Physics Mixing Geometry

Mixing from a cube

In particle physics, mixing describes how a particle that is created with one flavor can later be measured as another. The CKM matrix does this for quarks, and the PMNS matrix for neutrinos. These are not arbitrary tables of numbers; their entries are constrained by experiment. The question is whether their structure follows from a deeper principle.

Recognition Science offers one answer: the geometry of a cube. The framework models the three generations of matter as states living on the different parts of a cube: the vertices, the edges, and the faces. The mathematics then forces the mixing angles from the counting of these parts and their symmetries. A cube has 12 edges, and each edge connects 2 vertices, giving 24 vertex-edge slots. This number 24, along with the golden ratio and the fine-structure constant, appears throughout the derived angles.

The framework's library of machine-checked theorems formalizes this. It defines the mixing weights directly from the cube's structure. The solar angle, which governs neutrino oscillations from the sun, comes from a phi^-2 scaling. The atmospheric angle, responsible for neutrinos passing through the Earth, comes from a maximal parity mix of 1/2. The reactor angle, measured in nuclear reactors, comes from a phi^-8 scaling. Each of these is a definition in the library, not a theorem.

The theorems in the library are about the radiative corrections. These are small adjustments to the mixing angles caused by quantum effects. The framework proves that the solar angle is the phi^-2 weight minus a correction of 10 times the fine-structure constant. It proves the atmospheric angle is the 1/2 weight plus a correction of 6 times the fine-structure constant. And it proves the Cabibbo angle, the largest quark mixing angle, is the phi^-3 weight minus a correction of 1.5 times the fine-structure constant. These are exact algebraic identities in the formal system.

What this establishes is a pattern. The mixing angles are not free parameters in this account; they are expressions built from the cube's topology and the two fundamental constants of the framework. The proofs are formal, meaning a computer checked every step. Whether these expressions match the measured values to high precision is a separate, empirical question. The framework's contribution is to show that the structure of mixing can be derived from a simple geometric starting point.

theorem vertex_edge_slots_eq_24 : vertex_edge_slots = 24 := by
  unfold vertex_edge_slots cube_edges
  norm_num

/-- **THEOREM: Solar Angle Forced**
    The solar mixing angle is derived from the phi^-2 scaling with a
    10-alpha radiative correction. -/
theorem solar_angle_forced :
    solar_weight - solar_radiative_correction = phi ^ (-2 : ℤ) - 10 * alpha_CODATA := by
  unfold solar_weight solar_radiative_correction
  ring

/-- **THEOREM: Atmospheric Angle Forced**
    The atmospheric mixing angle is derived from the maximal parity mix
    with a face-mediated radiative correction. -/
theorem atmospheric_angle_forced :
    atmospheric_weight + atmospheric_radiative_correction = 1 / 2 + 6 * alpha_CODATA := by
  unfold atmospheric_weight atmospheric_radiative_correction
  ring

/-- **THEOREM: Cabibbo Scaling Forced**
    The Cabibbo scaling factor is forced by the torsion overlap and the
    face-mediated radiative corrections. -/
theorem cabibbo_scaling_forced :
    torsion_overlap - cabibbo_radiative_correction = phi ^ (-3 : ℤ) - (3/2) * alpha_CODATA := by
  unfold torsion_overlap cabibbo_radiative_correction
  ring

What this page does not claim

This page does not claim the derived angles match experimental measurements. This page does not claim the framework has derived the value of the fine-structure constant itself. This page does not claim a physical mechanism connecting the cube's geometry to real particles has been established.

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/Physics/MixingGeometry.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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