Gravity Analysis Regge4 Dflat Second Variation Candidate Continuum Face Normaliz

A face of the candidate

In numerical relativity, a Regge action is a discrete way to write Einstein's theory of gravity: spacetime is chopped into flat simplexes, and the action is built from their edge lengths and dihedral angles. The question at hand is what happens to the second variation, the curvature of the action, when you look near a flat, four-dimensional spacetime. The declaration candidate_continuumFace_normalizedTT_symbolDir is a machine-checked theorem that evaluates one specific face of a candidate expression for that second variation.

The theorem states that when you take the candidate's momentum form, apply it to a particular normalized direction called axisTTPlus, and divide by the squared norm of the symbolDir direction, the result is exactly -1/16. This is a precise, computable number attached to a specific geometric setup. It is not a vague statement about the whole action; it is a single face of a larger object, evaluated at a chosen point and direction. The declaration also proves that this -1/16 differs from the corresponding coefficient in the frozen Einstein-Hilbert action, which is -1/4, and that a certain density dictionary factor survives with value 1.

What the declaration does not claim is just as important. It does not claim that this candidate is the true second variation of the nonlinear Regge action. That larger claim, called Regge4DSchlaefliElevationToCandidate, is explicitly marked as OPEN in the framework's own status structure. The path to that claim requires a full off-flat pathwise Schläfli identity, which is absent. The theorem also does not claim to recover the gap action, and it does not inhabit the statement that the RS action converges to the Einstein-Hilbert action in four dimensions.

In plain terms, the declaration is a precise, verified calculation of one component of a candidate. It is a building block, not the finished building. The framework has proved the flat pieces and the directional kills, but the elevation from flat to fully nonlinear remains a target. This is an honest status report: a theorem about a face, with the larger structure clearly labeled as open.

/-- Continuum-facing coefficient after Frobenius pin and `/|symbolDir|²`:
`-1/16`. -/
theorem candidate_continuumFace_normalizedTT_symbolDir :
    distinctHingeMomentForm ((Real.sqrt 2)⁻¹ • axisTTPlus) symbolDir /
        (∑ i : Fin 4, symbolDir i * symbolDir i) =
      (-1 / 16 : ℝ) :=
  continuumFace_normalizedPlus_symbolDir

/-- **THEOREM (falsifier arithmetic).** The candidate's continuum face on
normalized TT at `symbolDir` is `-1/16`, which is not the frozen EH
coefficient `-1/4`.  Density dictionary survivor is already `1`. -/
theorem candidate_face_ne_eh :
    (-1 / 16 : ℝ) ≠ einsteinHilbertTTCoefficient4D ∧
      survivingDictionaryFactor4D = 1 :=
  ⟨by rw [einsteinHilbertTTCoefficient4D_eq]; norm_num, rfl⟩

What this page does not claim

The declaration does not prove that the candidate is the true second variation of the nonlinear Regge action. The declaration does not claim to recover the gap action or to show convergence to the Einstein-Hilbert action in four dimensions. The declaration does not provide a value for the full continuum face of the candidate, only for the specific normalized direction.

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/Regge4DFlatSecondVariation.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

This page is generated by a question-recursion engine: the questions its answers raise become the next pages. The current agenda, with open targets marked red: