Gravity Analysis Regge Exact Flat Hessian Norm Gate4 D Discrete Bookkeeping Fact

The bookkeeping factor

In numerical work on Einstein's equations, a discrete approximation to a continuous action can differ from its continuum limit by a constant factor. This page concerns one such factor, the number 2, that appears in the Recognition Science framework's analysis of the Regge action, a discrete version of gravity built from flat simplices. The framework's machine-checked library of formal theorems establishes that its discrete bookkeeping factor, a numerical multiplier used to reconcile discrete and continuum expressions, equals the factor 2 in its exact discrete Regge action. The proof is a direct definitional equality: the factor is defined to be 2, and the theorem confirms that this matches the factor appearing in the exact action symbol.

The substance of the result is algebraic, not geometric. The framework's library shows that multiplying the exact unit-Frobenius coefficient, -1/8, by this factor 2 yields -1/4, which is the frozen coefficient used in an earlier preflight check. That identity, 2 * (-1/8) = -1/4, is a statement about numbers. The library explicitly records that this identity does not inhabit the geometric continuum limit, does not inhabit the ledger convergence statement, and does not flip the action recovery gap. In plain terms, the factor 2 is a bookkeeping device that reconciles two algebraic faces of the same discrete action; it is not a claim about the continuum limit of Regge gravity.

The distinction matters because the framework's own history records a failure when this distinction was blurred. A preflight check once demanded the frozen coefficient -1/4 on a unit-Frobenius transverse-traceless tensor, while the exact algebraic m² gives -1/8 per unit Frobenius. The factor 2 was banked as a non-ledger algebraic identity to recover the frozen value. The library now carries a certificate of this historical fail: the exact unit-Frobenius coefficient is not equal to the frozen preflight coefficient, and the frozen coefficient is the factor 2 times the exact one. The factor 2 is thus a reconciliation, not a derivation of the continuum value.

What the declaration does not claim is as important as what it proves. It does not claim that the continuum Einstein-Hilbert action is -1/4 times the squared Frobenius norm. The library states plainly that the continuum face is scale-explicit at -1/8 times the squared norm, and that the -1/4 value is the axisTTPlus face with squared norm 2. The factor 2 identity also does not establish convergence of the discrete Regge action to the Einstein-Hilbert action in any geometric sense; that convergence remains a target requiring a mesh limit. The declaration is a narrow algebraic fact, and the framework's own documentation flags the name "Discrete" as a misnomer: the discrete side is the factor applied, not the value returned.

theorem discreteBookkeepingFactor_eq_exactAction :
    discreteBookkeepingFactor =
      Regge4DExactActionSymbol.discreteBookkeepingFactor := by
  simp [discreteBookkeepingFactor, Regge4DExactActionSymbol.discreteBookkeepingFactor]

theorem exact_unitFrobenius_ne_frozen_preflight_EH :
    exactUnitFrobeniusTTCoefficient ≠ frozenPreflightEHCoefficient := by
  unfold exactUnitFrobeniusTTCoefficient frozenPreflightEHCoefficient
    exactHessianM2UnitFrobeniusTTCoeff einsteinHilbertTTCoefficient4D
  norm_num

/-- Frozen `-1/4` is discrete bookkeeping times the unit-F m² face. -/
theorem frozen_EH_is_discrete_bookkeeping_times_unitF :
    frozenPreflightEHCoefficient =
      discreteBookkeepingFactor * exactUnitFrobeniusTTCoefficient := by
  unfold frozenPreflightEHCoefficient exactUnitFrobeniusTTCoefficient
    discreteBookkeepingFactor exactHessianM2UnitFrobeniusTTCoeff
    einsteinHilbertTTCoefficient4D
  norm_num

What this page does not claim

This does not claim the continuum Einstein-Hilbert action is -1/4 times the squared Frobenius norm. This does not claim the factor 2 identity establishes geometric convergence of the discrete Regge action to the continuum action. This does not claim the declaration flips the action recovery gap or inhabits the ledger convergence statement.

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/ReggeExactFlatHessianNormGate4D.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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