Gravity Analysis Regge Bloch M2 Symbol4 D Fold Along M2 Tendsto Gauge Iff

The gauge equivalence

In the Recognition Science framework, gravity is studied through a discrete model: a four-dimensional lattice of hinges whose geometry is encoded in a matrix H. The framework's machine-checked library of formal theorems contains a declaration, FoldAlongM2Tendsto_gauge_iff, that establishes an equivalence for a specific configuration called decoyGauge. This configuration is one of two special test cases (the other is axisTTPlus) used to probe how the model behaves as momentum approaches zero.

The theorem states that the general limit condition FoldAlongM2Tendsto decoyGauge holds if and only if the specialized condition FoldAlongM2Tendsto_decoyGauge holds. Both conditions describe the same limit: as a momentum parameter μ tends to zero, the ratio foldAlong decoyGauge μ / μ² approaches the value 0. The equivalence is a formal rewriting: it shows that the general definition, which references the coefficient m2Symbol decoyGauge, reduces to the explicit target value 0 for this configuration. The proof is a direct simplification using the already-established fact that m2Symbol decoyGauge equals 0.

The result matters because it closes one step in a larger verification program. The framework aims to show that its discrete lattice model reproduces the Einstein-Hilbert action of continuum gravity in a suitable limit. This theorem confirms that for the gauge configuration, the small-momentum behavior is exactly what the framework's definitions require, with no hidden discrepancy. It is one of several similar checks: the axis configuration has an analogous equivalence with target value -3, and both configurations have been shown to have vanishing value at zero momentum.

What the theorem does not claim is equally important. It does not prove the full convergence of the framework's action to the Einstein-Hilbert action; that remains an open target. It does not establish the limit for arbitrary configurations H, only for the two named test cases. It does not address the recovery of the gravitational gap action, a separate open question. The theorem is a local, specialized check, not a global derivation.

theorem FoldAlongM2Tendsto_gauge_iff :
    FoldAlongM2Tendsto decoyGauge ↔ FoldAlongM2Tendsto_decoyGauge := by
  constructor <;> intro h <;>
    simpa [FoldAlongM2Tendsto, FoldAlongM2Tendsto_decoyGauge,
      m2Symbol_decoyGauge] using h

def FoldAlongM2Tendsto_decoyGauge : Prop :=
  Tendsto (fun μ : ℝ => foldAlong decoyGauge μ / μ ^ 2) (𝓝[≠] (0 : ℝ))
    (𝓝 (0 : ℝ))

def m2Symbol (H : Mat4) : ℝ :=
  ∑ s : Fin 24, ∑ t : Fin 10, m2SlotCoeff H s t

What this page does not claim

The theorem does not prove convergence of the framework's action to the Einstein-Hilbert action. The theorem does not establish the limit for arbitrary matrices H, only for the decoyGauge and axisTTPlus configurations. The theorem does not address recovery of the gravitational gap action.

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/ReggeBlochM2Symbol4D.lean
expected axiom basis: [propext, Classical.choice, Quot.sound] (the Lean kernel's standard three; no RS-specific axioms)

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