Gravity Weak Field Conformal Regge Weak Field Conformal Reduction Laplacian Data

The weak-field reduction

Regge calculus is a way of doing general relativity without coordinates, by replacing smooth spacetime with a network of flat pieces joined along edges. The action, the quantity whose variation gives the field equations, is a sum over the hinges of the network of the area of each hinge times the deficit angle, the angle by which the surrounding flat pieces fail to close flat. In the weak-field limit, one perturbs the edge lengths slightly away from a uniform background and keeps only terms up to second order in the perturbation. The theorem weak_field_conformal_reduction_laplacianData_kappa states that for a symmetric matrix of coefficients A, the second-order Regge action divided by the coupling constant κ equals (1/κ) times one half times the Dirichlet form of A, which is a sum over pairs of sites of A_ij times the squared difference of the perturbation values at those sites.

The statement is a formal identity in the machine-checked library of formal theorems. It is a conditional theorem: it holds for any symmetric matrix A, any nonzero κ, and any perturbation field ε on a finite set of n sites. The proof is a short algebraic manipulation that rewrites the action using the definition of the Laplacian data, then applies the previously proved reduction for general weak-field data, and finally clears denominators. The theorem does not depend on any specific geometry, only on the algebraic structure of the Laplacian coefficients, which are defined so that the sum of each row is zero.

What the theorem establishes is a bridge between two ways of writing the same second-order quantity. On the left stands the Regge action, built from areas and deficit angles. On the right stands the Dirichlet form, the discrete analog of the integral of the squared gradient, familiar from the theory of electrical networks and random walks. The equality says that, in the conformal sector and to second order, the gravitational action is exactly a Dirichlet energy with weights given by the matrix A. This is the algebraic core of a reduction: once the geometric coefficients are supplied, the physics of weak-field gravity on a lattice becomes the physics of a discrete diffusion problem.

In Recognition Science, this theorem is one step in a program that treats the ledger, a discrete record of events, as the primitive object from which geometry emerges. The Dirichlet form is the natural energy for a field on a ledger, and the theorem shows how such an energy arises from the Regge action without any additional assumption beyond the conformal ansatz and the row-sum condition. The framework's library proves the identity once and for all, so that any future computation of the geometric coefficients can rely on it as a lemma.

The theorem does not compute the matrix A from any particular lattice. That computation, from Cayley–Menger determinants or dihedral angles, remains a separate geometric task. The theorem also does not establish that the row-sum condition, the Schläfli condition, holds for any physical lattice; that is a further geometric hypothesis. What is proved is the algebraic reduction: if the row-sum condition holds, then the action is exactly the Dirichlet form.

theorem weak_field_conformal_reduction_laplacianData_kappa {n : ℕ}
    (A : Fin n → Fin n → ℝ) (hA : ∀ i j, A i j = A j i)
    (κ : ℝ) (hκ : κ ≠ 0) (ε : LogPotential n) :
    secondOrderReggeAction (laplacianReggeData A hA) ε / κ
      = (1 / κ) * (1 / 2) * dirichletForm A ε := by
  rw [weak_field_conformal_reduction_laplacianData A hA ε]
  field_simp

theorem weak_field_conformal_reduction_laplacianData_kappa {n : ℕ}
    (A : Fin n → Fin n → ℝ) (hA : ∀ i j, A i j = A j i)
    (κ : ℝ) (hκ : κ ≠ 0) (ε : LogPotential n) :
    secondOrderReggeAction (laplacianReggeData A hA) ε / κ
      = (1 / κ) * (1 / 2) * dirichletForm A ε := by
  rw [weak_field_conformal_reduction_laplacianData A hA ε]
  field_simp

theorem weak_field_conformal_reduction_laplacianData_kappa {n : ℕ}
    (A : Fin n → Fin n → ℝ) (hA : ∀ i j, A i j = A j i)
    (κ : ℝ) (hκ : κ ≠ 0) (ε : LogPotential n) :
    secondOrderReggeAction (laplacianReggeData A hA) ε / κ
      = (1 / κ) * (1 / 2) * dirichletForm A ε := by
  rw [weak_field_conformal_reduction_laplacianData A hA ε]
  field_simp

What this page does not claim

This theorem does not compute the geometric coefficients A from any specific lattice geometry. This theorem does not prove the Schläfli row-sum condition holds for any physical lattice. This theorem does not establish the physical validity of the conformal ansatz for edge lengths.

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

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