predictions
RS predictions for neutrino mixing angles and phases are collected in this definition. A neutrino phenomenologist would reference these when testing Recognition Science against oscillation data. The list assembles results from 8-tick symmetry for maximal theta23, phi-connections for theta12, and Berry phase arguments for delta_CP.
claimThe Recognition Science predictions for the Pontecorvo-Maki-Nakagawa-Sakata matrix parameters are $θ_{23} ≈ 45°$ from eight-tick symmetry, $sin²θ_{12} ∈ [0.276, 0.307]$ from the golden ratio connection, $θ_{13}$ small from $φ$-hierarchy, $δ_{CP} ≈ π + O(φ^{-1})$ near 190-200°, and normal mass ordering preferred.
background
The PMNSMatrix module derives neutrino flavor mixing from RS, contrasting large angles with the CKM sector. Upstream, EightTick.phase defines the 8-tick phases as $kπ/4$ for $k=0..7$, while F2Power.axis1 and axis2 supply the weight-1 vectors $(true,false,false)$ and $(false,true,false)$ for Berry phase calculations. Constants.tick fixes the fundamental time quantum $τ_0=1$.
proof idea
This definition is a direct list literal of five qualitative statements. It draws the maximal $θ_{23}$ from eight-tick symmetry and the $sin²θ_{12}$ interval from phi-ladder scaling. The $δ_{CP}$ entry incorporates the structural Berry phase equality on axis1 and axis2 with torsion corrections from generation indices.
why it matters in Recognition Science
This definition supplies the observable targets for the PMNS derivation in StandardModel.PMNSMatrix. It connects to the eight-tick octave (T7) and phi fixed point (T6) from the forcing chain. The module targets a PRD paper on neutrino mixing from golden ratio geometry; quantitative precision of the torsion correction for $δ_{CP}$ remains open.
scope and limits
- Does not provide numerical fits to current experimental data.
- Does not derive the full PMNS matrix elements.
- Does not prove the predictions from first principles within this definition.
- Does not address inverted mass ordering possibilities.
formal statement (Lean)
252def predictions : List String := [
proof body
Definition body.
253 "θ₂₃ ≈ 45° from 8-tick symmetry",
254 "sin²θ₁₂ ≈ 0.276-0.307 from φ-connection",
255 "θ₁₃ small but nonzero from φ-hierarchy",
256 "δ_CP ≈ π + O(φ-1) ~ 190-200°",
257 "Normal mass ordering preferred"
258]
259
260/-! ## δ_CP(PMNS) from Q₃ Berry Phase — Structural Derivation
261
262In the CKM sector: δ_CKM = π/2 from the [4,2,2] Gray code Berry phase
263 Berry(gen1) = flipCount(axis0) × π/4 = 4π/4 = π
264 Berry(gen2) = flipCount(axis1) × π/4 = 2π/4 = π/2
265 δ_CKM = Berry(gen1) − Berry(gen2) = π/2
266
267In the PMNS (lepton) sector: neutrinos are in the axes-1 and axes-2 sub-space
268 Berry(ν_2) = flipCount(axis1) × π/4 = 2π/4 = π/2
269 Berry(ν_3) = flipCount(axis2) × π/4 = 2π/4 = π/2
270 Structural δ_CP(PMNS) = Berry(ν_2) − Berry(ν_3) = 0 [axes 1 and 2 are symmetric]
271
272The non-zero experimental δ_CP ≈ 197° ≈ π + π/9 comes from sub-leading
273corrections involving the generation torsion {0, 11, 17}. To leading order
274in torsion: δ_CP(PMNS) = π + Δτ₂₃/(Δτ₁₂) × (π/4) = π + (6/11) × (π/4) ≈ π + 0.428 ≈ 3.57 rad ≈ 204°.
275-/
276
277/-- The Berry phases for the neutrino sector are equal:
278 axis 1 and axis 2 both have flipCount = 2, giving the same Berry phase.
279 This is proved by the [4,2,2] Gray code structure. -/