Gravity Cubic Regge Proof Cubic Regge Convergence Cert

The convergence certificate

In numerical physics, a lattice is a discrete grid of points used to approximate a continuous space. The cubic lattice is the simplest kind: points arranged in a regular three-dimensional grid, like the corners of stacked sugar cubes. A lattice approximation is judged by whether its behavior converges to the continuous behavior as the grid spacing shrinks. The declaration cubic_regge_convergence_cert is a machine-checked certificate, a formal object in the Recognition Science library, that this convergence holds for a specific cost function on a cubic lattice.

The cost function in question is the recognition cost, a quantity the framework defines as J(x) = (x + 1/x)/2 - 1. On a lattice, this cost is assigned to each bond, the link between neighboring grid points. The certificate proves that as the lattice spacing a tends to zero, the discrete equations of motion approach the continuous equations. The key estimate is that the difference between the full cost and its quadratic approximation is bounded by a term proportional to the fourth power of the spacing, so the relative error vanishes as a squared. The Euler-Lagrange equations of the discrete system linearize to the lattice Laplacian, which in turn converges to the continuous Laplacian at order a squared. The certificate bundles these steps into a single structure, ProvedConvergenceChain, and then into CubicReggeConvergenceCert.

The proof is direct and specific. It does not invoke the general Cheeger-Müller-Schrader theorem for arbitrary simplicial complexes. Instead, it exploits the special features of the cubic lattice: the cost function is known, the Taylor expansion is fixed, and the hyperbolic sine that appears in the Euler-Lagrange equation has derivative 1 at zero. The certificate also includes the closed form for the coupling constant, kappa = 8 times the fifth power of the golden ratio, and the fact that the flat lattice has zero deficit angle. These are the ingredients that make the convergence proof tractable.

What the certificate does not claim is broader significance. It establishes convergence of the variational principle to the linearized Einstein field equations at order a squared on a cubic lattice in any dimension D. It does not prove convergence for arbitrary simplicial complexes, nor does it handle the full nonlinear Einstein equations. It does not claim that the lattice itself is a physical model of spacetime; it is a mathematical approximation scheme. The certificate is a theorem about a specific discrete system, not a statement about the universe.

/-- The J-cost action on each bond approximates the quadratic action
    with error ≤ |ε|⁴/20. This is J_log_quadratic_approx. -/
theorem action_per_bond (ε : ℝ) (hε : |ε| < 1) :
    |J_log ε - ε ^ 2 / 2| ≤ |ε| ^ 4 / 20 :=
  J_log_quadratic_approx ε hε

/-- The linearized EL equation equals minus the lattice Laplacian. -/
theorem linearized_el_eq_neg_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 := by
  linarith [linearized_el_plus_laplacian_zero f x]

/-- The lattice Laplacian (scaled by 1/a²) converges to the continuous
    Laplacian ∇² at O(a²). This is the standard finite-difference result,
    already proved in ContinuumLimit.lean. -/
theorem laplacian_continuum_limit (f : ℝ → ℝ) (x a : ℝ)
    (ha : a ≠ 0) (hf : ContDiff ℝ 4 f) :
    ∃ C : ℝ,
      |(f (x + a) + f (x - a) - 2 * f x) / a ^ 2 -
        deriv (deriv f) x| ≤ C * a ^ 2 := by
  obtain ⟨C, _hC_nn, hC⟩ := continuum_limit_second_order f x a ha hf
  exact ⟨C, hC⟩

/-- **THE CERTIFICATE**: all proved. Zero axioms. Zero sorry. -/
theorem cubic_regge_convergence_cert : CubicReggeConvergenceCert where
  action_quadratic := J_log_quadratic_approx
  action_symmetric := J_log_symmetric
  action_vacuum := J_log_zero
  el_is_laplacian := fun D f x => linearized_el_eq_neg_laplacian f x
  flat_solution := fun D c x => flat_satisfies_el c x
  linearization_coeff := sinh_deriv_at_zero
  laplacian_converges := fun a ha f hf x => by
    obtain ⟨C, _hC_nn, hC⟩ := continuum_limit_second_order f x a ha hf
    exact ⟨C, hC⟩
  kappa_derived := ZeroParameterGravity.kappa_rs_closed_form
  kappa_positive := ZeroParameterGravity.kappa_pos
  flat_deficit := cubic_flat_deficit
  shape_optimal := cubic_shape_bound_positive
  second_order := fun _ _ _ => by ring
  error_vanishes := NonlinearConvergence.error_vanishes
  quartic_fixed := quartic_coeff
  sextic_fixed := sextic_coeff
  relative_rate := fun M a ha ha1 hM =>
    relative_convergence_rate M a ha ha1 hM

/-- In the gravitational sector, the lattice field f encodes the
    metric perturbation h_μν (in harmonic/Lorenz gauge).

    The EL equation lattice_laplacian(h) = 0 in the continuum limit
    gives ∇²h = 0, which IS the linearized vacuum EFE in harmonic gauge.

    The sourced case: ∇²h = −2κT with κ = 8φ⁵ (derived). -/
theorem kappa_derived : ZeroParameterGravity.kappa_rs = 8 * phi ^ 5 :=
  ZeroParameterGravity.kappa_rs_closed_form

What this page does not claim

The certificate does not prove convergence for arbitrary simplicial complexes with varying mesh quality. The certificate does not handle the full nonlinear Einstein field equations. The certificate does not claim the cubic lattice is a physical model of spacetime.

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

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