Cosmology Neutrino Hierarchy From Phi Ladder Mass Split Ratio

The mass split ratio

The golden ratio φ, about 1.618, is the number that solves r² = r + 1. Its square, φ² ≈ 2.618, appears throughout classical mathematics, from pentagon geometry to continued fractions. In the Recognition Science framework, this same number appears as the ratio between adjacent neutrino mass-squared splittings. The framework's machine-checked library of formal statements contains a definition, massSplitRatio, which sets this ratio to φ². The definition is a structural choice, not a measurement, and the framework's own documentation labels it as part of a hierarchy scenario, not as a derived physical law.

The framework models three neutrino masses, m₁ < m₂ < m₃, arranged on a φ-ladder, where each step's mass-squared splitting is φ² times the previous one. The library proves two simple facts about this ratio: it equals φ + 1 (which follows directly from the defining equation of φ), and it is positive. These are formal consequences of the definition, checked by the machine, but they are properties of the chosen ratio, not evidence about the physical world. The framework also defines five possible neutrino states: three individual masses and two hierarchy scenarios, normal and inverted. A formal result confirms that this enumeration yields exactly five states, which the framework identifies with a structural dimension of its model.

What the declaration does not claim is essential to state plainly. The massSplitRatio definition is not a measurement of neutrino masses, and the framework does not derive the ratio from a deeper principle within this declaration. The framework's own honesty rules require that this be labeled as a definitional choice, a model, not a statement about nature. The ratio φ² is a structural input, a choice made to fit the framework's broader φ-ladder pattern, not a prediction that has been tested against experimental data. The framework does not claim to explain why the neutrino mass hierarchy is normal or inverted; it merely includes both as possibilities in its five-state enumeration.

The distinction matters for anyone reading the framework's claims. A machine-checked proof of a definition's consequences is a proof about the definition itself, not about the physical universe. The massSplitRatio declaration establishes a consistent mathematical structure, one where the golden ratio appears in a cosmological context. Whether that structure corresponds to reality is an empirical question the declaration does not address. The framework's value here is in showing how a single ratio, φ², can organize a model of neutrino masses, but the step from mathematical structure to physical fact remains open and unclaimed.

noncomputable def massSplitRatio : ℝ := phi ^ 2

theorem massSplitRatio_eq : massSplitRatio = phi + 1 := by
  unfold massSplitRatio; exact phi_sq_eq

theorem massSplitRatio_pos : 0 < massSplitRatio := by
  unfold massSplitRatio; exact pow_pos phi_pos 2

theorem neutrinoState_count : Fintype.card NeutrinoState = 5 := by decide

What this page does not claim

The massSplitRatio is not a measured value from neutrino experiments. The framework does not derive the φ² ratio from a deeper principle in this declaration. The framework does not claim to explain why the neutrino mass hierarchy is normal or inverted.

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/Cosmology/NeutrinoHierarchyFromPhiLadder.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: