Gravity Analysis Regge Bloch Local Incidence M2 Eval4 D M2 Path B Mean Local Axi

Averaging over orbits

In the Recognition Science framework's study of discrete gravity, the declaration m2PathB_meanLocal_axisTTCross_symbolDir is a proved theorem about a particular averaged quantity. The quantity is the mean local all-orbit moment, computed for a specific 4-dimensional configuration: the axis labeled axisTTCross with the direction symbolDir. The theorem states that this mean local moment equals exactly negative one quarter (-1/4). This is a precise, machine-checked result within the framework's library of formal theorems.

The computation is part of a broader comparison between two ways of defining a moment in the framework's discrete geometry: Path B and the distinct-hinge witness. The theorem shows that, for this particular axis and direction, the mean-local Path B value equals the distinct-hinge value. This agreement holds on the tested rays, as part of a witness table that also includes other axis-direction pairs with different values (0 and -1/8). The equality is proved by rewriting the mean-local definition in terms of the distinct-hinge definition, a standard proof step in the framework's library.

What the declaration does not claim is more important than what it does. It does not claim that the averaged Path B moment recovers the Einstein-Hilbert coefficient in 4 dimensions. A separate theorem in the same file proves that the normalized mean-local value for a related configuration is not equal to the Einstein-Hilbert coefficient. The framework's status record confirms this: position-resolved Path B does not close the Einstein-Hilbert gap, and the gap-action recovery flag remains false. The anisotropy that appears in the e0 direction also persists under the mean-local averaging, meaning the averaging does not restore isotropy.

In plain terms, this declaration is a small, exact brick in a larger negative result. It establishes one specific numerical equality, and it participates in a table that shows Path B and distinct-hinge agree on these witnesses. But the framework explicitly records that this line of investigation does not achieve the goal of recovering Einstein-Hilbert action from the discrete model. The theorem is a precise statement about a particular computation, not a claim about the success of the overall approach.

theorem m2PathB_meanLocal_axisTTCross_symbolDir :
    m2MeanLocalAllOrbitMoment axisTTCross symbolDir = (-1 / 4 : ℝ) := by
  rw [m2MeanLocalAllOrbitMoment_eq_distinctHinge,
    m2TransportedAllOrbitMomentDistinctHinge_axisTTCross_symbolDir]

theorem pathB_vs_distinctHinge_witness_table :
    m2MeanLocalAllOrbitMoment axisTTPlus symbolDir =
        m2TransportedAllOrbitMomentDistinctHinge axisTTPlus symbolDir ∧
      m2MeanLocalAllOrbitMoment axisTTCross symbolDir =
        m2TransportedAllOrbitMomentDistinctHinge axisTTCross symbolDir ∧
      m2MeanLocalAllOrbitMoment axisTTPlus e0Dir =
        m2TransportedAllOrbitMomentDistinctHinge axisTTPlus e0Dir ∧
      m2MeanLocalAllOrbitMoment axisTTCross e0Dir =
        m2TransportedAllOrbitMomentDistinctHinge axisTTCross e0Dir ∧
      m2MeanLocalAllOrbitMoment axisTTPlus symbolDir = (-1 / 4 : ℝ) ∧
      m2MeanLocalAllOrbitMoment axisTTCross symbolDir = (-1 / 4 : ℝ) ∧
      m2MeanLocalAllOrbitMoment axisTTPlus e0Dir = (0 : ℝ) ∧
      m2MeanLocalAllOrbitMoment axisTTCross e0Dir = (-1 / 8 : ℝ) := by
  refine ⟨?_, ?_, ?_, ?_, ?_, ?_, ?_, ?_⟩
  · exact m2MeanLocalAllOrbitMoment_eq_distinctHinge _ _
  · exact m2MeanLocalAllOrbitMoment_eq_distinctHinge _ _
  · exact m2MeanLocalAllOrbitMoment_eq_distinctHinge _ _
  · exact m2MeanLocalAllOrbitMoment_eq_distinctHinge _ _
  · exact m2PathB_meanLocal_axisTTPlus_symbolDir
  · exact m2PathB_meanLocal_axisTTCross_symbolDir
  · exact m2PathB_meanLocal_axisTTPlus_e0Dir
  · exact m2PathB_meanLocal_axisTTCross_e0Dir

/-- Position-resolved Path B does not close EH (status false / OPEN). -/
theorem pathB_positionResolved_does_not_close_eh :
    Regge4DPathBPositionResolvedClosesEH = False :=
  Regge4DPathBPositionResolvedClosesEH_status_open

What this page does not claim

The declaration does not claim that the mean-local Path B moment recovers the Einstein-Hilbert coefficient in 4 dimensions. The declaration does not claim that averaging restores isotropy; the e0 anisotropy persists. The declaration does not claim that Path B and distinct-hinge agree on all configurations, only on the tested witnesses.

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/ReggeBlochLocalIncidenceM2Eval4D.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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