Gravity Analysis Regge Bloch Orbit Transport4 D Orbit Covering Perm Spec

The covering rule

In Regge calculus, spacetime is chopped into flat 4-simplexes glued along triangular hinges. The framework's recognition ledger, a discrete record of events, treats each hinge as a slot in a 24-element symmetry group. The theorem orbitCoveringPerm_spec states a covering rule: for every slot (s,t), the first permutation in a fixed order that matches the slot's orbit type also satisfies the exact coordinate transport equation. The proof is a finite case check over all 24 by 10 slots, with no axioms added beyond the kernel's three standard ones.

The practical content is that the covering permutation exists on every slot and agrees with the legacy hand table on the special (1,1) slots. The hand table, slotTransportPerm, only covers that single case. The theorem proves the general rule for all slots, so the framework's transport is uniform. The equation itself is: permDiffPair (coordPermOf p) (orbitRep ty) = (diffMaskA s t, diffMaskB s t), where p is the covering permutation.

What the theorem does not claim is any physical content about gravity itself. It is a combinatorial statement about the symmetry group acting on a fixed lattice. It does not assert that the covering permutation is unique, only that the first one found works. It also does not say the legacy table is wrong, only that it is incomplete for non-(1,1) slots. The theorem's scope is the algebraic transport rule, not the dynamics of the gravitational field.

/-- Packaging: when `ty` is the slot's orbit type, the covering equation holds. -/
theorem orbitCoveringPerm_spec (ty : HingeOrbitType) (s : Fin 24) (t : Fin 10)
    (h : hingeOrbitType s t = ty) :
    permDiffPair (coordPermOf (orbitCoveringPerm ty s t))
        (orbitRep ty).1 (orbitRep ty).2 =
      (diffMaskA s t, diffMaskB s t) := by
  have hc := orbitCoveringPerm_covers s t
  subst h
  simpa [coversOrbitSlot, decide_eq_true_iff] using hc

set_option maxRecDepth 8000 in
set_option maxHeartbeats 400000 in

/-- Every lattice slot is covered by its own orbit representative. -/
theorem orbitCoveringPerm_covers (s : Fin 24) (t : Fin 10) :
    coversOrbitSlot (hingeOrbitType s t) s t
        (orbitCoveringPerm (hingeOrbitType s t) s t) = true := by
  fin_cases s <;> fin_cases t <;> decide

/-- On `(1,1)` slots the covering perm agrees with the legacy table. -/
theorem orbitCoveringPerm_t11_eq_slotTransportPerm (s : Fin 24) (t : Fin 10) :
    orbitCoveringPerm .t11 s t = slotTransportPerm s t := by
  fin_cases s <;> fin_cases t <;> decide

What this page does not claim

The theorem does not claim uniqueness of the covering permutation. The theorem does not claim the legacy table is incorrect, only incomplete. The theorem does not address the dynamics or physical content of gravity.

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/ReggeBlochOrbitTransport4D.lean
expected axiom basis: [propext, Classical.choice, Quot.sound] (the Lean kernel's standard three; no RS-specific axioms)

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