Foundation Mass Weak Bases Ckm Hierarchy From Torsion Gaps

The hierarchy from torsion gaps

The CKM matrix describes how quarks change flavor when they interact via the weak force. Its off-diagonal entries are not all the same size: the mixing between the first and second generations (the Cabibbo angle) is much larger than the mixing between the second and third, which is in turn larger than the first-to-third mixing. This ordering, |V_ub| << |V_cb| << |V_us|, is a measured fact of particle physics.

The Recognition Science framework derives this ordering from a structural number, not from a fit. In the framework's model, the three quark generations correspond to three torsion values on a discrete structure called Q₃: 0, 11, and 17. The theorem ckm_hierarchy_from_torsion_gaps proves, in the machine-checked library of formal theorems, that the absolute values of these gaps satisfy 0 < |11 - 17| < |0 - 17|. That is, the gap between the first and third generations (17) is larger than the gap between the second and third (11). The framework identifies the size of the mixing angle with the overlap between generations, and a smaller torsion gap means a larger overlap, hence a larger mixing angle. So the theorem establishes the qualitative hierarchy of the CKM matrix from the torsion gap hierarchy alone.

This is a structural result, not a numerical prediction. The theorem does not compute the actual values of the CKM entries; it proves an ordering. The framework's own documentation states that the precise matrix elements come from overlap integrals in a separate module, CKMFromCube. The theorem also does not derive the torsion values themselves; they are inputs from prior framework results. What the theorem adds is the logical link: given those torsion values, the observed hierarchy of mixing angles follows as a matter of proof, not of parameter adjustment.

In Recognition Science, this is one step in a larger program. The framework models both the mass and weak eigenstates as different decompositions of the same Q₃ structure, and the CKM matrix as the change of basis between them. The theorem shows that a purely structural quantity, the torsion gap, already encodes the most striking qualitative feature of quark mixing: its strong hierarchy. It does not claim to derive the full CKM matrix, nor to explain why the torsion values are what they are. It establishes one clean consequence of the framework's structural assumptions.

/-- The CKM hierarchy |V_ub| << |V_cb| << |V_us| follows from the
    torsion gap hierarchy 17 > 6 > ... (with flip-count modulation). -/
theorem ckm_hierarchy_from_torsion_gaps :
    (0 : ℤ).natAbs < (11 - 17 : ℤ).natAbs ∧
    (11 - 17 : ℤ).natAbs < (0 - 17 : ℤ).natAbs := by
  norm_num

/-- 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

/-- The CKM hierarchy |V_ub| << |V_cb| << |V_us| follows from the
    torsion gap hierarchy 17 > 6 > ... (with flip-count modulation). -/
theorem ckm_hierarchy_from_torsion_gaps :
    (0 : ℤ).natAbs < (11 - 17 : ℤ).natAbs ∧
    (11 - 17 : ℤ).natAbs < (0 - 17 : ℤ).natAbs := by
  norm_num

What this page does not claim

The theorem does not derive the numerical values of the CKM matrix elements. The theorem does not derive the torsion values 0, 11, and 17 themselves. The theorem does not establish the full CKM matrix, including the CP-violating phase.

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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