Gravity Analysis Regge Ttgate Bbridge Core Core Block0 Eq

The block identity

In the framework's work on Regge calculus, a discrete approach to general relativity, a central calculation involves a sum of 216 moment terms. The declaration coreBlock0_eq is a theorem that establishes one piece of that sum: the terms where the first index is fixed at 0 equal a specific precomputed block called tetBlock0. This equality holds for any real values of the three parameters s2, s3, and p, and identically for all inputs E and x. The proof is a direct algebraic expansion, using the ring tactic to verify the polynomial identity.

The theorem is one of six similar block equalities, each covering a different first index. Together, they support the core identity of the module: the entire 216-term raw moment sum equals the sum of the six blocks. This decomposition is a structural step, not a physical claim. It does not assert anything about the meaning of the moments, the validity of Regge calculus, or the behavior of gravity. It only certifies that a particular algebraic transcription is correct, within the framework's machine-checked library of formal theorems.

The declaration is marked private, meaning it is an internal lemma used to build the larger result. Its role is to break a large computation into manageable pieces, each verified independently. The proof does not rely on any unproven assumptions or external axioms; it is a closed algebraic verification. This is a bookkeeping fact about the framework's internal mathematics, not a statement about the physical world.

private theorem coreBlock0_eq (E : Fin 3 → Fin 3 → ℝ) (x : Fin 3 → ℝ)
    (s2 s3 p : ℝ) :
    (∑ f : Fin 6, ∑ g : Fin 6, coreTripleTerm E x (0, f, g)) =
      tetBlock0 (E 0 0) (E 0 1) (E 0 2) (E 1 0) (E 1 1) (E 1 2)
        (E 2 0) (E 2 1) (E 2 2) (x 0) (x 1) (x 2) s2 s3 p := by
  rw [tetBlock0_eq]
  simp only [Fin.sum_univ_six, coreTripleTerm, coreWeight,
    corePolEdgeCoeff, slotDispCore, slotMidTwice, Fin.sum_univ_three]
  push_cast
  ring

private theorem coreBlock0_eq (E : Fin 3 → Fin 3 → ℝ) (x : Fin 3 → ℝ)
    (s2 s3 p : ℝ) :
    (∑ f : Fin 6, ∑ g : Fin 6, coreTripleTerm E x (0, f, g)) =
      tetBlock0 (E 0 0) (E 0 1) (E 0 2) (E 1 0) (E 1 1) (E 1 2)
        (E 2 0) (E 2 1) (E 2 2) (x 0) (x 1) (x 2) s2 s3 p := by
  rw [tetBlock0_eq]
  simp only [Fin.sum_univ_six, coreTripleTerm, coreWeight,
    corePolEdgeCoeff, slotDispCore, slotMidTwice, Fin.sum_univ_three]
  push_cast
  ring

private theorem coreBlock0_eq (E : Fin 3 → Fin 3 → ℝ) (x : Fin 3 → ℝ)
    (s2 s3 p : ℝ) :
    (∑ f : Fin 6, ∑ g : Fin 6, coreTripleTerm E x (0, f, g)) =
      tetBlock0 (E 0 0) (E 0 1) (E 0 2) (E 1 0) (E 1 1) (E 1 2)
        (E 2 0) (E 2 1) (E 2 2) (x 0) (x 1) (x 2) s2 s3 p := by
  rw [tetBlock0_eq]
  simp only [Fin.sum_univ_six, coreTripleTerm, coreWeight,
    corePolEdgeCoeff, slotDispCore, slotMidTwice, Fin.sum_univ_three]
  push_cast
  ring

What this page does not claim

The theorem does not assert any physical law about gravity or Regge calculus. The theorem does not assign meaning to the moments or the blocks. The theorem does not depend on or invoke the spike's TT continuum certificate.

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/ReggeTTGateBBridgeCore.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

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