Cosmology Neutrino Hierarchy From Phi Ladder Neutrino State

Neutrino states and the phi ladder

Neutrinos come in three mass states, usually labeled m₁, m₂, and m₃, ordered from lightest to heaviest. The Recognition Science framework's declaration NeutrinoState encodes exactly those three states, plus two possible orderings of them: the normal hierarchy (m₁ lightest) and the inverted hierarchy (m₃ lightest). That gives five objects in total, and the framework's machine-checked library of formal theorems proves the count is five by a direct computation.

The framework also defines a ratio between adjacent mass-squared splittings. It sets this ratio to φ², where φ is the golden ratio, approximately 1.618. The library proves φ² equals φ + 1, a standard identity, and that the ratio is positive. So the declaration establishes a structural claim: if the three masses sit on a ladder where each step's squared splitting is φ² times the previous one, then the enumeration of five states is complete and consistent.

In Recognition Science, this is a definitional choice, not a measurement. The framework models the neutrino sector as a five-state system with a golden-ratio splitting rule. It does not claim that measured neutrino masses actually follow this ladder, nor does it predict which hierarchy is realized in nature. The declaration is a formal scaffold: it says what the states are and what the ratio would be, and it proves the arithmetic is coherent.

What the declaration does not claim is just as important. It does not derive the neutrino masses from first principles, and it does not fit them to experimental data. The golden-ratio splitting is an assumption, not a theorem about the physical world. The framework's library proves the internal consistency of the model, but that proof does not make the model true. A reader should take this as a structural proposal, clearly labeled, not as a confirmed result in particle physics.

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

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

noncomputable def massSplitRatio : ℝ := phi ^ 2

What this page does not claim

The declaration does not claim that measured neutrino masses follow the golden-ratio ladder. The declaration does not predict which mass hierarchy nature realizes. The declaration does not derive neutrino masses 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.

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