Standard Model Pmns Atmospheric Theta23 Rs

What the module proves

The PMNS matrix describes how neutrino flavors mix; its atmospheric angle theta_23 is measured near maximal, with sin^2(theta_23) close to 0.5. In Recognition Science, the framework models the cost of a recognition event with the function J(x) = (x + 1/x)/2 - 1. The module PMNS_Atmospheric_Theta23_RS.lean defines a domain cost, a discrete record of the recognition cost between two positive quantities m and e, as J(m/e).

What the machine-checked library of formal theorems proves is three general facts about that cost. First, when the two quantities are equal, the cost is zero. Second, for positive inputs the cost is never negative. Third, the threshold phi - 3/2 is positive, where phi is the golden ratio. These are all consequences of the general cost function J, and the proofs are formal.

In Recognition Science, the module does not establish that theta_23 equals 0.5. The docstring records a research note that maximal mixing would follow from mu-tau symmetry, but the formal code defines m and e without reference to neutrinos. The module proves no statement specific to the PMNS matrix. It is a template shared with 2383 sibling modules, and the cost facts are stated once, universally, in DomainCostTemplate.

What the module does establish, in plain language, is that the cost function used throughout the framework has the basic properties any sensible cost should have: zero when nothing changes, nonnegative when something does, and a positive threshold that separates regimes. These properties are proved from the definition of J, not assumed. The step from these general facts to a prediction about theta_23 remains open.

def domainCost (m e : ℝ) : ℝ := Jcost (m / e)

theorem domainCost_at_eq (r : ℝ) (h : r ≠ 0) : domainCost r r = 0 := by
  unfold domainCost; rw [div_self h]; exact Jcost_unit0

theorem domainCost_nonneg (m e : ℝ) (hm : 0 < m) (he : 0 < e) : 0 ≤ domainCost m e := by
  unfold domainCost; exact Jcost_nonneg (div_pos hm he)

theorem canonicalThreshold_pos : 0 < canonicalThreshold := by
  unfold canonicalThreshold; linarith [phi_gt_onePointFive]

theorem cert_inhabited : Nonempty PMNSTheta23Cert := ⟨cert⟩

What this page does not claim

The module predicts sin^2(theta_23) = 0.5. The module proves maximal mixing from mu-tau symmetry. The module derives the measured value of theta_23.

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/StandardModel/PMNS_Atmospheric_Theta23_RS.lean
expected axiom basis: [propext, Classical.choice, Quot.sound] (the Lean kernel's standard three; no RS-specific axioms)

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Derived articles

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