Physics Pmnsmixing Angles From Rs

The PMNS angles

The PMNS matrix is the standard object particle physicists use to describe how neutrinos oscillate between three flavors: electron, muon, and tau. It is a 3x3 unitary matrix parameterized by three mixing angles and a CP-violating phase, plus two extra phases if neutrinos are Majorana particles. The three angles have measured values: the solar angle θ₁₂ is about 33.4°, the atmospheric angle θ₂₃ is close to 45°, and the reactor angle θ₁₃ is small, about 8.5°. The surprising fact is that 45° is exactly maximal mixing, and 33.4° is close to arctan(1/φ), where φ is the golden ratio.

The golden ratio φ ≈ 1.618 satisfies φ² = φ + 1, so 1/φ ≈ 0.618. The solar mixing angle's tangent, tan(33.4°), is approximately 0.66, which sits near 1/φ. The atmospheric angle being exactly π/4 means tan(π/4) = 1, a value the framework's machine-checked library proves formally. The reactor angle is small, and the framework notes it without forcing a specific value.

In Recognition Science, a framework that derives physical structure from a forced cost of recognition events, the PMNS parameters appear as a counting fact. The framework models the five independent PMNS parameters (three angles, one CP phase, one Majorana phase) as the dimension of a configuration space. Its library proves that this count is exactly 5, matching the dimension D = 5 that the framework's forcing chain derives for configuration spaces. The module also proves two analytic facts: tan(π/4) = 1, which pins the maximal mixing angle, and that 1/φ lies in the interval (0.617, 0.623), which brackets the solar mixing tangent.

What the module does not do is derive the measured values of θ₁₂, θ₂₃, or θ₁₃ from first principles. It proves structural facts: the count is 5, the maximal angle has tangent 1, and the golden-ratio inverse sits in a narrow band near the solar tangent. The comparison to measured PDG values is an empirical check, not a theorem. The framework's contribution is to show that these angles are not arbitrary: they sit at points where the framework's structural constants, like φ, appear naturally.

The practical consequence is a sharper question for neutrino physics. If the solar angle is exactly arctan(1/φ), then future precision measurements could test that prediction against the current best fit. The module gives a clean target: measure tan(θ₁₂) to see whether it converges to 0.618 or drifts away. That is a falsifiable statement the framework puts on the table.

/-- Maximal mixing angle: tan(π/4) = 1. -/
theorem maximal_mixing : Real.tan (Real.pi / 4) = 1 := by
  simp [Real.tan_pi_div_four]

theorem solarTangent_band :
    (0.617 : ℝ) < solarTangent ∧ solarTangent < 0.623 := by
  unfold solarTangent
  have h1 := phi_gt_onePointSixOne
  have h2 := phi_lt_onePointSixTwo
  constructor
  · rw [lt_inv_comm₀ (by norm_num) phi_pos]
    linarith
  · rw [inv_lt_comm₀ phi_pos (by norm_num)]
    linarith

theorem pmnsParameterCount : Fintype.card PMNSParameter = 5 := by decide

What this page does not claim

The module does not derive the measured values of θ₁₂, θ₂₃, or θ₁₃ from first principles. The framework does not prove that the solar angle is exactly arctan(1/φ); it proves that 1/φ lies in a narrow band near the measured tangent. The CP-violating phase δ_CP and Majorana phases are counted but not assigned 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/Physics/PMNSMixingAnglesFromRS.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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