Gravity Unified Lattice Manifold Correspondence

The lattice bridge to curved space

General relativity describes gravity as the curvature of spacetime: a smooth, continuous fabric that bends around mass and energy. But a competing picture has existed since the 1960s, when physicists Tullio Regge and John Wheeler showed that curved space can be approximated by a lattice, a discrete grid of straight edges meeting at hinges. The idea is that a curved surface, viewed closely enough, looks like a patchwork of flat triangles; the curvature lives in the angles where the pieces meet. Regge's formulation replaces Einstein's smooth equations with a set of algebraic rules for the edge lengths, and the approximation improves as the grid spacing shrinks.

The correspondence in question is the claim that this lattice picture, in the limit of infinitely fine spacing, recovers the full smooth theory. The classical result, proved by Regge and refined by others through the 1970s and 1980s, states that the lattice action, a sum over hinges, converges to the Einstein-Hilbert action, and the discrete equations of motion converge to the Einstein field equations. The convergence is quadratic in the lattice spacing: if you halve the grid size, the error drops by a factor of four. This is the standard bridge between discrete and continuous gravity, and it underpins numerical relativity, where computers simulate black holes and gravitational waves on grids.

In Recognition Science, the framework models this bridge as a theorem about its own foundations. The framework's machine-checked library of formal theorems packages the correspondence into a single certificate, ledger, a discrete record of events, in this case the lattice itself. The certificate takes a smooth metric perturbation on a finite box, meaning a small deviation from flat spacetime, and constructs a sequence of cubic lattices with edge lengths prescribed by the metric. It then proves two convergence results: the lattice action differs from the smooth action by a term of order the spacing squared, and the discrete Regge equations converge pointwise to the linearized vacuum Einstein equations, which describe weak gravitational fields.

The framework's contribution is not new geometry but a packaging of existing proofs. The certificate composes earlier formal results: the edge-length rule from Regge calculus, the quadratic action convergence, the identification of the lattice Laplacian with the continuum Laplacian, and a coupling identity. That identity is striking: the Regge coupling constant equals the Einstein coupling constant, both equal to 8 times the fifth power of the golden ratio, a number that emerges from the framework's forcing chain rather than being fitted. The theorem is unconditional in the weak-field regime, where the perturbation is small, covering solar-system tests, galaxy rotation, gravitational-wave strain, and cosmological perturbations. The nonlinear extension, for strong fields like black-hole interiors, is provided conditionally on an external classical result, the Cheeger-Müller-Schrader convergence theorem, which the library takes as a labelled axiom.

What this means in practice is that the framework's discrete starting point does not abandon general relativity; it reproduces it in the limit. The lattice is not a metaphor but a mathematical object whose behavior provably matches the smooth theory where the smooth theory is well tested. The certificate gives a single theorem one can cite for the claim that weak-field gravity emerges from a discrete grid with no free parameters. The nonlinear case remains conditional, but the linearized regime, which covers almost all observed gravitational physics, is closed.

/-- Per-bond action deviation, bounded by `|ε|⁴/20`. This is the per-bond
    statement underlying the `O(a²)` total convergence rate. -/
theorem perBondActionDeviation (ε : ℝ) (hε : |ε| < 1) :
    |J_log ε - ε ^ 2 / 2| ≤ |ε| ^ 4 / 20 :=
  CubicReggeProof.cubic_regge_convergence_cert.action_quadratic ε hε

/-- **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 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

What this page does not claim

The module does not prove the nonlinear convergence without the Cheeger-Müller-Schrader axiom. The module does not derive the value of the gravitational constant from first principles; it identifies the coupling through the framework's constants.

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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