Geometry Regge Action Second Variation Regge Action Remainder Second Variation Z

The flat-space remainder

Regge calculus is a discrete approach to general relativity where spacetime is approximated by a network of flat simplices, typically tetrahedra in three dimensions. The Regge action is the sum over all edges of the edge length times a deficit angle, a measure of how much the geometry bends at that edge. In the Recognition Science framework, this action is studied as a function of the vertex positions, which can be varied to find the geometry that minimizes the action, analogous to the Einstein field equations in the continuum.

The declaration reggeActionRemainder_secondVariation_zero in the machine-checked library of formal theorems states a precise property of the remainder term in a Taylor expansion of the nonlinear Regge action. When the action is expanded around a flat configuration, the remainder is what is left after subtracting the linear and quadratic terms. The theorem says that the second derivative of this remainder, taken along any conformal line through the flat configuration, is exactly zero at that flat point. In plainer terms, the remainder has no quadratic curvature contribution at flat space; the curvature information is fully captured by the canonical Hessian, the matrix of second derivatives of the action.

The theorem is conditional. It does not prove that the remainder itself is zero, nor that its second variation vanishes everywhere. It requires an input structure, ReggeActionRemainderSecondVariationInput, which packages the assumption that the second variation is zero. The declaration simply extracts that assumption as a theorem. The real analytic work, the large Cayley-Menger and arccos chain-rule expansion that would establish this property from first principles, is not yet materialized in the library. The theorem is a formal bridge, not the final calculation.

What the declaration does not claim is equally important. It does not claim the remainder is identically zero, which would make the action exactly quadratic. It does not claim the second variation vanishes for arbitrary directions, only along the one-dimensional conformal lines defined by scaling a single potential. It does not claim the flat configuration is a local minimum of the action, which would require the Hessian to be positive definite. And it does not claim the remainder has no cubic or higher-order terms; a separate, still-conditional theorem reggeActionRemainder_cubic_bound bounds the remainder by a constant times the cube of the perturbation norm, but only near the flat point and only if a Taylor bound input is supplied.

The practical significance is that the framework can now state, in a formally checkable way, that the second-order behavior of the nonlinear Regge action at flat space is governed entirely by the canonical Hessian. This is a necessary step toward proving that the discrete action behaves like its continuum counterpart near flat geometries. But the proof of the underlying analytic fact remains an open target, and the theorem should be read as a scaffold for that future work, not as the work itself.

theorem reggeActionRemainder_secondVariation_zero
    (K : Triangulation3D) (hK : IncidenceConsistent K)
    (h_rem : ReggeActionRemainderSecondVariationInput K hK) :
    CanonicalRemainderSecondVariationZero K hK :=
  h_rem.remainder_secondVariation_zero

structure ReggeActionRemainderSecondVariationInput
    (K : Triangulation3D) (hK : IncidenceConsistent K) where
  remainder_secondVariation_zero :
    CanonicalRemainderSecondVariationZero K hK

What this page does not claim

The remainder term is identically zero, making the action exactly quadratic. The second variation of the remainder vanishes for arbitrary directions, not just conformal lines. The flat configuration is a local minimum of the action. The theorem provides the analytic proof of the second variation property; that proof remains an open target.

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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Derived articles

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