Gravity Analysis Regge Ttsymbol Preflight Conformal Tet Sq Edges Eq Typed Field

The typed field identity

In the Regge calculus approach to general relativity, spacetime is built from flat tetrahedra glued together, and the gravitational action is a sum over edges of a length factor times a deficit angle, a measure of how much geometry fails to be flat around that edge. The declaration conformalTetSqEdges_eq_typedField is a formal statement in the framework's machine-checked library of formal theorems. It establishes, as a proved theorem, that the squared edge lengths of a tetrahedron, as read off from a certain typed field, agree with the same lengths computed by the standard conformal ansatz. In plain terms, it confirms that two different ways of writing down the edge lengths, one from the field directly and one from the conformal recipe, give the same numbers.

The identity is part of a larger preflight check for a program that studies gravitational waves on a lattice. The framework defines a true nonlinear Regge action on a periodic torus, where each edge has a squared length, and the action is the sum over edges of the square root of that length times the deficit angle. The conformal ansatz, by contrast, writes each edge length as the flat length times an exponential factor built from vertex functions. The theorem conformalTetSqEdges_eq_typedField says that for any vertex function, the squared edge lengths produced by the conformal recipe are exactly the same as the lengths obtained by evaluating the true action's field at those edges. This is a real identity between two independently defined expressions, not a tautology.

What the declaration does not claim is equally important. It does not prove that the true action equals the conformal action for all fields; it only proves equality on the restricted family of conformal edge fields. It does not prove that the conformal ansatz is a good approximation to the true action, nor that the quadratic model derived from it is correct. The theorem says nothing about the value of the action, only about the edge lengths. It is a structural identity, a bridge between two formalisms, not a physical result about gravity.

The practical consequence is that when the framework later analyzes the conformal action, it can be sure it is studying the true action on a specific, well-defined subset of edge fields. This makes the conformal model a genuine restriction of the full theory, not a separate or ad hoc construction. The identity is a foundation stone for the preflight program, allowing subsequent work to compare the conformal model against the full nonlinear action on equal footing.

/-- The six local squared-edge coordinates of the tetrahedron
`(cell, tet)`, read off an edge field through the canonical local-edge
tables. -/
def tetSqEdgesOfField (ℓ : EdgeField N) (cellTet : PeriodicTet N N N) :
    Geometry.CayleyMengerPolynomial.SqEdges :=
  fun f => ℓ (localEdgeOf cellTet.1 cellTet.2 f)

/-- The conformal edge field induced by a vertex potential `u`: edge `e`
carries `ℓ²_flat(e) · exp(u(e₊) + u(e₋))`.  This is EXACTLY the family of
edge fields the frozen wave analyzed (through
`ReggeActionConcrete.conformalLocalSqEdge`). -/
def typedConformalEdgeField (u : Vertex N N N → ℝ) : EdgeField N :=
  fun e =>
    periodicDispSqEdge e.disp *
      Real.exp (u e.endpoints.1 + u e.endpoints.2)

/-- THE TRUE NONLINEAR 3D REGGE ACTION on the side-`N` periodic Freudenthal
torus, as a function of the edge squared-length field:
`S(ℓ) = Σ_e √(ℓ_e) · deficit_e`.  This is the object whose TT Bloch symbol
the C10 probe measured. -/
def trueReggeAction (ℓ : EdgeField N) : ℝ :=
  ∑ e : PeriodicEdge N N N, Real.sqrt (ℓ e) * deficitOfField N ℓ e

What this page does not claim

The theorem does not prove that the conformal action equals the true action for all edge fields. The theorem does not prove that the conformal ansatz is a good approximation to the true action. The theorem says nothing about the value of the action, only about edge lengths.

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

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