Foundation Cycle Operator Gray Order Inv Right Inv

The inverse ordering

A Gray code is a way of ordering binary numbers so that consecutive values differ by only one bit. The Recognition Science framework uses the 3-bit Gray code to order the eight vertices of a cube, visiting them in the sequence 0, 1, 3, 2, 6, 7, 5, 4. The declaration grayOrderInv_right_inv proves, in the framework's machine-checked library of formal theorems, that this ordering has a perfect inverse: if you take any vertex, look up its position in the sequence, and then ask which vertex sits at that position, you get the vertex you started with. In symbols, for every vertex j, grayOrder (grayOrderInv j) = j.

This is a statement about the mechanics of the ordering, not about physics. It says the mapping from vertices to positions and back is consistent, so the Gray code cycle can be treated as a well-defined permutation of the eight vertices. The proof is computational: the library checks all eight cases directly. The result matters because the cycle operator, built from this ordering, is the algebraic object from which the framework derives the CKM matrix, the matrix that describes how quark types mix. Without a verified inverse, the cycle would not be a clean permutation, and the later structure would lack a solid footing.

The declaration does not claim that the Gray code ordering is the only possible one, nor that it has any physical meaning on its own. It does not assert that the cycle operator's eigenvalues are the eighth roots of unity, that the operator has period eight, or that the framework's derivation of the CKM matrix is complete. Those are separate theorems in the library. The right inverse statement is a local, technical fact: the reverse lookup is exact. Its role is to make the cycle a genuine permutation, a necessary condition for the operator to be well-defined, not a sufficient one for any physical conclusion.

theorem grayOrderInv_right_inv : ∀ j, grayOrder (grayOrderInv j) = j := by
  intro j; fin_cases j <;> native_decide

What this page does not claim

The declaration does not claim the Gray code ordering is unique or physically meaningful on its own. It does not claim the cycle operator has period eight or that its eigenvalues are the eighth roots of unity. It does not claim the framework's derivation of the CKM matrix is complete.

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/Foundation/CycleOperator.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.

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