Cosmology Neutrino Hierarchy From Phi Ladder

The neutrino ladder

Neutrinos are elementary particles that come in three flavors, and they have small but nonzero masses. Physicists measure the differences between the squares of these masses, not the masses themselves. The puzzle of the neutrino mass hierarchy asks whether the third mass is much heavier than the other two (normal hierarchy) or whether the first two are close together with the third lighter (inverted hierarchy). The standard model does not predict these masses; they are free parameters fixed by experiment.

In Recognition Science, the framework models this situation with a discrete ladder. The golden ratio φ, about 1.618, is the unique self-similar scaling that the framework's forcing chain derives. The framework models the three neutrino masses m₁ < m₂ < m₃ as sitting on this ladder, with the ratio of adjacent mass-squared splittings equal to φ², which is also φ + 1, about 2.618. This means the gaps between the squared masses grow by a fixed factor as you climb the ladder. The framework then adds the two known hierarchy scenarios, normal and inverted, as separate states.

Together, the three masses and the two hierarchy scenarios form a five-state structure. The framework's machine-checked library of formal theorems proves that this structure has exactly five states, that the splitting ratio equals φ + 1, and that this ratio is positive. The proof is complete: zero unproved assumptions and zero axioms beyond the ambient type theory. This is a structural enumeration, not a measurement. The framework does not predict which hierarchy nature chose; it certifies that the ladder plus the two scenarios closes into a five-state configuration.

The consequence is a compact combinatorial object where a cosmology question becomes a counting statement. The framework shows that the neutrino hierarchy question, phrased as a choice among three masses and two orderings, has a definite shape: five states, one ratio. This does not tell you which hierarchy is real, but it gives a precise target for experiment. If future measurements find a mass-squared splitting ratio that is not φ², that would falsify this particular ladder model.

noncomputable def massSplitRatio : ℝ := phi ^ 2

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

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 neutrinoHierarchyCert : NeutrinoHierarchyCert where
  five_states := neutrinoState_count
  split_ratio_phi_sq := massSplitRatio_eq
  split_ratio_pos := massSplitRatio_pos

What this page does not claim

The framework does not predict which neutrino hierarchy nature realizes. The φ² splitting ratio is a model choice, not a derived consequence of the forcing chain. No measurement of neutrino mass-squared splittings is cited or claimed as evidence.

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)

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