Physics Neutrino Mass From Phi Ladder

The phi ladder

Neutrinos are elementary particles that come in three known types, or flavors: electron, muon, and tau. They have tiny but nonzero masses, and the three masses are not equal. The lightest is m₁, the middle m₂, and the heaviest m₃. In the standard model of particle physics, these masses are free parameters: experiment measures them, but theory does not predict them.

In Recognition Science (RS), the framework models the three neutrino masses as a geometric ladder based on the golden ratio φ ≈ 1.618. The model defines m_k = φ^k for k = 0, 1, 2. So m₁ = φ⁰ = 1, m₂ = φ¹ = φ, and m₃ = φ² = φ + 1 (since φ² = φ + 1). The ratio between adjacent masses is constant: m₂/m₁ = m₃/m₂ = φ. This is the phi ladder. The framework predicts the normal ordering, meaning m₁ < m₂ < m₃, which is consistent with current experimental preference.

The cost function in Recognition Science leads to this ladder, and the ledger of formal results includes two key facts. First, the normal ordering theorem: for any k, m_k < m_{k+1}. Second, the mass ratio theorem: m_{k+1}/m_k = φ. Both are proved from the definition using the identity φ² = φ + 1, with no unproved assumptions. The results are also packaged into a certificate structure that bundles the ordering and ratio properties.

The framework's claim is not that it predicts the absolute scale of neutrino masses. The ladder gives ratios, not absolute values. The numbers are in arbitrary units; the framework does not say what m₁ is in electronvolts. What is established is a structural prediction: if neutrino masses lie on this ladder, then the adjacent ratios are exactly φ and the ordering is normal. This is a clean, testable consequence, though the connection to measured masses is an empirical check, not a theorem.

The phi ladder also connects to a broader RS idea: the number 5 appears as a configuration dimension. The framework notes that extending to five flavors, including sterile neutrinos, matches this dimension. But that extension is not formalized; it remains a stated idea, not a proved theorem.

For a reader, the practical takeaway is simple: RS offers a concrete, falsifiable prediction for neutrino mass ratios. If future experiments measure the ratio m₂/m₁ and find it close to 1.618, that would support the ladder. If they find a different ratio, the ladder would be falsified. The recognition of this pattern is made precise and machine-checked.

/-- Three active neutrino mass eigenstates. -/
noncomputable def neutrinoMass (k : ℕ) : ℝ := phi ^ k

/-- Mass ratio between adjacent eigenstates = phi. -/
theorem mass_ratio (k : ℕ) : neutrinoMass (k + 1) / neutrinoMass k = phi := by
  unfold neutrinoMass
  have hpos := pow_pos phi_pos k
  rw [pow_succ, div_eq_iff hpos.ne']
  ring

/-- Normal ordering: m_1 < m_2 < m_3. -/
theorem normal_ordering (k : ℕ) : neutrinoMass k < neutrinoMass (k + 1) := by
  unfold neutrinoMass
  have hphi_sq : phi ^ 2 = phi + 1 := phi_sq_eq
  have hpos := pow_pos phi_pos k
  rw [pow_succ]
  linarith [mul_lt_mul_of_pos_left one_lt_phi hpos]

noncomputable def neutrinoMassCert : NeutrinoMassCert where
  normal_ordering := normal_ordering
  phi_ratio := mass_ratio

What this page does not claim

The absolute scale of neutrino masses is not predicted by this module. The five-flavor extension is not formalized as a theorem. The phi ladder is not claimed to be the only possible mass pattern.

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/Physics/NeutrinoMassFromPhiLadder.lean
expected axiom basis: [propext, Classical.choice, Quot.sound] (the Lean kernel's standard three; no RS-specific axioms)

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