Geometry Regge Action Second Variation Regge Action Second Variation Input

The second variation input

The Regge action is a discrete version of the Einstein-Hilbert action used in simplicial quantum gravity: it assigns a number to a triangulated 3-dimensional space by summing, over each edge, the product of the edge length and the deficit angle (the amount by which the surrounding triangles fail to lie flat). The second variation of an action is the quadratic term in its Taylor expansion around a stationary point; it controls whether the stationary point is a local minimum, maximum, or saddle. For the Regge action, the natural stationary point is the flat configuration, where all deficit angles vanish.

In the machine-checked library of formal theorems, the declaration ReggeActionSecondVariationInput is a structure: a named package that bundles a hypothesis. It states, as an assumption, that the nonlinear Regge action, restricted to any one-dimensional conformal line through the flat potential, has second derivative at zero equal to the canonical incidence Hessian evaluated on the same line. In plain language, it assumes that the quadratic part of the action's Taylor expansion at the flat configuration is exactly the canonical Hessian that the framework already defines. The declaration does not prove this fact; it packages it as an input for later theorems.

The library also provides a companion structure, ReggeActionCubicRemainderInput, which assumes a local cubic bound on the remainder: there exist positive radius r and constant C such that for all perturbations of norm less than r, the remainder is bounded by C times the cube of the norm. This is the standard estimate needed for a Taylor expansion with a controlled error term. Together, the two inputs supply the analytic facts that a full second-variation theorem would require: the quadratic term is canonical, and the error beyond quadratic is cubic.

In Recognition Science, this declaration belongs to the framework's geometry layer, where the Regge action is a candidate for the discrete gravitational action. The framework models the action as a function on vertex potentials, and the flat configuration is the point where the action is stationary. The declaration is a bridge: it states the analytic property that the nonlinear action is expected to have, so that downstream theorems can be proved conditional on it. The lower-level calculation, a large Cayley-Menger and arccos chain-rule expansion, is not yet materialized in the library; the input structure names the required fact without proving it.

What the declaration does not claim is as important as what it claims. It does not assert that the second variation of the nonlinear Regge action is actually equal to the canonical Hessian; that remains an open target pending the full expansion. It does not assert that the flat configuration is a local minimum, since the Hessian's positive definiteness is a separate matter. And it does not claim that the cubic remainder bound holds for the actual action; that too is an input, not a theorem. The declaration is a scaffold, not a result.

The practical consequence is that the library can proceed with theorems that depend on the second-variation structure before the full analytic calculation is finished. The theorems reggeAction_secondVariation_eq_canonicalHessian and reggeActionRemainder_cubic_bound are proved conditional on the respective inputs: given the input, the conclusion follows. This is a standard modular pattern in formal mathematics: name the assumption, prove what follows from it, and leave the assumption as a target for future work. The reader who wants to know what the framework has established about the Regge action's second variation should look at the input structures and the conditional theorems, not at an unconditional result that does not yet exist.

/-- Named second-variation input for the full nonlinear action. -/
structure ReggeActionSecondVariationInput
    (K : Triangulation3D) (hK : IncidenceConsistent K)
    (_h_flat : FlatConfiguration K hK) where
  canonical_secondVariation : CanonicalHessianSecondVariationAtZero K hK

/-- Phase-D second-variation theorem, conditional on the named nonlinear
second-variation input. -/
theorem reggeAction_secondVariation_eq_canonicalHessian
    (K : Triangulation3D) (hK : IncidenceConsistent K)
    (h_flat : FlatConfiguration K hK)
    (h_second : ReggeActionSecondVariationInput K hK h_flat) :
    CanonicalHessianSecondVariationAtZero K hK :=
  h_second.canonical_secondVariation

/-- Named Taylor-remainder input.  This is the exact place where Mathlib's
multivariate Taylor theorem, or an `IsBigO` fallback, should be connected. -/
structure ReggeActionCubicRemainderInput
    (K : Triangulation3D) (hK : IncidenceConsistent K)
    (_h_flat : FlatConfiguration K hK) where
  cubic_bound : LocalCubicRemainderBound K hK

/-- Phase-E cubic remainder theorem, conditional on the named Taylor input. -/
theorem reggeActionRemainder_cubic_bound
    (K : Triangulation3D) (hK : IncidenceConsistent K)
    (h_flat : FlatConfiguration K hK)
    (h_cubic : ReggeActionCubicRemainderInput K hK h_flat) :
    LocalCubicRemainderBound K hK :=
  h_cubic.cubic_bound

What this page does not claim

The declaration does not prove that the nonlinear Regge action's second variation equals the canonical Hessian; it only names that equality as an input. The declaration does not assert that the flat configuration is a local minimum of the action. The declaration does not establish the cubic remainder bound for the actual action; that bound is a separate named input.

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

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