Gravity Analysis Regge Exact Flat Hessian Norm Gate4 D Proposed Unit Frobenius E

The normalization gate

In numerical work on gravity, the Einstein-Hilbert action often appears with a coefficient that depends on how the metric perturbation is normalized. The Recognition Science framework's machine-checked library of formal theorems contains a declaration, proposed_unitFrobenius_EH_eq_exact, that pins down one such coefficient. It proves that the proposed coefficient for the Einstein-Hilbert transverse-traceless face, when the perturbation has unit Frobenius norm, equals the exact coefficient derived from the Hessian of the Regge action. Concretely, the theorem states that the value -(1/8) equals the exact unit-Frobenius transverse-traceless coefficient.

The theorem is a definitional equality: it holds by unfolding definitions and simplifying, so it is a formal identity within the framework's library, not a physical derivation. It resolves a historical mismatch in the framework's own development. An earlier preflight check demanded a frozen coefficient of -1/4 on unit-Frobenius transverse-traceless perturbations, while the exact algebraic computation gave -1/8. The new theorem confirms that the proposed -1/8 is the correct exact value for this normalization, and the frozen -1/4 is recovered only by multiplying by a discrete bookkeeping factor of 2.

What the theorem does not claim is as important as what it proves. It does not assert that the continuum Einstein-Hilbert action converges to this value in any geometric limit. The docstring is explicit: the identity does not inhabit the geometric ContinuumSymbolIs Tendsto statement, does not inhabit the ledger convergence statement S_RS_converges_EH_4d, and does not flip the gap_action_recovery flag. Those require a separate geometric mesh Tendsto argument. The theorem also does not claim that the Regge action equals the Einstein-Hilbert action; the library notes that the Regge transverse-traceless face is actually ρ times the Einstein-Hilbert one with ρ = 1/2, so the convergence result targets the Regge face, not the Einstein-Hilbert one.

In plain terms, the declaration is a normalization honesty gate. It certifies that a proposed coefficient matches the exact algebraic value under a specific normalization, and it documents that a previously frozen coefficient was a bookkeeping artifact, not a geometric fact. The reader can now see that the framework's internal consistency check passes for this coefficient, while the physical convergence claim remains a separate, open target.

theorem proposed_unitFrobenius_EH_eq_exact :
    einsteinHilbertTTCoefficient4D_unitFrobenius_proposed =
      exactUnitFrobeniusTTCoefficient :=
  unitFrobenius_EH_eq_exact

theorem unitFrobenius_EH_eq_exact :
    einsteinHilbertTTCoefficient4D_unitFrobenius =
      exactUnitFrobeniusTTCoefficient := rfl

/-- 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

def typedBlocker_preflight_EH_unitF_mismatch : String :=
  "Algebraic face banked: discreteBookkeepingFactor * unitF = 2*(-1/8)=-1/4 on unit-Frobenius TT (EH audit §2.3). Constant-face ContinuumSymbolIs inhabit REVERTED; ledger S_RS / gap_action_recovery require geometric mesh Tendsto (finiteExactReggeSymbol / |k|^2)."

What this page does not claim

The theorem does not prove any continuum limit or convergence of the discrete action to the Einstein-Hilbert action. The theorem does not claim the Regge action equals the Einstein-Hilbert action; the library notes they differ by a factor of ρ = 1/2. The theorem does not establish that the frozen -1/4 coefficient is physically correct for any geometric setting.

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.

Derived articles

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