Gravity Analysis Regge Exact Flat Hessian Norm Gate4 D Continuum Ehdiscrete Face

The normalization gate

The declaration continuumEHDiscreteFace_on_unitF is a small but precise result in a machine-checked library of formal theorems. It states that a certain function, when evaluated with its input set to 1, returns the value -1/4. The function is called continuumEHDiscreteFace, and it multiplies a bookkeeping factor of 2 by an exact coefficient of -1/8 and by the square of a Frobenius norm, a measure of the size of a matrix. Setting that norm square to 1, the product is 2 times -1/8, which is -1/4. The theorem is a definitional equality, meaning it holds by the very definitions involved, and it is checked by the Lean proof assistant.

The context of this result is a normalization gate in a discrete model of gravity. The number -1/4 was a frozen preflight coefficient, a value that a previous version of the framework required. The exact algebraic computation, however, gives -1/8 per unit Frobenius norm. The factor of 2 is a discrete bookkeeping factor, a dimension-independent constant that appears in the discrete action. The theorem continuumEHDiscreteFace_on_unitF shows that the frozen -1/4 is exactly the product of this bookkeeping factor and the exact unit-Frobenius coefficient. It thus reconciles the frozen value with the exact computation, but only as an algebraic identity, not as a geometric limit.

The declaration explicitly does not claim that the discrete bookkeeping factor of 2 is a geometric continuum limit. It does not inhabit the geometric convergence statement S_RS_converges_EH_4d, which requires a mesh sequence tending to a continuum. The identity 2 times -1/8 equals -1/4 is banked only as a non-ledger algebraic identity. It does not flip the gap action recovery flag. The name Discrete in the function is flagged as a misnomer: the discrete side is the factor that was applied, not the value returned. The declaration is a normalization gate pass, a check that the frozen coefficient is consistent with the exact algebraic value, but it is not a proof of convergence to the Einstein-Hilbert action.

The result matters because it clarifies a historical failure in the framework. A preflight check once demanded a frozen coefficient that did not match the exact algebraic value. This declaration records that mismatch and shows how the frozen value arises from the exact one through the discrete bookkeeping factor. It is a certificate of honesty: the framework states plainly what the identity does and does not establish. The reader can now see that the coefficient -1/4 is the product of a discrete factor and an exact unit-Frobenius face, and that this product is not a geometric limit. This is a small but load-bearing piece of the framework's self-audit.

theorem continuumEHDiscreteFace_on_unitF :
    continuumEHDiscreteFace (1 : ℝ) = frozenPreflightEHCoefficient := by
  unfold continuumEHDiscreteFace
  rw [mul_one, frozen_EH_is_discrete_bookkeeping_times_unitF]

/-- 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 declaration does not prove convergence to the Einstein-Hilbert action; it only records an algebraic identity. The declaration does not establish that the discrete bookkeeping factor of 2 is a geometric continuum limit. The declaration does not flip the gap action recovery flag.

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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