Cosmology Neutrino Hierarchy From Phi Ladder Mass Split Ratio Pos

The positive mass split

The neutrino mass hierarchy is the observed ordering of the three neutrino masses, from lightest to heaviest. In the standard model of particle physics, neutrinos have tiny but nonzero masses, and their squared-mass differences have been measured by solar and atmospheric neutrino experiments. The Recognition Science framework proposes that these masses lie on a ladder built from the golden ratio φ, the number approximately 1.618, which satisfies φ² = φ + 1. The declaration massSplitRatio_pos is a formal statement within that framework's machine-checked library of formal theorems: it proves that the proposed ratio of adjacent mass-squared splittings, defined as φ², is greater than zero.

What this establishes is mathematically modest but logically precise. The definition massSplitRatio sets the ratio to φ², and the theorem massSplitRatio_pos proves 0 < φ². This is not a measurement of neutrino masses, nor a prediction of their values. It is a consistency check: if one assumes the φ-ladder structure, then the ratio is a well-defined positive real number, which is a necessary condition for any physical mass-squared splitting. The proof itself is short, relying on the positivity of φ and the fact that a positive number raised to a power remains positive. The theorem also shows φ² equals φ + 1, a defining property of the golden ratio, so the ratio is not an arbitrary constant but one tied to the self-similar structure the framework derives elsewhere.

In Recognition Science, the framework models the neutrino sector as a discrete enumeration: three mass states plus two hierarchy scenarios (normal and inverted), totaling five configurations, which matches a structural dimension called configDim. The massSplitRatio_pos theorem is one component of a certificate structure that bundles the count, the ratio identity, and its positivity into a single object. But the certificate does not assert that the measured neutrino mass splittings actually follow this ratio. That would be an empirical claim requiring experimental data, which the framework does not provide here. The declaration only guarantees the internal consistency of the proposed model, not its agreement with nature.

The practical consequence is that the framework's neutrino hierarchy model is at least formally coherent: it does not stumble on a negative or undefined mass ratio. This matters because a proposed physical law must first be mathematically sound before it can be tested against experiment. The positivity proof removes one potential objection, but it leaves the central question open: whether the φ-ladder ratio actually matches the measured solar and atmospheric mass-squared differences. That comparison is not part of this declaration and would require a separate empirical study.

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

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

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

What this page does not claim

The theorem does not assert that measured neutrino masses follow the φ-ladder. The theorem does not provide any experimental data or empirical validation. The theorem does not prove that the ratio φ² is the unique possible mass-splitting ratio.

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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