Gravity Analysis Regge4 Dcontinuum Preflight Discrete Exact Regge Continuum Face

The discrete face coefficient

In numerical relativity, Regge calculus approximates spacetime by a mesh of flat simplices, and the Einstein-Hilbert action of general relativity is replaced by a sum over the mesh. The central question is whether the discrete action, as the mesh is refined, converges to the continuum Einstein-Hilbert action. The declaration discreteExactReggeContinuumFaceCoeff names the coefficient that the discrete face contribution must equal for that convergence to hold.

The declaration establishes a precise algebraic identity: the discrete face coefficient equals the Einstein-Hilbert transverse-traceless coefficient, which is -(1/4). This is a definitional fact, not a derived theorem about convergence. It pins down the target value that the discrete expression must match, using the independently frozen Einstein-Hilbert quadratic functional with kappa_einstein, a nonzero coupling constant. The identity is checked by the machine-checked library of formal theorems as a definitional equality, meaning it holds by the way the objects are defined.

What the declaration does not claim is the harder result: it does not prove that the discrete Regge action, as the mesh side N grows, actually converges to the continuum Einstein-Hilbert action. That convergence remains an OPEN target. The declaration also does not reverse-engineer lattice weights from the Einstein-Hilbert answer; the Einstein-Hilbert quadratic is frozen independently, and the later algebraic closer must observe equality, never fit a scale. The wrong mesh power N⁻² instead of N⁻⁴ is explicitly rejected as a decoy, and arbitrary pullbacks are excluded from the action theorem.

In plain terms, the declaration fixes the target number on the wall before the computation begins. It says: the discrete face coefficient must be -(1/4), and here is the definitional identity that states it. The convergence itself, the Tendsto property that the discrete sequence approaches the continuum value, is a separate, still-open problem that this declaration does not settle.

/-- Banked algebraic continuum face: `2 · (-1/8) · ‖E‖_F²` (EH audit §2.3).
Not the ledger ContinuumSymbolIs binder. -/
def discreteExactReggeContinuumFaceCoeff (E : Mat4) : ℝ :=
  ReggeExactFlatHessianNormGate4D.continuumEHDiscreteFace (frobeniusNormSq E)

theorem einsteinHilbertTTCoefficient4D_eq :
    einsteinHilbertTTCoefficient4D = -(1 / 4 : ℝ) := rfl

theorem decoy_wrong_mesh_power {N : ℕ} (hN : 2 ≤ N) :
    wrongMeshPowerWeight N ≠ correctTorusDensityWeight N := by
  unfold wrongMeshPowerWeight correctTorusDensityWeight
  have hNne : (N : ℝ) ≠ 0 := by exact_mod_cast (ne_of_gt (lt_of_lt_of_le (by norm_num : 0 < 2) hN))
  intro h
  have hclear :
      ((N : ℝ)⁻¹) ^ 2 * (N : ℝ) ^ 4 = ((N : ℝ)⁻¹) ^ 4 * (N : ℝ) ^ 4 :=
    congrArg (fun t : ℝ => t * (N : ℝ) ^ 4) h
  have hNsq : (N : ℝ) ^ 2 = 1 := by
    field_simp [hNne] at hclear
    exact hclear
  have hcast : (2 : ℝ) ≤ (N : ℝ) := by exact_mod_cast hN
  nlinarith [sq_nonneg ((N : ℝ) - 1)]

What this page does not claim

The declaration does not prove that the discrete Regge action converges to the continuum Einstein-Hilbert action. The declaration does not fit or reverse-engineer lattice weights from the Einstein-Hilbert answer. The declaration does not establish that the discrete face coefficient is the continuum limit, only that it is the target value.

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/Regge4DContinuumPreflight.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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