Gravity Analysis Regge Bloch Local Incidence4 D Regge4 Dpath Bposition Resolved

A status flag, not a result

In the Recognition Science framework, a ledger, a discrete record of events, is the primitive object from which physical structure is derived. The framework's machine-checked library of formal theorems contains a declaration named Regge4DPathBPositionResolvedClosesEH_status_open. In plain language, this declaration establishes that a specific proposal, called Path B with position-resolved kernels, does not close the Einstein-Hilbert action. The declaration is a theorem in the formal library, but its content is a negative status flag: it proves that the proposition Regge4DPathBPositionResolvedClosesEH is false.

The context is a search for how the Einstein-Hilbert action, the classical starting point for general relativity, might emerge from the ledger's recognition costs. Path B is one of two attempts to build local interaction kernels in a four-dimensional continuum. The first layer, mean-local, is vacuous: it merely reproduces the distinct-hinge result by linearity. The second layer, position-resolved, is the substantive attempt. The formal library proves that the mean-local construction equals the distinct-hinge construction on all inputs, and that the position-resolved construction is well-defined for the t11 orbit. But the key negative result is that extending the position-resolved construction to the t12 orbit breaks the agreement between the plus and cross polarization symbols and does not produce the required factor of -1/4 for the Einstein-Hilbert action. The status structure records pathBClosesEH := false and gapActionRecovery := false.

The declaration does not claim that the Einstein-Hilbert action cannot be derived from the ledger at all. It only records that this particular Path B, with position-resolved kernels, fails. Nor does it claim that the t11 position-resolved construction is undefined or useless; the library proves it is defined and agrees with the distinct-hinge result on tested rays. The declaration is a checkpoint in an ongoing search, a formal record of a dead end on one route, not a statement about the impossibility of the destination.

theorem Regge4DPathBPositionResolvedClosesEH_status_open :
    Regge4DPathBPositionResolvedClosesEH = False :=
  rfl

/-- 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 reggeBlochLocalIncidence4DStatus_flags :
    reggeBlochLocalIncidence4DStatus.meanLocalEqualsDistinctHinge = true ∧
      reggeBlochLocalIncidence4DStatus.positionResolvedT11Defined = true ∧
        reggeBlochLocalIncidence4DStatus.pathBClosesEH = false ∧
          reggeBlochLocalIncidence4DStatus.gapActionRecovery = false := by
  decide

What this page does not claim

The Einstein-Hilbert action cannot be derived from the ledger at all. The t11 position-resolved construction is undefined or useless. Path A or any other route to the Einstein-Hilbert action has been ruled out.

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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