Foundation Mass Weak Bases Weak Complement Is Identity

The weak complement map

In the Standard Model of particle physics, quarks come in three generations, and the CKM matrix describes how they mix when they interact through the weak force. The matrix is the overlap between two different ways of labeling the same quantum states: the mass basis, where each generation has a definite mass, and the weak basis, where each generation has a definite weak interaction. The Recognition Science framework builds both bases from the same underlying structure, an eight-dimensional space Q₃, and the declaration weakComplement_is_identity is a small but necessary piece of that construction.

In the framework, the weak basis is built from the action of the SU(2) gauge subgroup, which flips pairs of axes in the eight-dimensional space. Each generation is assigned to one of three such flip generators, and the "complement axis" is the one axis that a given generator does not flip. The theorem weakComplement_is_identity states, in formal terms, that for every one of the three generations, the complement axis is simply the generation's own index: generation 1 has complement axis 0, generation 2 has complement axis 1, and generation 3 has complement axis 2. In plain language, the map that assigns a complement axis to each generation is the identity map.

This is a deliberately modest result. It does not, by itself, produce any mixing. The framework's own documentation is explicit that the mass and weak axis assignments are both the identity for this simple labeling, and that the actual CKM mixing comes from the internal structure of the states, not from the axis labels. The mass states are eigenstates of a cost operator weighted by flip counts [4,2,2], while the weak states are eigenstates of the even-sign-flip generators. These have different internal structure even when the axis labels coincide. The theorem is a foundation stone: it pins down the weak-basis labeling so that the later, more substantial work of computing overlap integrals can proceed.

In Recognition Science, this lemma sits inside a larger program that derives the CKM matrix from the geometry of Q₃. The framework's library contains theorems asserting that the Cabibbo angle is the largest mixing angle, that V_ub is the smallest, and that the hierarchy of mixing angles follows from the torsion gaps between generations. The identity lemma is not one of those headline results. It is the kind of small, machine-checked fact that a formal proof needs to be complete: a precise statement about how the weak basis is labeled, verified for each of the three cases by direct computation.

The distinction between the axis labels and the internal structure is the key to understanding what this theorem does and does not claim. The identity map says nothing about the physics of mixing; it only fixes a convention. The framework's own documentation warns against reading too much into it. The real content, the part that would connect to measured CKM elements, lives in the overlap between the differently-structured eigenstates, which is a separate and much harder problem. This lemma is a necessary but not sufficient step toward that goal.

/-- The weak complement axis assignment is the identity. -/
theorem weakComplement_is_identity :
    ∀ i : Fin 3, weakComplementAxis i = i := by
  intro i; fin_cases i <;> rfl

/-- The mass and weak axis assignments are BOTH the identity for this
    simple axis labeling. The actual mixing comes from the INTERNAL
    structure: the mass states are eigenstates of the J-cost operator
    weighted by flip counts [4,2,2], while the weak states are
    eigenstates of the even-sign-flip generators. These have different
    internal structure even when the axis labels coincide.

    The precise CKM matrix elements come from the overlap integrals
    between these differently-structured eigenstates (see CKMFromCube). -/
theorem both_bases_label_axes : ∀ i, massBasisAxis i = weakBasisAxis i := by
  intro i; fin_cases i <;> rfl

/-- Structural mixing data: the ingredients that determine the CKM matrix.
    All values are RS-derived (zero free parameters). -/
structure MixingAngleData where
  flipCounts : Fin 3 → ℕ
  flipCounts_values : flipCounts 0 = 4 ∧ flipCounts 1 = 2 ∧ flipCounts 2 = 2
  torsion : Fin 3 → ℤ
  torsion_values : torsion 0 = 0 ∧ torsion 1 = 11 ∧ torsion 2 = 17
  faceCount : ℕ
  faceCount_value : faceCount = 6
  edgeCount : ℕ
  edgeCount_value : edgeCount = 12
  totalFlips : flipCounts 0 + flipCounts 1 + flipCounts 2 = 8

What this page does not claim

This theorem does not by itself produce any quark mixing or any CKM matrix element. The identity map on axis labels does not imply that the mass and weak bases are the same physical basis. The framework does not claim to have computed the numerical values of the CKM matrix elements from this lemma alone.

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)

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: