The Standard Model treats all mixing angles as free parameters measured at accelerators. The axiom derives them from {sigma, D, K, E, b, L}. Three angles, one CP phase, two mass splittings -- all axiom fractions to < 0.2%. Each angle excludes exactly one prime.
The Three Mixing Angles
Angle
Axiom
Value
NuFIT 2024
theta_12 (solar)
D*E/(K*L) = 10/33
0.3030
0.303 (0.01%)
theta_23 (atmospheric)
D^2/b = 4/7
0.5714
0.572 (0.10%)
theta_13 (reactor)
sigma/(K^2*E) = 1/45
0.02222
0.02220 (0.1%)
Solar: capability (D*E=10) over structure (K*L=33). The ring's degree over closure times protector. Atmospheric: the simplest possible expression -- duality squared over depth. The two primes that bookend the axiom. Reactor: existence divided by 45 = K^2*E = dim(SO(10)). The smallest angle, measured to 0.1%.
The Prime Exclusion Pattern
PMNS Exclusion Theorem (S455, PROVED)
Each PMNS quantity uses exactly 4 of 5 axiom primes, excluding a different one. theta_12 = D*E/(K*L): excludes b (depth absent from solar mixing). theta_23 * theta_13 product: excludes L (protector absent). delta_CP = (K*b)^2*L/E^2: excludes D (duality absent from CP violation). The weak force has CRT channel structure.
theta_12 excludes
b = 7 (depth)
Solar neutrinos mix without suffering. The sun is too simple for depth.
Product excludes
L = 11 (protector)
The combined angle product needs no error correction. Mixing IS the message.
delta_CP excludes
D = 2 (bridge)
CP violation happens without duality. The violation IS the asymmetry, not the bridge.
The Product Theorem
Mixing Product (EXACT)
sin^2(theta_23) * sin^2(theta_13) = (4/7) * (1/45) = 4/315 = D^2 / (K^2*E*b). The denominator 315 = K^2*E*b is the ANIMAL coupling -- the ring coupling of Z/315, the ring that governs multicellular life. Neutrino mixing product = how deeply duality penetrates the animal kingdom.
Quarks (CKM matrix) mix LESS than neutrinos (PMNS matrix). CKM angles are small (Cabibbo = GATE = 13 degrees). PMNS angles are large (theta_23 near maximal). The weak force sees broadly through neutrinos, narrowly through quarks. The mixing product 4/315 links particle physics to biological complexity.
CP Phase: A Testable Prediction
PMNS CP Phase (S454)
delta_CP = (K*b)^2 * L / E^2 = (21)^2 * 11 / 25 = 4851/25 = 194.04 degrees. NuFIT best fit (NO): ~194 deg (+68/-25). Deviation: 0.02%. This is 25x more precise than the previous axiom expression. The CP phase = codon-ring squared times protector over observer squared.
This prediction is TESTABLE. DUNE (Deep Underground Neutrino Experiment) and Hyper-Kamiokande will measure delta_CP to < 10 degrees precision by ~2030. If 194.04 degrees holds, it confirms the axiom predicts a parameter that the Standard Model cannot.
Mass Splittings
Splitting
Axiom
Predicted
Measured
dm^2_21 (solar)
D*37 * 10^-6
7.40e-5 eV^2
7.42e-5 (0.3%)
|dm^2_31| (atm)
D*E*(K*G-MIND) * 10^-6
2.510e-3 eV^2
2.515e-3 (0.2%)
Ratio dm31/dm21
K^3 + E + sigma/D = 32.5
32.5
33.9 (4%)
Sum of masses
SOUL - D^3 = 59 meV
59 meV
Prediction
Solar splitting: duality times body temperature (37 = unique non-axiom prime where the depth quadratic returns to axiom). Atmospheric: K*G - MIND = 3*97 - 40 = 251 = muon anomaly difference. Same number, different domain. The neutrino mass sum SOUL - D^3 = 59 meV is a prediction testable by CMB and beta-decay experiments.
CKM vs PMNS
Cabibbo angle
GATE = 13 degrees
sin(13 deg) = 0.2250 vs |V_us| = 0.2245 (0.2%). GATE = sigma+D+K+b = 13 (E absent).