Physics Quark Mass Hierarchy From Phi Ladder

The φ-ladder mass scale

The standard model of particle physics recognises six quark flavours: up, down, strange, charm, bottom, and top. They group naturally into three generations, each containing one up-type and one down-type quark. The masses of these quarks span roughly five orders of magnitude, from the light up quark at about 2.2 MeV/c² to the heavy top quark at about 173 GeV/c². A central question in particle physics is whether this hierarchy of masses follows any simple pattern.

Recognition Science offers one such pattern: a geometric ladder where each flavour's mass is φ times the mass of the previous flavour. The golden ratio φ, approximately 1.618, is the unique positive solution to the equation r² = r + 1. On this ladder, if the lightest quark has mass m₀, then the next has mass m₀·φ, the next m₀·φ², and so on. The ratio between any two adjacent rungs is exactly φ.

In Recognition Science, the framework models quark masses as powers of φ. The definition sets quarkMass(k) = φ^k for a natural number k. The framework proves, in its machine-checked library of formal theorems, that the ratio of successive masses is exactly φ, and that every mass is positive. It also proves that there are exactly six quark flavours, and that this six equals three generations times two charge types. These are structural facts about the ladder, not empirical measurements.

The framework's contribution is to show that the six flavours fit a single geometric scale with one ratio, φ. This is a definitional choice, not a derivation from first principles. The framework does not predict the absolute masses, nor does it explain why the top quark is so much heavier than the up quark in absolute terms. It only asserts that, on this ladder, the relative spacing is uniform. The framework's library proves the internal consistency of this ladder: the count of flavours, the partition into generations and charge types, the ratio between adjacent masses, and the positivity of all masses. These four facts together form a certificate that the φ-ladder description is coherent.

The practical consequence is a compact way to think about the quark mass hierarchy: one number, φ, replaced the six seemingly arbitrary mass values with a single ratio. Whether this ratio matches the measured masses is an empirical question, not a theorem. The framework's ladder is a structural model, and the measured masses are a separate check.

noncomputable def quarkMass (k : ℕ) : ℝ := phi ^ k

theorem quarkMass_ratio (k : ℕ) : quarkMass (k + 1) / quarkMass k = phi := by
  unfold quarkMass
  have hpos : (0 : ℝ) < phi ^ k := pow_pos phi_pos k
  rw [div_eq_iff hpos.ne', pow_succ]
  ring

theorem quarkFlavour_count : Fintype.card QuarkFlavour = 6 := by decide

/-- 6 = 3 generations × 2 charge types. -/
theorem six_partition : 6 = 3 * 2 := by decide

theorem quarkMass_pos (k : ℕ) : 0 < quarkMass k := pow_pos phi_pos k

What this page does not claim

The φ-ladder is not a derivation of quark masses from first principles; it is a definitional model. The framework does not predict the absolute masses of the quarks, only their relative spacing on the ladder. No claim is made that the measured quark masses exactly follow the φ ratio; that is an empirical check, not a theorem.

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/QuarkMassHierarchyFromPhiLadder.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: