Gravity Analysis Regge Bloch Local Incidence M2 Eval4 D Path B Vs Distinct Hinge

The witness table

The declaration pathB_vs_distinctHinge_witness_table is a compact table of eight equalities, each one a computed number attached to a geometric test case. The table compares two ways of averaging a quantity called the all-orbit moment, a measure of how a geometric structure curves when transported around a loop. The first method, Path B, averages over all orbits in a way that keeps position information. The second, distinct-hinge, averages over a set of distinct hinge points. The table records that on four test cases, the two methods produce identical numbers.

Those numbers are not arbitrary. For two of the test cases, the shared value is -1/4; for the other two, the values are 0 and -1/8. The table also shows that the two methods disagree on one test case, the e0 direction, where Path B gives 0 and distinct-hinge gives -1/8. This disagreement is the heart of the matter: it means the two averaging methods are not interchangeable in general, even though they coincide on the four witness cases.

In Recognition Science, the framework's machine-checked library of formal theorems uses this table to record a status report. The report states that the mean-local Path B equals distinct-hinge on the four witnesses, but that the position-resolved Path B does not close the Einstein-Hilbert gap and does not restore isotropy. The framework models these as boolean flags in a structure: three flags are true, two are false. The false flags are the ones that matter for physics.

The table does not claim that Path B and distinct-hinge are the same method. It does not claim that the position-resolved Path B matches Einstein gravity, and it does not claim that the gap-action recovery flag is flipped. The table is a witness in the mathematical sense: it exhibits specific cases where two methods agree, and one case where they disagree, so that a reader can see exactly what the framework has and has not established.

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

theorem m2PathB_meanLocal_plus_cross_disagree_e0Dir :
    m2MeanLocalAllOrbitMoment axisTTPlus e0Dir ≠
      m2MeanLocalAllOrbitMoment axisTTCross e0Dir := by
  rw [m2PathB_meanLocal_axisTTPlus_e0Dir, m2PathB_meanLocal_axisTTCross_e0Dir]
  norm_num

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

theorem does_not_flip_gap_action_recovery :
    reggeBlochLocalIncidenceM2Eval4DStatus.gapActionRecovery = false :=
  rfl

What this page does not claim

Path B and distinct-hinge are the same averaging method in general. The position-resolved Path B matches Einstein gravity on any test case. The witness table establishes a physical law; it only records computed values.

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)

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Derived articles

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