Gravity Analysis Regge Bloch Local Incidence4 D Regge Bloch Local Incidence4 Dst

Status flags

The declaration reggeBlochLocalIncidence4DStatus_flags is a machine-checked status report. It records, in four boolean flags, where a specific research path stands. The path is called Path B, and its goal is to derive the Einstein-Hilbert action, the starting point of general relativity, from a discrete combinatorial structure. The report is a theorem in the framework's machine-checked library of formal theorems, meaning its content is verified by a computer.

Two of the four flags are set to true, marking successes. The first, meanLocalEqualsDistinctHinge, states a certain averaging procedure, a mean-local kernel, produces exactly the same result as a previously defined distinct-hinge kernel. The second, positionResolvedT11Defined, confirms that a more detailed, position-resolved version of the calculation for a specific component, t11, is well-defined. These are positive results, but they are technical steps, not the final goal.

The other two flags are false, marking open problems. The flag pathBClosesEH is false, meaning this Path B does not, in its current form, close the gap to produce the Einstein-Hilbert action. The final flag, gapActionRecovery, is also false, confirming there is no recovery of the missing action term. The status report is a precise, honest ledger of what has been achieved and what remains incomplete.

In Recognition Science, this declaration does not claim to have derived gravity's equations. It does not claim the path is complete. It establishes a clear, verified checkpoint: two technical definitions work as intended, but the central goal of deriving the Einstein-Hilbert action from this particular local-incidence approach remains open. The value of the declaration is its clarity; it prevents a false impression of progress and directs attention to the specific, unresolved step.

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

theorem blochFoldAllMeanLocal_eq_distinctHinge (H : Mat4) (m : Fin 4 → ℝ) :
    blochFoldAllMeanLocal H m = blochFoldAllDistinctHinge H m := by
  unfold blochFoldAllMeanLocal blochFoldAllDistinctHinge
  refine Finset.sum_congr rfl fun ty _ =>
    blochFoldOrbitMeanLocal_eq_scaled ty H m

theorem Regge4DPathBPositionResolvedClosesEH_status_open :
    Regge4DPathBPositionResolvedClosesEH = False :=
  rfl

theorem does_not_flip_gap_action_recovery :
    reggeBlochLocalIncidence4DStatus.gapActionRecovery = false :=
  rfl

What this page does not claim

This declaration does not claim that the Einstein-Hilbert action has been derived from Path B. This declaration does not claim that the position-resolved t11 calculation is correct for all components, only that it is defined. This declaration does not claim that the mean-local equivalence holds for any kernel other than the one specified.

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)

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

This page is generated by a question-recursion engine: the questions its answers raise become the next pages. The current agenda, with open targets marked red: