Geometry Regge Rigorous Foundation Regge Rigorous Foundation Cert

A certified first step

Regge calculus is a way of doing general relativity without coordinates: chop spacetime into flat tetrahedral pieces and let the edge lengths carry the geometry. The component theorem `M_ij = -area(f_ij)` is the key identity that connects this discrete picture back to the continuous Einstein equations, but proving it rigorously requires a solid foundation. The declaration `reggeRigorousFoundationCert` is a machine-checked certificate that such a foundation now exists for the purely geometric part of the problem.

The certificate bundles eight proven facts about the Cayley-Menger polynomial, the degree-3 polynomial in the six squared edge lengths whose positive values describe the volume of a tetrahedron. It proves the polynomial is fully explicit and smooth everywhere, that it equals 4 for the unit regular tetrahedron and 8 for the right-angle unit tetrahedron, and that a polynomial Taylor identity holds at any base point. It also proves the conformal edge map, which stretches edges according to vertex potentials, is smooth, and that composing it with the Cayley-Menger polynomial remains smooth. These are the analytic tools needed to take derivatives of volume with respect to edge lengths.

What the certificate does not do is prove the component theorem itself. The full result requires three external classical hypotheses: Schläfli's identity (Regge 1961), the dihedral cosine formula via Cayley-Menger cofactors (Berger 1987), and smoothness of the dihedral angle on the realisability cone. The certificate records these as named assumptions, imported from the standard Regge calculus literature, not as consequences of the framework. This is the honest formal-math practice: the geometry is nailed down, but the physics that connects it to the Einstein equations remains a stated, multi-month formalisation project.

In Recognition Science, this certificate is the first rung of a ladder. It establishes that the framework's machine-checked library can handle the concrete geometry of Regge calculus, and it names precisely what must be added to climb to the component theorem. The payoff for the reader is a clear map: the smooth polynomial layer is done, the dihedral and Schläfli layers are the next target, and the component theorem itself is a defined goal, not a claimed result.

theorem reggeRigorousFoundationCert : ReggeRigorousFoundationCert where
  cm3_polynomial_explicit := fun a => by unfold cm3; ring
  cm3_smooth := cm3_contDiff
  cm3_regular := cm3_regular_unit
  cm3_rightAngle := cm3_rightAngle_unit
  cm3_taylor_identity := cm3_taylor
  cm3_update_form := cm3_update_polyform
  conformal_smooth := conformalSqEdge_contDiff
  cm3_conformal_smooth := cm3_conformal_contDiff

theorem reggeRigorousFoundationCert : ReggeRigorousFoundationCert where
  cm3_polynomial_explicit := fun a => by unfold cm3; ring
  cm3_smooth := cm3_contDiff
  cm3_regular := cm3_regular_unit
  cm3_rightAngle := cm3_rightAngle_unit
  cm3_taylor_identity := cm3_taylor
  cm3_update_form := cm3_update_polyform
  conformal_smooth := conformalSqEdge_contDiff
  cm3_conformal_smooth := cm3_conformal_contDiff

theorem reggeRigorousFoundationCert : ReggeRigorousFoundationCert where
  cm3_polynomial_explicit := fun a => by unfold cm3; ring
  cm3_smooth := cm3_contDiff
  cm3_regular := cm3_regular_unit
  cm3_rightAngle := cm3_rightAngle_unit
  cm3_taylor_identity := cm3_taylor
  cm3_update_form := cm3_update_polyform
  conformal_smooth := conformalSqEdge_contDiff
  cm3_conformal_smooth := cm3_conformal_contDiff

/-- Schläfli identity (3D, Euclidean tetrahedral form), stated as a named
external classical hypothesis.  See module-doc references. -/
def Schlaefli3DIdentity : Prop :=
  ∀ (T : NonDegenerateTet) (dihedralDeriv : Fin 6 → Fin 6 → ℝ)
    (_volumeDeriv : Fin 6 → ℝ),
    -- `dihedralDeriv e e'` represents `∂θ_e^{(T)} / ∂L_{e'}` at `T`.
    -- `_volumeDeriv e` represents `∂V / ∂L_e` at `T` and is auxiliary;
    -- Euclidean Schläfli itself is the vanishing of the angle term.
    -- Schläfli says: for every edge e', the sum of L_e · ∂θ_e/∂L_{e'} over e
    -- vanishes.
    (∀ e' : Fin 6,
      (∑ e : Fin 6, Real.sqrt (T.sqEdge e) * dihedralDeriv e e')
        = 0)

What this page does not claim

The certificate does not prove the component theorem M_ij = -area(f_ij). The certificate does not derive Schläfli's identity from the Recognition Science framework. The certificate does not establish the smoothness of the dihedral angle on the realisability cone.

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

A page whose claims cannot be reproduced this way does not ship. In production, every anchor links to the exact declaration in the public source release, and this block carries the build receipt for the page itself.

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