Cosmology Neutrino Hierarchy From Phi Ladder Neutrino Hierarchy Cert

The neutrino hierarchy certificate

Neutrinos are elementary particles that come in three types, or flavors, and each type has a mass. Physicists know these masses differ, but the exact values and ordering are not fully pinned down. The two leading arrangements are called the normal hierarchy, where the masses increase in a known pattern, and the inverted hierarchy, where the ordering is reversed. The question of which one nature uses is one of the open problems in particle physics.

Within the Recognition Science (RS) framework, a formal object called NeutrinoHierarchyCert packages a specific claim about these masses. The claim is that the three neutrino masses lie on a ladder based on the golden ratio phi, approximately 1.618. Specifically, the ratio of adjacent mass-squared splittings equals phi squared, which is the same as phi plus one. The certificate also counts the total number of structural states: three masses plus two hierarchy scenarios, for a total of five. All of this is encoded in a machine-checked library of formal theorems, meaning the logical structure is verified by a computer.

What the certificate actually establishes is mathematical, not physical. It proves, within its formal system, that the number of states is five, that the splitting ratio is positive, and that the ratio equals phi plus one. These are theorems about a defined structure, not measurements of the real world. The certificate does not prove that real neutrinos follow this pattern. The connection between the formal ladder and actual neutrino masses is a hypothesis, and it predicts that the mass-squared splitting ratio will be measured at phi squared. The falsifier is a precise measurement of the three neutrino masses that shows the ratio is not phi squared.

The value of the certificate is that it makes the RS claim precise and checkable. It turns a vague idea into a concrete statement that can be tested against data. It also demonstrates the framework's method: define a structure, prove internal consistency, and then compare against experiment. The physics remains open, but the formal groundwork is laid.

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

What this page does not claim

The certificate does not claim that real neutrinos follow the phi-ladder pattern. The certificate does not establish which hierarchy, normal or inverted, nature uses. The certificate does not provide measured values for the neutrino masses.

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)

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

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