Cosmology Neutrino Hierarchy From Phi Ladder Mass Split Ratio Eq

The neutrino mass ratio

The golden ratio φ is the number that satisfies φ² = φ + 1, approximately 1.618. In the Recognition Science framework, a declaration named massSplitRatio_eq establishes that a proposed ratio of adjacent neutrino mass-squared splittings equals φ², which is the same as φ + 1. The statement is a theorem in the framework's machine-checked library of formal theorems, meaning it is proved from definitions with no unproved assumptions.

The context is a model of neutrino masses arranged on a φ-power ladder, with three masses m₁ < m₂ < m₃. The framework defines massSplitRatio as φ² and then proves it equals φ + 1. The proof is a direct consequence of the defining property of φ. The framework also proves the ratio is positive, and the whole construction is bundled into a structure called NeutrinoHierarchyCert that records these facts.

This declaration does not claim that the ratio matches any measured neutrino data. It is a definitional identity within the framework, not an empirical result. The model also includes two hierarchy scenarios, normal and inverted, and counts the total structural states as five, but this count is a modeling choice, not a physical measurement.

What the theorem does is fix a precise mathematical relationship that the framework's neutrino model relies on. It gives a clean algebraic identity: the proposed splitting ratio is exactly φ² = φ + 1 ≈ 2.618. This is a building block for further structural claims, not a statement about the physical world.

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

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

What this page does not claim

No claim that the φ² ratio matches any measured neutrino data. No claim that the five-state count is a physical fact rather than a modeling choice. No claim that the framework derives the neutrino mass hierarchy from first principles.

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: