Foundation Mass Weak Bases Even Flip Involution

The even flip

In the Recognition Science framework, the basic objects of the generation space are eight vertices of a cube, each labeled by three bits that record which of three axes are flipped. An even flip is a transformation that changes the sign of exactly two axes at once, leaving the third alone. The declaration evenFlip_involution proves a simple property of this operation: applying any even flip twice returns every vertex to its original state. In the language of group theory, each even flip is an involution, an operation whose square is the identity.

The proof is computational and exhaustive. The declaration checks all three possible even flips (axes 0 and 1, axes 0 and 2, axes 1 and 2) against all eight vertices, confirming the involution property in every case. This is a theorem in the machine-checked library of formal theorems, meaning the result is verified by a computer program that checks each step of the reasoning. The theorem is stated for all three generations and all eight vertex states, so it is a universal claim about the even-flip operation, not a special case.

The significance of this result is structural rather than numerical. The even flips form the basis for the weak eigenstates in the framework's account of quark mixing. The fact that they are involutions means they generate a group of order four, the Klein four-group, which is the symmetry group of the SU(2) doublet structure. This group structure is what allows the framework to define a weak basis that differs from the mass basis, producing the CKM matrix that describes how quarks change flavor in weak interactions.

What the theorem does not claim is important. It does not say that the even flips are the only operations on the cube, nor that they commute with each other. It does not establish that the weak basis is physically realized in nature, nor that the CKM matrix computed from this structure matches measured values. The theorem is a purely mathematical statement about a defined operation; its physical interpretation is a separate step that the framework makes elsewhere, with its own evidence and its own limitations.

In Recognition Science, this involution property is a small but necessary piece of the larger structure. It guarantees that the weak basis is well-defined and consistent, that flipping two axes twice always returns to the starting point. Without this property, the entire edifice of weak eigenstates and mixing angles would collapse. The theorem is a load-bearing brick in the framework's account of particle physics, even though it says nothing by itself about the masses or mixing angles of real quarks.

/-- Each even flip is an involution (applying it twice gives identity). -/
theorem evenFlip_involution (gen : Fin 3) (v : Fin 8) :
    evenFlipOnVertex gen (evenFlipOnVertex gen v) = v := by
  fin_cases gen <;> fin_cases v <;> native_decide

What this page does not claim

The theorem does not claim that even flips commute with each other. The theorem does not claim that the weak basis is physically realized. The theorem does not claim that the CKM matrix computed from this structure matches measured values.

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

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