Gravity Analysis Regge Exact Flat Hessian Norm Gate4 D Frozen Eh Is Discrete Boo

A normalization gate

In numerical relativity, the Einstein-Hilbert action is often evaluated on a transverse-traceless (TT) perturbation, a symmetric tensor that is divergence-free and trace-free. The action's coefficient is then proportional to the squared Frobenius norm of that tensor. The Recognition Science framework's machine-checked library of formal theorems contains a declaration, frozen_EH_is_discrete_bookkeeping_times_unitF, that pins down a specific algebraic relationship between two ways of writing this coefficient.

The declaration states that a frozen preflight coefficient, equal to -1/4, is the product of a dimensionless discrete bookkeeping factor of 2 and an exact unit-Frobenius coefficient of -1/8. In plain arithmetic: 2 multiplied by -1/8 equals -1/4. The theorem proves this identity by unfolding the definitions and simplifying. A companion theorem confirms that the exact unit-Frobenius coefficient is not equal to the frozen preflight coefficient, so the two are genuinely different numbers that happen to be related by the factor of 2.

The framework's own documentation is explicit about what this identity is not. It is not a statement about a geometric limit, and it does not inhabit the ledger's convergence theorem for the Einstein-Hilbert action. The identity is banked only as a non-ledger algebraic fact. The framework's library also records that a proposed scale-explicit form, -(1/8) times the squared Frobenius norm, matches the exact coefficient, and that the frozen -1/4 corresponds to the case where that squared norm equals 2.

What this means in practice is that the framework has a normalization gate that passes. The gate checks that the frozen coefficient used in a preflight check is consistent with the exact algebraic value, once the discrete bookkeeping factor is accounted for. The theorem does not claim that the discrete factor arises from a continuum limit, nor that the frozen coefficient is the exact one. It simply certifies the algebraic bridge between the two, and the library flags the historical failure that motivated this precise statement.

/-- 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

theorem exact_unitFrobenius_ne_frozen_preflight_EH :
    exactUnitFrobeniusTTCoefficient ≠ frozenPreflightEHCoefficient := by
  unfold exactUnitFrobeniusTTCoefficient frozenPreflightEHCoefficient
    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 discrete bookkeeping factor of 2 is not derived from a geometric continuum limit. The frozen coefficient -1/4 is not the exact unit-Frobenius coefficient. The identity does not establish convergence of the discrete action to the Einstein-Hilbert 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/ReggeExactFlatHessianNormGate4D.lean
expected axiom basis: [propext, Classical.choice, Quot.sound] (the Lean kernel's standard three; no RS-specific axioms)

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