Gravity Analysis Regge Bloch Orbit Transport4 D

Orbit transport

In the Recognition Science framework, gravity analysis works on a discrete lattice of hinge slots, each indexed by a pair (s, t) with s ranging over 24 positions and t over 10. The framework models these slots as belonging to orbit types, which are symmetry classes of the lattice. The module ReggeBlochOrbitTransport4D addresses a specific transport problem: for each slot and each orbit type, find a coordinate permutation that maps the orbit's representative to that slot's exact position.

The classical setting is Regge calculus, where spacetime is discretized into simplices and curvature lives on hinges. Bloch transport refers to how a wavefunction or geometric quantity picks up a phase or permutation when moved around a lattice. In this 4D context, the framework defines a covering permutation as the first of 24 possible permutations (in a fixed order) that satisfies the covering equation: applying the permutation to the orbit representative's difference pair yields the slot's diff masks. The definition is precise: permDiffPair (a function comparing two coordinate pairs) must equal the target masks.

The module proves three theorems. First, covering existence: every lattice slot is covered by its own orbit representative, meaning a valid permutation always exists. Second, the specification theorem: when the orbit type matches the slot's type, the covering equation holds exactly. Third, agreement with the legacy table: on (1,1) slots, the covering permutation matches the hand-written slotTransportPerm table. The proofs are computational, using fin_cases and decide, with no sorry, admit, new axioms, or native_decide shells.

The practical consequence is that transport is now uniform across all orbit types. The older hand table was (1,1)-only and could not be used for non-(1,1) slots; the new covering construction works for all slots. This is a measured lesson from factorized versus transported fold experiments: the covering approach generalizes where the table did not. The framework's library, a machine-checked collection of formal theorems, establishes that the transport mechanism is complete and consistent, removing a gap that previously required case-by-case handling.

/-- 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

No claim that this module derives the fine-structure constant or any physical coupling. No claim that the transport mechanism is physically realized in spacetime, only that it is formally consistent. No claim that the legacy table is wrong, only that it was limited to (1,1) slots.

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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