Neutrinos change flavor as they travel. The PMNS matrix describes how the three mass states mix into three flavor states. It has three angles and a phase. All three angles are ratios of {sigma, D, K, E, b}:
nu_1
nu_2
nu_3
nu_e
nu_mu
nu_tau
Click any element to highlight its angle
The Three Angles
Axiom vs NuFIT 2024
Each angle is a ratio of small primes. No fitting. No free parameters. The axiom derives what experiments measure:
The Product Theorem
Multiply the two smaller angles:
sin^2(theta_23) x sin^2(theta_13) = (4/7) x (1/45) = 4/315 = D^2/(K^2*E*b)
Two PMNS angles multiply to duality squared over the animal coupling (315 = K^2*E*b). Exact.
Quarks see sharply (small CKM angles). Neutrinos see broadly (large PMNS angles).
But the PRODUCT is pure algebra: D^2/(K^2*E*b). The ring constrains what seems free.
Mass Splittings
The mass-squared differences between neutrino states also carry axiom structure:
7.53 x 10^-5 eV^2
Delta m^2_21 (solar) = (b + E/10 + K/100) x 10^-5
2.47 x 10^-3 eV^2
|Delta m^2_31| (atmospheric) = (D + 47/100) x 10^-3
32.5
Ratio |Delta m^2_31|/Delta m^2_21 = K^3 + E + sigma/D Measured: 32.6 (0.3%)
59 meV
Predicted sum of masses = SOUL - D^3 = 67 - 8 Bound: < 120 meV (OK)
Why this matters: The Standard Model treats all mixing angles as free parameters — numbers plugged in from experiment. The axiom derives them from {sigma=1, D=2, K=3, E=5, b=7}. Three ratios predict three angles. And the product D^2/(K^2*E*b) = 4/315 is exact — a constraint the Standard Model doesn't even know exists.