evaluated at one, equals that same frozen coefficient. The theorem then asserts that a normalization gate passes, meaning the discrete and continuum computations are consistent at this specific point.

The context is a lattice model of gravity. The framework models the curvature of space-time as a discrete structure, a lattice of points with couplings. The declaration concerns a specific object: the exact midpoint Bloch symbol, a trigonometric polynomial that represents the Hessian, or second-order variation, of the action on a unit cell. The theorem proves that two ways of computing a particular coefficient, one discrete and one continuum, agree. This is a check that the discrete model reproduces a known continuum result at a single point, not a proof that the two theories are equivalent everywhere.

The declaration is part of a larger effort to show that a discrete, bookkeeping-based model of gravity converges to the classical Einstein-Hilbert action. The theorem proves a specific algebraic identity and a limit statement. It does not prove that the discrete model is fully equivalent to general relativity. The library's own status record marks two related goals as open: the existence of a certain structure and the recovery of a gap action. The theorem is a gate that passes, but it is not the final gate.

The practical consequence is that the framework has a verified, machine-checked point of contact between its discrete model and the continuum theory. This is a necessary but not sufficient step. It shows that the discrete model is not obviously wrong at this point, but it does not show that the model is correct everywhere. The open questions remain targets for future work.

theorem gate_passes_with_discrete_bookkeeping :
    NormalizationGatePass = true ∧
      discreteBookkeepingFactor * exactUnitFrobeniusTTCoefficient =
        frozenPreflightEHCoefficient ∧
          continuumEHDiscreteFace (1 : ℝ) = frozenPreflightEHCoefficient :=
  ⟨normalizationGatePass_true, frozen_EH_is_discrete_bookkeeping_times_unitF.symm,
    continuumEHDiscreteFace_on_unitF⟩

theorem gate_passes_with_discrete_bookkeeping :
    NormalizationGatePass = true ∧
      discreteBookkeepingFactor * exactUnitFrobeniusTTCoefficient =
        frozenPreflightEHCoefficient ∧
          continuumEHDiscreteFace (1 : ℝ) = frozenPreflightEHCoefficient :=
  ⟨normalizationGatePass_true, frozen_EH_is_discrete_bookkeeping_times_unitF.symm,
    continuumEHDiscreteFace_on_unitF⟩

theorem gate_passes_with_discrete_bookkeeping :
    NormalizationGatePass = true ∧
      discreteBookkeepingFactor * exactUnitFrobeniusTTCoefficient =
        frozenPreflightEHCoefficient ∧
          continuumEHDiscreteFace (1 : ℝ) = frozenPreflightEHCoefficient :=
  ⟨normalizationGatePass_true, frozen_EH_is_discrete_bookkeeping_times_unitF.symm,
    continuumEHDiscreteFace_on_unitF⟩

What this page does not claim

The theorem does not prove the discrete model is equivalent to general relativity. The theorem does not establish the existence of the structure marked as open in the status record. The theorem does not prove the recovery of the gap action.

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/ReggeExactFlatHessianBlochSymbol4D.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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