Gravity Unified Lattice Manifold Correspondence Discrete Regge Eq Neg Lattice La

The discrete equation

The declaration discreteRegge_eq_neg_lattice_laplacian is a theorem in the framework's machine-checked library of formal theorems. It states that, on a finite lattice, the discrete Regge equation of motion equals the negative lattice Laplacian. In plain terms, the variation of the Regge action with respect to an edge length, set to zero, produces exactly the same expression as the negative discrete Laplacian acting on the field. The equation is Σ_k [(f x - f (x - e_k)) - (f (x + e_k) - f x)] = -Δ f x, where the sum runs over the lattice directions and Δ denotes the lattice Laplacian.

This identity is a discrete analogue of the continuum fact that the Euler-Lagrange equation for a free scalar field is the wave equation, □φ = 0, whose spatial part is the Laplacian. The theorem is proved for any lattice field f on a lattice of dimension D, with no restriction on the field values or the lattice size. It is a purely algebraic identity, derived from the definition of the Regge action and the lattice Laplacian, and it holds for every lattice point.

The importance of this identity is that it connects the discrete geometry of the Regge calculus to the continuum Laplacian. In the framework's broader correspondence, this discrete Laplacian converges to the continuum Laplacian as the lattice spacing goes to zero, at a rate proportional to the square of the spacing. That convergence is what links the discrete Regge equations to the linearized Einstein field equations in the weak-field limit. The identity is a necessary step in that chain, but it is not the whole chain.

In Recognition Science, this theorem is part of a larger packaged statement that a sequence of deformed cubic lattices can approximate a smooth Lorentzian manifold. The framework models gravity as a discrete lattice theory, and this identity is one of the certificates that the discrete theory reproduces the continuum equations in the appropriate limit. The framework's library proves this identity from the definitions, with no additional axioms beyond the standard three.

What the theorem does not claim is that the Regge equations are exactly the Einstein field equations. It only establishes the discrete identity and its convergence to the linearized equations. The full nonlinear extension, covering strong fields and singularities, is a separate conditional statement that depends on an external result. The theorem also does not claim that the lattice is the fundamental nature of spacetime; it only shows that the discrete theory converges to the continuum one in the weak-field regime.

/-- The discrete Regge equation (linearised EL) at site `x` equals minus
    the lattice Laplacian of `f`:
    `Σ_k [(f(x) − f(x−eₖ)) − (f(x+eₖ) − f(x))] = −Δ_lat f(x)`.

    This is the algebraic identity in
    `CubicReggeProof.linearized_el_eq_neg_laplacian`. -/
theorem discreteRegge_eq_neg_lattice_laplacian {D : ℕ}
    (f : LatticeField D) (x : Fin D → ℤ) :
    (∑ k : Fin D,
      ((f x - f (shift_minus k x)) -
       (f (shift_plus k x) - f x))) =
    -lattice_laplacian f x :=
  CubicReggeProof.cubic_regge_convergence_cert.el_is_laplacian D f x

/-- The lattice Laplacian (scaled by 1/a²) converges to the continuum
    Laplacian `∇²` at `O(a²)`. This is the standard finite-difference
    statement, here supplied for the smooth field `h`. -/
theorem latticeLaplacian_to_continuum (W : WeakFieldData) (x a : ℝ)
    (ha : a ≠ 0) :
    ∃ C : ℝ, |(W.h (x + a) + W.h (x - a) - 2 * W.h x) / a ^ 2 -
              deriv (deriv W.h) x| ≤ C * a ^ 2 :=
  CubicReggeProof.cubic_regge_convergence_cert.laplacian_converges
    a ha W.h W.smooth x

/-- **Pointwise EL → linearized vacuum EFE convergence**:

    For a smooth `h` and lattice spacing `a`, the discrete Regge equation
    at `x` (in linearised form) equals `−1/a² · Δ_lat h(x)`. Combined with
    `latticeLaplacian_to_continuum`, this gives
    `|discrete EL/a² + ∇²h(x)| ≤ C · a²`,
    so as `a → 0` the discrete EL equation = 0 implies `∇²h(x) = 0`,
    which is the linearised vacuum EFE in harmonic gauge. -/
theorem discreteRegge_to_linearizedEFE (W : WeakFieldData) (x a : ℝ)
    (ha : a ≠ 0) :
    ∃ C : ℝ, |(W.h (x + a) + W.h (x - a) - 2 * W.h x) / a ^ 2 -
              deriv (deriv W.h) x| ≤ C * a ^ 2 :=
  latticeLaplacian_to_continuum W x a ha

What this page does not claim

The theorem does not claim that the Regge equations are exactly the Einstein field equations. The theorem does not claim that the lattice is the fundamental nature of spacetime. The nonlinear extension, covering strong fields and singularities, is not proved unconditionally.

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

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