Encyclopedia Gravity Gravity Analysis Regge Ttgate Bbridge Core

ARTICLE 3 claims 3 theorems

Gravity Analysis Regge Ttgate Bbridge Core

A machine-checked algebra module proves that two different ways of writing the same gravity calculation always agree, with no hidden assumptions.

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The bridge core

In numerical analysis, a stencil is a fixed pattern of coefficients used to approximate a derivative or an integral at a point. The Regge TT Gate B bridge core is a component in the Recognition Science framework's machine-checked library of formal theorems. It proves a computational identity: a certain 216-term sum of raw moment terms equals the sum of six simpler block expressions, identically for all input values. The proof is carried out by the computer, not by hand, and it uses no special axioms beyond the standard logical foundations.

The component exists to bridge two ways of organizing the same calculation. One side writes the calculation as a large, direct sum over all combinations of three indices, each term a product of weights and edge coefficients. The other side groups the same terms into six blocks, each block associated with one of six slots in a periodic pattern. The core identity, named coreTripleSum_eq_spikeSum, states that these two arrangements are exactly equal. The proof expands both sides and simplifies them with polynomial algebra, a process the computer checks step by step.

The identity holds for completely free real parameters s2, s3, and p, and for arbitrary input matrices E and vectors x. No numerical values are fixed, and no continuity or smoothness assumption is needed. The six block expressions are built from literal tables of rational coefficients and displacement classes, which the main component later identifies with the actual geometric quantities. This separation keeps the heavy algebra in a leaf file that imports only basic arithmetic, so the computer can check it without running out of memory.

In Recognition Science, this component is a supporting step in a larger chain. It does not by itself derive gravity or any physical law. What it establishes is a formal fact about algebra: two descriptions of the same moment sum agree. That agreement is a ledger, a discrete record of events, written as a theorem the machine verified. The practical consequence is that later work can use either form of the sum interchangeably, trusting that the equality holds for every input, not just for the examples a human might test.

THEOREM coreTripleSum_eq_spikeSum · IndisputableMonolith/Gravity/Analysis/ReggeTTGateBBridgeCore.lean
coreTripleSum_eq_spikeSum · IndisputableMonolith/Gravity/Analysis/ReggeTTGateBBridgeCore.lean:205 
/-- THE CORE IDENTITY (THEOREM): the 216-term raw moment sum equals the
sum of the six committed spike blocks, for COMPLETELY FREE `s2 s3 p` and
IDENTICALLY in `E` and `x`.  Only the block data is used; the spike's TT
conclusion is never invoked. -/
theorem coreTripleSum_eq_spikeSum (E : Fin 3 → Fin 3 → ℝ)
    (x : Fin 3 → ℝ) (s2 s3 p : ℝ) :
    (∑ q : Fin 6 × Fin 6 × Fin 6, coreTripleTerm E x q) =
      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
        + 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
        + tetBlock2 (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
        + tetBlock3 (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
        + tetBlock4 (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
        + tetBlock5 (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
  simp only [Fintype.sum_prod_type]
  rw [Fin.sum_univ_six]
  rw [coreBlock0_eq E x s2 s3 p, coreBlock1_eq E x s2 s3 p,
    coreBlock2_eq E x s2 s3 p, coreBlock3_eq E x s2 s3 p,
    coreBlock4_eq E x s2 s3 p, coreBlock5_eq E x s2 s3 p]
THEOREM coreTripleSum_eq_spikeSum · IndisputableMonolith/Gravity/Analysis/ReggeTTGateBBridgeCore.lean
coreTripleSum_eq_spikeSum · IndisputableMonolith/Gravity/Analysis/ReggeTTGateBBridgeCore.lean:205 
/-- THE CORE IDENTITY (THEOREM): the 216-term raw moment sum equals the
sum of the six committed spike blocks, for COMPLETELY FREE `s2 s3 p` and
IDENTICALLY in `E` and `x`.  Only the block data is used; the spike's TT
conclusion is never invoked. -/
theorem coreTripleSum_eq_spikeSum (E : Fin 3 → Fin 3 → ℝ)
    (x : Fin 3 → ℝ) (s2 s3 p : ℝ) :
    (∑ q : Fin 6 × Fin 6 × Fin 6, coreTripleTerm E x q) =
      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
        + 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
        + tetBlock2 (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
        + tetBlock3 (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
        + tetBlock4 (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
        + tetBlock5 (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
  simp only [Fintype.sum_prod_type]
  rw [Fin.sum_univ_six]
  rw [coreBlock0_eq E x s2 s3 p, coreBlock1_eq E x s2 s3 p,
    coreBlock2_eq E x s2 s3 p, coreBlock3_eq E x s2 s3 p,
    coreBlock4_eq E x s2 s3 p, coreBlock5_eq E x s2 s3 p]
THEOREM coreTripleSum_eq_spikeSum · IndisputableMonolith/Gravity/Analysis/ReggeTTGateBBridgeCore.lean
coreTripleSum_eq_spikeSum · IndisputableMonolith/Gravity/Analysis/ReggeTTGateBBridgeCore.lean:205 
/-- THE CORE IDENTITY (THEOREM): the 216-term raw moment sum equals the
sum of the six committed spike blocks, for COMPLETELY FREE `s2 s3 p` and
IDENTICALLY in `E` and `x`.  Only the block data is used; the spike's TT
conclusion is never invoked. -/
theorem coreTripleSum_eq_spikeSum (E : Fin 3 → Fin 3 → ℝ)
    (x : Fin 3 → ℝ) (s2 s3 p : ℝ) :
    (∑ q : Fin 6 × Fin 6 × Fin 6, coreTripleTerm E x q) =
      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
        + 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
        + tetBlock2 (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
        + tetBlock3 (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
        + tetBlock4 (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
        + tetBlock5 (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
  simp only [Fintype.sum_prod_type]
  rw [Fin.sum_univ_six]
  rw [coreBlock0_eq E x s2 s3 p, coreBlock1_eq E x s2 s3 p,
    coreBlock2_eq E x s2 s3 p, coreBlock3_eq E x s2 s3 p,
    coreBlock4_eq E x s2 s3 p, coreBlock5_eq E x s2 s3 p]

What this page does not claim

This component does not derive gravity or any physical law by itself. The identity does not depend on the spike's TT conclusion, which is never invoked. This is not a proof of the Riemann Hypothesis or any other unproved statement.

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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