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

A witness against symmetry

In the discrete geometry studied here, space is built from a finite set of points and links, and physical quantities are computed by summing over local configurations. A central question is whether different ways of averaging those local contributions give the same answer, especially along special directions. The theorem m2PathB_meanLocal_plus_cross_disagree_e0Dir (a machine-checked statement in the framework's library) establishes a precise disagreement: along the direction called e0, the mean-local moment computed on the plus-oriented axis equals 0, while the same quantity on the cross-oriented axis equals -1/8. These two values are not equal, so the theorem proves that the two choices of axis orientation yield different averaged results.

This disagreement is not an accident of one calculation. The same library proves that along a different direction, called symbolDir, both orientations give the same value, -1/4. The contrast is the point: averaging does not erase the difference between the two orientations in every direction. The theorem is a witness, a concrete numerical example showing that a proposed symmetry, the restoration of isotropy under this averaging procedure, fails. It is one of several results in the same file that together form a status report on the model: the mean-local procedure equals the distinct-hinge procedure on these test cases, but the e0 anisotropy persists, and the factor-4 residual also persists.

The theorem does not claim that the model is wrong, nor that the disagreement is physically meaningful. It does not say that the e0 direction is special in any absolute sense; it only shows that the two orientations differ there under this specific averaging rule. It does not establish that the model fails to reproduce general relativity, only that this particular averaging procedure does not restore the symmetry one might have hoped for. The library explicitly records that the position-resolved version of this path does not close the Einstein-Hilbert gap, and that the gap-action recovery flag remains false. These are separate, additional results.

What the theorem does establish is a precise, machine-checked fact about the discrete model: the mean-local moment is not isotropic under the two axis orientations. This is a concrete obstruction, a number that any future attempt to restore symmetry in this model must confront. It is a small but definite step in mapping which symmetries hold and which fail in this discrete setting, and it provides a test case for any proposed modification of the averaging rule.

theorem m2PathB_meanLocal_axisTTPlus_e0Dir :
    m2MeanLocalAllOrbitMoment axisTTPlus e0Dir = (0 : ℝ) := by
  rw [m2MeanLocalAllOrbitMoment_eq_distinctHinge,
    m2TransportedAllOrbitMomentDistinctHinge_axisTTPlus_e0Dir]

theorem m2PathB_meanLocal_axisTTCross_e0Dir :
    m2MeanLocalAllOrbitMoment axisTTCross e0Dir = (-1 / 8 : ℝ) := by
  rw [m2MeanLocalAllOrbitMoment_eq_distinctHinge,
    m2TransportedAllOrbitMomentDistinctHinge_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

theorem m2PathB_meanLocal_axisTTPlus_symbolDir :
    m2MeanLocalAllOrbitMoment axisTTPlus symbolDir = (-1 / 4 : ℝ) := by
  rw [m2MeanLocalAllOrbitMoment_eq_distinctHinge,
    m2TransportedAllOrbitMomentDistinctHinge_axisTTPlus_symbolDir]

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

/-- 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

The theorem does not claim the model is physically wrong or that the disagreement indicates a failure of the framework. It does not claim that all averaging procedures are anisotropic, only that this specific one differs along e0. It does not claim that the e0 direction is special in an absolute sense, only that the two orientations differ there.

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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