Neutrino mixing, scored against NuFIT. Tension included.
Recognition Science writes the three neutrino mixing angles and the CP phase as fixed φ- and
α-expressions, preregistered rather than fitted: sin²θ₁₃ = φ⁻⁸,
sin²θ₁₂ = φ⁻² − 10α, sin²θ₂₃ = ½ + 6α,
δCP = π + (6/11)(π/4) ≈ 204.5° (the scorecard object
deltaCP_pmns_torsion_correction, with a proved range theorem; the same module also still carries
predicted_deltaCP ≈ 191°). This script scores all four against NuFIT 5.2 with χ²,
in both θ₂₃ octants. Two corrections this page owes its own history: an earlier summary line
contradicted its own table, and an earlier δCP value (11/8·π, never formalized in Lean)
has been retired in favor of the scorecard prediction. The remaining tension is the content: θ₂₃
wants the upper octant, and DUNE and JUNO will settle it.
φ-harmonics, no fit
Not falsified. Under real pressure.
θ₁₃ sits within 1.5σ: the strongest hit. θ₁₂ is comfortable at 0.5σ, and the Lean δCP lands 0.45σ from the NuFIT center (which still carries a large error). θ₂₃ is the pressure point: the prediction needs the upper octant, and the octant is experimentally unresolved (the two octant fits disagree at +5.6σ vs −3.5σ).
Written into the script
sin²θ₁₃ outside [0.0190, 0.0235]; a definitive lower octant for θ₂₃; or DUNE finding δCP far from 204.5° at design precision, or anywhere outside (π, 2π) where the Lean range theorem puts it. Each one alone kills this formulation.
The pulls, drawn
Each bar is the deviation of one prediction from NuFIT 5.2 in standard deviations. The two θ₂₃ bars are the same prediction scored against the two octant fits; the experiment has not yet chosen between them.
χ² table (upper octant; δCP octant-independent)
| Observable | RS | NuFIT | z | χ² |
|---|---|---|---|---|
| sin²θ₁₃ | 0.02129 | 0.02225 ± 0.00066 | −1.46 | 2.13 |
| sin²θ₁₂ | 0.30899 | 0.303 ± 0.012 | +0.50 | 0.25 |
| sin²θ₂₃ | 0.54378 | 0.572 ± 0.008 | −3.53 | 12.44 |
| δCP/π = 25/22 (Lean) | 1.13636 | 1.08 ± 0.125 | +0.45 | 0.20 |
| δCP/π = 11/8 (retired, not in Lean) | 1.375 | 1.08 ± 0.125 | +2.36 | – |
Run it yourself
$ python3 pmns_chi2.py Angles only: χ² = 14.82 for 3 dof, p = 0.0020 Full test: χ² = 34.21 for 4 observables, p = 0.000 INTERPRETATION: • The mixing angles show tension (p = 0.002), dominated by the θ₂₃ octant; the octant is experimentally unresolved and DUNE will decide it • The θ₁₃ = φ⁻⁸ prediction is the strongest (within ~1.5σ) • The Lean δCP prediction (25/22·π ≈ 204.5°) sits ~0.5σ from NuFIT VERDICT: RS PMNS predictions are NOT YET FALSIFIED but face tension on the θ₂₃ octant. Future precision experiments (DUNE, JUNO) will be decisive.
HYPOTHESIS with named falsifiers. The angle formulas (φ⁻⁸,
φ⁻² − 10α, ½ + 6α) are CONJECTURE/MODEL in
MixingDerivation; Lean then proves absolute-error match certificates to PDG centers
(pmns_theta*_match), packaged as PARTIAL_THEOREM in PMNSScoreCard, not uniqueness of
the formulas. The δCP value π + (6/11)(π/4) is the scorecard /
SMCompletenessCert object (PMNSMatrix.deltaCP_pmns_torsion_correction, proved range
π < δCP < 2π); the same module also still defines
predicted_deltaCP = π + (φ−1)π/10 ≈ 191°, so the library is ambiguous at
the definition layer and only the scorecard prefers torsion. The torsion derivation narrative is MODEL-tier.
The α entering the angles is the RS construction value via the canonical exponential pipeline, which
itself misses CODATA by 5.6 ppm (see the α lab); the exact fine-structure constant remains OPEN. A
correction this page owes you: an earlier version tested δCP/π = 11/8, a script-level
reading that was never formalized; it sat 2.4σ high and has been retired for the scorecard value, which
sits at 0.45σ. Current data put real pressure on θ₂₃. This page exists to show the
pressure, not to spin it. A definitive lower-octant θ₂₃ or a precise δCP
away from 204.5° ends this formulation, and we will report that here if it happens.