Geometry Discrete Bianchi Schlafli Regge Data Inhabited

The flat witness

In the mathematics of discrete gravity, Regge calculus builds spacetimes from flat triangular pieces, like a curved surface approximated by flat tiles. A central identity in this setting is the Schläfli identity, a relation between how the area of a triangle changes when its vertices move. In the Recognition Science framework, the machine-checked library of formal theorems contains a declaration named SchlafliReggeData_inhabited. This declaration establishes that there exists at least one configuration of such discrete geometric data, called Regge data, that satisfies the Schläfli identity at every vertex.

The construction works by giving a specific example: a flat substrate where all deficit angles, bone areas, and gradients are zero. In this trivial case, the Schläfli identity holds by simple arithmetic, since every term in the sum is zero. The declaration establishes that the space of valid Schläfli Regge data is not empty. This is a non-vacuous witness, meaning the identity is not just a formal statement with no examples behind it. The proof is complete in the machine-checked library, with no gaps and no framework-specific axioms.

What this declaration does not claim is more important than what it does. It does not establish the general Schläfli identity for arbitrary triangulations. That would require a full proof of the simplex volume-area relation and its kinematic consequences, which remains future work. The declaration only shows that one particular flat example satisfies the identity. It does not show that the identity holds for curved or non-trivial geometries. The general case, where deficit angles and areas vary, is an open target within the framework.

The practical consequence is a structural scaffold. The framework can state the contracted discrete Bianchi identity, which is equivalent to the Schläfli identity, and certify that this statement is not vacuous. This is a necessary first step, but it is not the full theorem. The honest summary is that the framework has a proof of existence for a flat case, and a clear statement of what remains to be established for the general case.

/-- The hypothesis space of Schläfli-satisfying Regge triangulations is
nonempty (witnessed by `flatSchlafliReggeData`). -/
theorem SchlafliReggeData_inhabited (V B : Type) [Fintype B] :
    Nonempty (SchlafliReggeData V B) :=
  ⟨flatSchlafliReggeData V B⟩

/-- The flat Regge data satisfies the Schläfli identity at every
vertex by construction (zero deficits → zero sum). -/
theorem flatReggeData_schlafli {V B : Type} [Fintype B] :
    ∀ v : V,
      SchlafliIdentityAtVertex (flatReggeData V B) v := by
  intro v i
  simp [flatReggeData]

What this page does not claim

The general Schläfli identity holds for all triangulations. The discrete Bianchi identity is proved for curved or non-trivial geometries. The declaration provides a physical interpretation of the flat substrate as a spacetime.

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/DiscreteBianchi.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.

Derived articles

This page is generated by a question-recursion engine: the questions its answers raise become the next pages. The current agenda, with open targets marked red: