Cosmology Neutrino Hierarchy From Phi Ladder Neutrino State Count

Counting the neutrino states

Neutrinos come in three mass states, usually labeled m₁, m₂, m₃, ordered from lightest to heaviest. Beyond those three, physicists also consider two possible mass orderings: the normal hierarchy, where m₃ is the heaviest, and the inverted hierarchy, where m₃ is the lightest. So the full structural enumeration, three masses plus two orderings, gives five distinct possibilities.

The Recognition Science framework encodes this enumeration in a machine-checked library of formal theorems. Its declaration neutrinoState_count proves, by direct computation, that the number of these five states is exactly 5. The same library also defines the mass-splitting ratio between adjacent masses as φ², where φ is the golden ratio, about 1.618. Since φ² equals φ + 1, the ratio is about 2.618. A separate theorem proves this ratio is positive, which is a formal way of saying the masses are ordered and distinct.

What the declaration does not claim matters as much as what it proves. neutrinoState_count only counts the states; it does not say which hierarchy is realized in nature. The normal and inverted scenarios are both included in the count, and the theorem is silent on which one is correct. The framework also does not derive the actual neutrino masses, only the ratio of their squared splittings. The golden-ratio spacing is a structural assumption, a definitional choice, not a measured value or a derived law.

The practical use of this theorem is organizational. It gives a compact, machine-checked way to refer to the five possibilities when building larger arguments. A physicist reading the formal result learns that the framework's neutrino model has exactly five configurational states, and that the mass-splitting ratio is fixed to φ². What remains open is the empirical question of which hierarchy the real universe follows, a question the formal theorem does not touch.

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

noncomputable def massSplitRatio : ℝ := phi ^ 2

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

What this page does not claim

The theorem does not determine which neutrino mass hierarchy is the real one. The framework does not derive the absolute values of the neutrino masses, only the ratio of their splittings. The φ² spacing is a definitional choice, not a measured or independently derived law.

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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