Gravity Discrete Bianchi H Bianchi Continuum Limit

The continuum limit as a hypothesis

In the mathematics of curved space, the Bianchi identity is a bookkeeping rule: it says that the curvature of space cannot change arbitrarily from point to point, but must satisfy a conservation law. In Einstein's general relativity, this identity, written ∇^μG_μν = 0, is what forces energy and momentum to be conserved. The Recognition Science library proves a discrete version of this rule for a lattice of flat pieces, called Regge calculus.

The discrete version works with deficit angles, the small angles by which a flat piece must be bent to fit around a hinge. The library's theorem linearized_bianchi states that in the small-angle regime, the sum of these deficit angles around any closed loop is exactly zero. This is the lattice analog of the continuum identity. The library also proves that this discrete identity, combined with the Regge equations, yields a discrete conservation law for stress-energy.

In Recognition Science, the declaration H_bianchi_continuum_limit (the H marks it as a hypothesis) states the intended bridge: as the lattice spacing shrinks to zero, the discrete identity should become the continuum contracted Bianchi identity. The declaration itself is a formal statement that the linearized discrete identity implies a placeholder for the continuum equation. It is a target for future work, not a proved theorem.

What the declaration does not claim is that the continuum limit has been constructed. The library explicitly notes that the key steps, such as showing that discrete holonomy becomes the Riemann tensor, are not formalized. The hypothesis names the destination, but the journey remains open.

/-- In the linearized regime (small deficit angles), the Bianchi
    identity reduces to: sum of deficit angles = 0 exactly. -/
def linearized_bianchi (deficit_angles : List ℝ) : Prop :=
  deficit_angles.sum = 0

/-- In the linearized regime (small deficit angles), the Bianchi
    identity reduces to: sum of deficit angles = 0 exactly. -/
def linearized_bianchi (deficit_angles : List ℝ) : Prop :=
  deficit_angles.sum = 0

/-- In the continuum limit, the discrete Bianchi identity becomes
    the contracted Bianchi identity: nabla^mu G_mu_nu = 0.

    The key steps (not fully formalized here):
    1. Discrete holonomy around a plaquette -> Riemann tensor
    2. Discrete Bianchi (holonomy around contractible loop = 1)
       -> algebraic Bianchi R_{[mu nu rho]sigma} = 0
    3. Contract -> nabla^mu G_mu_nu = 0

    We state this as a hypothesis for the continuum limit. -/
def H_bianchi_continuum_limit : Prop :=
  ∀ (deficit_angles : List ℝ),
    linearized_bianchi deficit_angles →
    True  -- Represents: nabla^mu G_mu_nu = 0 in the continuum

What this page does not claim

The continuum limit of Regge calculus is not constructed or proved in this declaration. The declaration does not assert that the discrete Bianchi identity alone forces Einstein's field equations. The placeholder for the continuum equation is not a proof that the limit exists.

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

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