Gravity Analysis Regge Exact Flat Hessian Norm Gate4 D Continuum Ehscale Explici

A normalization gate

The declaration continuumEHScaleExplicit_axisTTPlus_face is a formal theorem in the Recognition Science framework's machine-checked library of formal theorems. It proves a numerical identity about the transverse-traceless (TT) face of the Einstein-Hilbert action in four spacetime dimensions. In plain terms, it shows that when the squared Frobenius norm of the relevant field tensor equals 2, the scale-explicit continuum expression evaluates to -1/4. This is a statement about algebraic bookkeeping, not about the physical continuum limit.

The context is a normalization gate. A preflight check once demanded a frozen coefficient of -1/4 on the unit-Frobenius TT face. The exact algebraic computation gives -1/8 per unit Frobenius norm. The theorem resolves the apparent mismatch by showing that -1/4 equals 2 times -1/8, where the factor 2 is a discrete bookkeeping factor. The identity is correct as algebra, but the library explicitly records that this discrete factor does not inhabit the geometric continuum symbol, does not inhabit the ledger convergence statement, and does not flip the action-recovery gap. It is kept for compatibility with consumers, not as a physical derivation.

The theorem also clarifies a naming subtlety. The declaration continuumEHScaleExplicit carries the letters EH, but the docstring flags that this is not the Einstein-Hilbert face; it is the Regge action's TT face, which is half the Einstein-Hilbert one. The convergence theorem for the dictionary mesh sequence establishes convergence to the Regge face, not to the Einstein-Hilbert face, which is twice it. The name overstates by a factor of 1/ρ with ρ = 1/2, and the library flags this rather than renaming to keep consumers stable.

What the identity does not claim is as important as what it proves. It does not claim that the discrete factor 2 arises from a geometric continuum limit; that would require a mesh refinement argument with a named Tendsto condition. It does not claim that the Regge action equals the Einstein-Hilbert action in the continuum; the theorem explicitly banks the 2× factor only as a non-ledger algebraic identity. And it does not claim that the normalization gate passing here settles any physical question about gravity; the gate is a bookkeeping check, and the library says so in its own words.

theorem continuumEHScaleExplicit_axisTTPlus_face :
    continuumEHScaleExplicit (2 : ℝ) = frozenPreflightEHCoefficient := by
  unfold continuumEHScaleExplicit einsteinHilbertTTCoefficient4D_unitFrobenius
    frozenPreflightEHCoefficient einsteinHilbertTTCoefficient4D
  norm_num

theorem continuumEH_unitF_matches_exact_m2 :
    continuumEHunitFrobeniusFromFirstPrinciples =
      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)."

/-- Value `-(1/8) · ‖E‖_F²`.  Despite the `EH` in the name this is **not** the
Einstein-Hilbert face: it is the **Regge action's** transverse-traceless face,
which is `ρ` times the Einstein-Hilbert one with `ρ = 1/2` derived in
`ReggeNormalizationDerived4D` (arc 2 step 7).  This matters because it is the
limit value that `SRSConvergesEH4D.S_RS_converges_EH_4d_closed` proves the
dictionary mesh sequence converges to, so that theorem establishes convergence
to the Regge face and not to the Einstein-Hilbert face, which is twice it.
Correct as a statement about the Regge action; the name overstates by `1/ρ`.
Flagged 2026-07-27 (arc 2 step 8) rather than renamed, to keep consumers
stable. -/
def continuumEHScaleExplicit (frobeniusSq : ℝ) : ℝ :=
  einsteinHilbertTTCoefficient4D_unitFrobenius * frobeniusSq

What this page does not claim

It does not claim that the discrete factor 2 arises from a geometric continuum limit. It does not claim that the Regge action equals the Einstein-Hilbert action in the continuum. It does not claim that the normalization gate passing settles any physical question about gravity.

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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