Gravity Unified Lattice Manifold Correspondence Regge Coupling Eq Einstein Coupl

The coupling identity

In general relativity, the Einstein field equations relate the curvature of spacetime to its energy and matter content. The equations contain a constant, the Einstein coupling κ, which sets how strongly matter bends spacetime. In the discrete approach called Regge calculus, spacetime is replaced by a lattice of flat pieces joined at edges; the Regge action, built from edge lengths and angles, has its own coupling constant, the Regge coupling. The theorem reggeCoupling_eq_einsteinCoupling proves that these two constants are equal: the Regge coupling equals 8φ⁵, and the Einstein coupling also equals 8φ⁵, where φ is the golden ratio. In plain language, the strength of gravity on the lattice is exactly the strength of gravity in the smooth continuum, with no free parameter left to adjust.

The equality is not a numerical coincidence. It is a proved statement in the framework's machine-checked library of formal theorems, derived from the same forcing chain that fixes the golden ratio as the fundamental scaling constant. The theorem appears in the module UnifiedLatticeManifoldCorrespondence, which packages a broader correspondence: given a smooth, weakly perturbed metric, one can build a sequence of cubic lattices whose edge lengths follow the metric, and as the lattice spacing shrinks to zero, the discrete Regge action converges to the Einstein–Hilbert action, and the discrete equations converge to the linearized vacuum Einstein field equations. The coupling identity is one component of that package, and it holds unconditionally in the linearized regime, where the metric perturbation is small.

What the theorem does not claim is just as important. It does not prove that the nonlinear regime, where gravity is strong, obeys the same convergence; that extension is a separate certificate, conditional on an external result from 1984. It does not claim that the coupling constant is measured or fitted; the value 8φ⁵ emerges from the framework's internal derivation, not from experiment. And it does not claim that the correspondence itself is a physical model of quantum gravity; the theorem is a statement about the mathematics of lattice approximations to general relativity, not a claim about the actual structure of spacetime at the Planck scale.

/-- The Regge coupling on the cubic lattice equals the Einstein coupling. -/
theorem reggeCoupling_eq_einsteinCoupling :
    ReggeCalculus.rs_kappa = 8 * phi ^ 5 :=
  ReggeCalculus.rs_kappa_value

/-- The Einstein coupling closed form `κ_Einstein = 8φ⁵`, restated for
    convenience and to make the chain self-contained. -/
theorem einsteinCoupling_closed_form :
    Constants.kappa_einstein = 8 * phi ^ (5 : ℝ) :=
  Constants.kappa_einstein_eq

/-- **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

/-- The nonlinear certificate is provable from the three exposed convergence hypotheses
    plus the (already-proved) coupling identity. The hypotheses are
    intentionally exposed as inputs, mirroring the existing architecture. -/
theorem nonlinearUnified_of_cms
    (h_action  : NonlinearConvergence.regge_to_eh_convergence_axiom)
    (h_ricci   : NonlinearConvergence.regge_ricci_convergence_axiom)
    (h_riemann : NonlinearConvergence.regge_riemann_convergence_axiom) :
    NonlinearUnifiedCert where
  cms_action := h_action
  cms_ricci := h_ricci
  cms_riemann := h_riemann
  coupling_identity := by
    have h1 : ReggeCalculus.rs_kappa = 8 * phi ^ 5 :=
      ReggeCalculus.rs_kappa_value
    have h2 : Constants.kappa_einstein = 8 * phi ^ (5 : ℝ) :=
      Constants.kappa_einstein_eq
    have h3 : phi ^ (5 : ℝ) = phi ^ (5 : ℕ) := by
      rw [show (5 : ℝ) = ((5 : ℕ) : ℝ) by norm_num, Real.rpow_natCast]
    rw [h1, h2, h3]
  coupling_closed_form :=
    ⟨ReggeCalculus.rs_kappa_value, Constants.kappa_einstein_eq⟩

What this page does not claim

The theorem does not prove the nonlinear convergence unconditionally. The theorem does not claim the coupling constant is measured from experiment. The theorem does not assert that the lattice is the actual physical structure 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/UnifiedLatticeManifoldCorrespondence.lean
expected axiom basis: [propext, Classical.choice, Quot.sound] (the Lean kernel's standard three; no RS-specific axioms)

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