Foundation Hierarchy Dissolution Hierarchy Dissolution Implies Rung Law

The rung mass law

The Standard Model of particle physics carries about two dozen free parameters, and among the most puzzling is the mass of the Higgs boson. Quantum corrections should drag it up to the Planck scale, roughly 10¹⁹ GeV, unless some mechanism holds it near 125 GeV. This mismatch is the hierarchy problem, and it has driven decades of speculative physics.

Recognition Science dissolves the problem by changing where masses come from. In its account, each particle's mass is set by a position on a discrete ladder of rungs, not by radiative corrections from loop integrals. The framework proves a theorem, hierarchy_dissolution_implies_rung_law, that states the mass on any rung r equals a fixed reference energy E_coh times phi raised to the power r. Here phi is the golden ratio, about 1.618, and E_coh is a framework-internal constant. The theorem is a formal identity in the machine-checked library of formal theorems: if a mass sits on a rung in the framework's mass hierarchy, then it obeys this geometric law.

The content of the theorem is deliberately narrow. It does not prove that the mass spectrum is geometric; that is a separate result, mass_ratio_geometric, which shows the muon-to-electron mass ratio equals phi to the 11th power. The rung law theorem takes the geometric form as a hypothesis and returns it as a conclusion. Its role is to make explicit that the framework's mass hierarchy is a geometric ladder, and that this structure, not renormalization, is what sets particle masses.

The framework's library also records what remains open. The mass function has no cutoff argument, meaning it has no dependence on an ultraviolet energy scale. But absence from a function signature is not yet a theorem comparing different regularizations. A proof that the mass is constant under changes of cutoff would require a cutoff-indexed mass model and a separate argument. That comparison remains a target, not a proved result.

/-- Hierarchy-dissolution structure implies the rung mass law. -/
theorem hierarchy_dissolution_implies_rung_law (r : ℤ)
    (h : Masses.MassHierarchy.mass_on_rung r = Masses.Anchor.E_coh * phi ^ r) :
    Masses.MassHierarchy.mass_on_rung r = Masses.Anchor.E_coh * phi ^ r :=
  h

/-- Hierarchy-dissolution structure implies the rung mass law. -/
theorem hierarchy_dissolution_implies_rung_law (r : ℤ)
    (h : Masses.MassHierarchy.mass_on_rung r = Masses.Anchor.E_coh * phi ^ r) :
    Masses.MassHierarchy.mass_on_rung r = Masses.Anchor.E_coh * phi ^ r :=
  h

/-- In RS, fermion mass ratios are geometric (powers of φ), not free parameters.
    This is the structural basis for hierarchy dissolution. -/
theorem mass_ratio_geometric :
    Masses.MassHierarchy.mass_on_rung (r_lepton "mu") / Masses.MassHierarchy.mass_on_rung (r_lepton "e") =
      phi ^ 11 :=
  Masses.MassHierarchy.lepton_hierarchy_geometric.1

/-- Hierarchy-dissolution structure implies the rung mass law. -/
theorem hierarchy_dissolution_implies_rung_law (r : ℤ)
    (h : Masses.MassHierarchy.mass_on_rung r = Masses.Anchor.E_coh * phi ^ r) :
    Masses.MassHierarchy.mass_on_rung r = Masses.Anchor.E_coh * phi ^ r :=
  h

What this page does not claim

The hierarchy problem is not proved absent; the framework's dissolution is a structural claim, and the cutoff-independence comparison remains open. The rung law theorem does not by itself prove the mass spectrum is geometric; it states the consequence of the geometric hypothesis. No claim is made that the framework's mass values match measured particle masses; that comparison 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/Foundation/HierarchyDissolution.lean
expected axiom basis: [propext, Classical.choice, Quot.sound] (the Lean kernel's standard three; no RS-specific axioms)

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