Gravity Analysis Regge Ttgate Bbridge Core Core Block1 Eq

A bridge of polynomial sums

In mathematics, two expressions that look completely different can turn out to be equal after careful rearrangement. The declaration coreBlock1_eq is a machine-checked proof of exactly this kind of identity. It shows that a certain sum of 36 terms, each built from a weight, a displacement, and a phase, equals a single polynomial expression called tetBlock1. The proof works for any choice of the real numbers s2, s3, and p, and for any real 3 by 3 matrix E and any real 3-dimensional vector x.

The identity is part of a larger project in the Recognition Science framework, which studies how physical structure might arise from a discrete record of recognition events. Here the framework is working on a technical bridge in a gravity calculation. The full calculation involves a sum of 216 such terms, and the framework splits this large sum into six blocks. coreBlock1_eq (a machine-checked theorem in the framework's library) proves that the first block of 36 terms matches its corresponding polynomial exactly. The other five blocks have their own matching theorems, and together they establish the whole 216-term identity.

What this theorem does not claim is just as important. It does not say anything about the physical meaning of the terms, nor does it assert that the gravity calculation itself is correct. It only says that one specific algebraic expression is equal to another. The theorem is a statement about polynomials, not about physics. It is a piece of bookkeeping, verifying that a complicated sum can be rewritten in a simpler form without changing its value.

This kind of verification matters because the larger calculation is too large to check by hand. A single transcription error in a 216-term sum would be nearly impossible to spot by eye. The machine-checked proof catches such errors automatically. It gives the framework's developers confidence that the algebraic foundation of their gravity bridge is sound, even before they interpret what the mathematics might mean physically.

private theorem coreBlock1_eq (E : Fin 3 → Fin 3 → ℝ) (x : Fin 3 → ℝ)
    (s2 s3 p : ℝ) :
    (∑ f : Fin 6, ∑ g : Fin 6, coreTripleTerm E x (1, f, g)) =
      tetBlock1 (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 [tetBlock1_eq]
  simp only [Fin.sum_univ_six, coreTripleTerm, coreWeight,
    corePolEdgeCoeff, slotDispCore, slotMidTwice, Fin.sum_univ_three]
  push_cast
  ring

private theorem coreBlock1_eq (E : Fin 3 → Fin 3 → ℝ) (x : Fin 3 → ℝ)
    (s2 s3 p : ℝ) :
    (∑ f : Fin 6, ∑ g : Fin 6, coreTripleTerm E x (1, f, g)) =
      tetBlock1 (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 [tetBlock1_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

This theorem does not assert any physical meaning for the terms. This theorem does not claim the gravity calculation itself is correct. This theorem does not derive any new physics; it only verifies an algebraic identity.

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)

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