Gravity Analysis Recognition Mesh Exact Jbridge4 D Exact Jequals True Regge Hess

The mesh bridge

In numerical relativity, the Regge action is a way to approximate the curvature of spacetime by triangulating it into flat, four-dimensional simplexes. The true Regge Hessian is the second derivative of that action with respect to the edge lengths, a quantity that controls how the action responds to small perturbations. The Recognition Science declaration exactJEqualsTrueReggeHessian_holds establishes a precise bridge between this discrete Regge object and a continuum object called the Option-C midpoint Bloch symbol, which appears in the framework's analysis of gravity.

The framework builds a canonical mesh on a four-dimensional torus, labeled by a natural number j. On this mesh it defines an action, exactJActionOnMesh, whose amplitude is a quadratic form in the perturbation amplitude ε. The declaration proves that the second difference of this action, its amplitude Hessian, exists and is exactly equal to the mesh true-Regge quadratic Hessian. This holds for all nonzero momentum modes and all polarizations that satisfy the transverse-traceless condition. The proof is a theorem in the framework's machine-checked library of formal theorems, meaning it is derived from the axioms of the underlying logic without additional assumptions.

The declaration does not claim that the full nonlinear Regge action is equal to the exact-J action. That elevation, via the Schläfli identity, remains open. It also does not claim that the iterated limit N → ∞ of the mesh Hessian converges to the Einstein-Hilbert action; that convergence is a separate theorem, recognitionExactJConvergesEH, which is closed at a scale-explicit face. The declaration specifically does not imply that the framework's action recovers the Einstein-Hilbert action in the continuum limit, nor does it establish any physical prediction about gravity. It is a statement about the equality of two mathematical objects at the level of their second-order variation on a discrete mesh.

What the declaration does establish is a consistency check: the framework's exact-J action, when restricted to a single mesh and a single mode, has the same curvature response as the standard Regge action. This is a necessary condition for the framework's gravity program to be viable, but it is not sufficient. The open questions are whether the full action, not just its Hessian, matches Regge, and whether the continuum limit reproduces general relativity. The declaration is a step, not a destination.

/-- **THEOREM:** exact-J amplitude Hessian equals the mesh true-Regge
Hessian by construction (MODEL action identification). -/
theorem exactJEqualsTrueReggeHessian_holds :
    ExactJEqualsTrueReggeHessian := by
  intro j m E _hm _hTT
  refine ⟨meshTrueReggeQuadraticHessian (canonicalRecognitionMesh j) m E,
    exactJAmplitudeHessian_eq_mesh _ _ _, rfl⟩

/-- **THEOREM:** exact-J amplitude Hessian equals the mesh true-Regge
Hessian by construction (MODEL action identification). -/
theorem exactJEqualsTrueReggeHessian_holds :
    ExactJEqualsTrueReggeHessian := by
  intro j m E _hm _hTT
  refine ⟨meshTrueReggeQuadraticHessian (canonicalRecognitionMesh j) m E,
    exactJAmplitudeHessian_eq_mesh _ _ _, rfl⟩

What this page does not claim

The full nonlinear Regge action equals the exact-J action. The continuum limit of the mesh action recovers the Einstein-Hilbert action. The declaration establishes any physical prediction about gravity. The declaration uses the ExactJRefinementFamilyLimit as a continuum premise.

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/RecognitionMeshExactJBridge4D.lean
expected axiom basis: [propext, Classical.choice, Quot.sound] (the Lean kernel's standard three; no RS-specific axioms)

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