Gravity Analysis Regge Bloch Local Incidence4 D Mean Local Inherits Distinct Hin

The mean-local equivalence

In the framework's study of gravity, a kernel is a rule that assigns a number to each pair of nearby points, describing how influence spreads. The declaration meanLocal_inherits_distinctHinge_on_any is a theorem: for any 4x4 matrix H and any direction vector dir, the quantity m2MeanLocalAllOrbitMoment H dir equals m2TransportedAllOrbitMomentDistinctHinge H dir. In plain words, it proves that a particular way of averaging over local slots, called mean-local, produces exactly the same second-moment value as the distinct-hinge method, for every choice of H and dir.

This is a formal result in the machine-checked library of formal theorems. The proof works by unfolding the definitions and showing that each orbit's mean-local term is just a scaled copy of the corresponding distinct-hinge term. The scaling factor, the inverse of the orbit's star size, cancels out when summed over all orbits. The theorem is a direct corollary of the earlier lemma m2MeanLocalAllOrbitMoment_eq_distinctHinge, which establishes the equality for all inputs.

The theorem does not claim that the mean-local approach works for the position-resolved case, where the kernel is evaluated at shifted positions rather than averaged. The pack shows a separate status flag pathBClosesEH is set to false, and the proposition Regge4DPathBPositionResolvedClosesEH is defined as False. This means the more detailed, non-vacuous version of the local-incidence path does not close the Einstein-Hilbert action and does not recover the gap action. The equality in meanLocal_inherits_distinctHinge_on_any holds only for the mean-local (vacuous) construction, not for the position-resolved expansion.

What this changes for the reader: the theorem gives a clean, fully general equivalence for the mean-local kernel, which is a useful simplification. But it also marks a boundary. The framework's own status record confirms that the position-resolved path remains open, with the flags for closing the Einstein-Hilbert action and recovering the gap action both false. This is not a failure of the theorem, but a precise statement of where the framework's derivation currently stops.

/-- Vacuous mean-local Path B cannot repair e0 anisotropy / factor 4,
because it equals distinct-hinge. -/
theorem meanLocal_inherits_distinctHinge_on_any (H : Mat4)
    (dir : Fin 4 → ℝ) :
    m2MeanLocalAllOrbitMoment H dir =
      m2TransportedAllOrbitMomentDistinctHinge H dir :=
  m2MeanLocalAllOrbitMoment_eq_distinctHinge H dir

theorem Regge4DPathBPositionResolvedClosesEH_status_open :
    Regge4DPathBPositionResolvedClosesEH = False :=
  rfl

theorem does_not_flip_gap_action_recovery :
    reggeBlochLocalIncidence4DStatus.gapActionRecovery = false :=
  rfl

What this page does not claim

The theorem does not apply to the position-resolved kernel, which remains an open problem. It does not establish that the mean-local construction closes the Einstein-Hilbert action; the status flag for that is false. It does not claim any recovery of the 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/ReggeBlochLocalIncidence4D.lean
expected axiom basis: [propext, Classical.choice, Quot.sound] (the Lean kernel's standard three; no RS-specific axioms)

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