Gravity Analysis Regge Ttbloch Interface Audit Raw Cell Stencil Term

The raw stencil term

A stencil is a fixed pattern of weights used to approximate a derivative on a discrete grid. In the Recognition Science framework's audit of Regge calculus, a discrete approach to general relativity built from flat tetrahedra, the declaration RawCellStencilTerm defines one raw term of such a stencil. The object is a triple: it names one of six tetrahedron types in a periodic cell, and an ordered pair of slots among six possible positions on that tetrahedron. The concrete finite sum that uses these terms has 216 summands per periodic cell type, the product of 6 tetrahedra, 6 left slots, and 6 right slots.

The term itself is a product of three factors. The first is a velocity associated with a plane wave on the tetrahedron, divided by twice the square root of a squared edge length. The second is the same plane-wave velocity evaluated at the right slot. The third is a flat-angle Jacobian, a derivative of the angle at a hinge with respect to edge lengths. The framework's machine-checked library of formal theorems proves one key identity about this object: the reduced second variation of the Regge action, written as canonicalFiniteH, equals the full 216-term sum built from these terms. The proof distributes a finite inner sum and follows the sign convention of the live A2 theorem, where the reduced second variation is the negative Schlaefli-reduced contraction.

What this term does not claim is as important as what it proves. The full rational bucket aggregation, which would group the 216 terms into rational weights and show that a constant block vanishes, is not shown here. The same-day symbolic diagnostic found that the stencil-only constant block does not vanish. The framework marks Gate A2-full, the aggregation over all buckets, as OPEN, and Gate A3, the hinge-aware zero-mode cancellation, as OPEN. The assembled symbol and moment definitions appear in the file, but the theorem that the raw stencil alone produces the correct continuum limit is not proved. The ContinuumLimit engine must use a different route, the cosine two-jet, after the hinge and diagonal constant block is formally connected.

The practical consequence is a narrow but real one. The framework has a precise, machine-checked handle on one piece of the Regge TT Bloch analysis: the raw stencil term is correctly defined and correctly equals the reduced second variation. That gives future work a solid foundation to build on. But the harder problem, showing that the full bucket aggregation cancels the constant block and yields the right continuum limit, remains open, and the framework says so explicitly.

/-- One raw term of the cell stencil: tetrahedron type and ordered slot pair.
The concrete finite sum below has `6 x 6 x 6 = 216` summands per periodic cell
type. -/
structure RawCellStencilTerm where
  tet : Fin 6
  left : Fin 6
  right : Fin 6

/-- One raw term of the cell stencil: tetrahedron type and ordered slot pair.
The concrete finite sum below has `6 x 6 x 6 = 216` summands per periodic cell
type. -/
structure RawCellStencilTerm where
  tet : Fin 6
  left : Fin 6
  right : Fin 6

/-- Gate A1, honest part: the A2 reduced finite value equals the literal
triple raw stencil.  The proof is finite distribution of the inner
`flatSlotAngleDeriv` sum, not a definitional alias between the two sides.
The panel's `hN` premise is not needed: the incidence identity holds for
every `N` with `[NeZero N]`, which is a strictly stronger statement. -/
theorem a2_reduced_eq_rawCellStencil (N : ℕ) [NeZero N]
    (E : Fin 3 → Fin 3 → ℝ) (m : Fin 3 → ℤ) :
    canonicalFiniteH N E m = rawCellStencil N E m := by
  unfold canonicalFiniteH rawCellStencil rawCellStencilTerm
  congr 1
  congr 1
  refine Finset.sum_congr rfl fun τ _ => ?_
  refine Finset.sum_congr rfl fun f _ => ?_
  unfold flatSlotSqrtDeriv flatSlotAngleDeriv
  rw [Finset.mul_sum]
  refine Finset.sum_congr rfl fun g _ => ?_
  ring_nf

What this page does not claim

The raw stencil term alone does not prove the full bucket aggregation or the continuum limit. The theorem does not establish the sign convention for the assembled symbol or moment definitions. The raw stencil term does not claim to be the complete Regge TT Bloch analysis.

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

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