Two-flavour oscillation: a neutrino born as flavour a is a superposition of two mass states with different phase velocities. The survival probability is P(a→a) = 1 − sin²2θ · sin²(1.267 Δm²[eV²] L[km] / E[GeV]) and the oscillation length Losc = πE / (1.267 Δm²). The check evolves i dpsi/dL = lambda [[-cos2θ, sin2θ],[sin2θ, cos2θ]] psi, lambda = 1.267 Δm²/E, with RK4 and compares it to the closed form.

Real beams are not monoenergetic: the energy-smearing button averages P over a 15 % wide energy band, which washes out fast oscillations, as in reactor and atmospheric data. Real neutrinos have three flavours, CP phase and matter effects; none is modelled here.

Halo: a circular-orbit speed follows v² = G M(<r) / r (spherical approximation). The halo is an NFW profile rho = rhos / [x(1+x)²] with M(<r) = M₂₀₀ m(x)/m(c), m(x) = ln(1+x) − x/(1+x), r₂₀₀ from 200 times the critical density (136 M☉ kpc³, h = 0.7). The disc is the spherical enclosed mass of an exponential disc, a crude approximation. A disc alone gives a falling curve; the halo keeps it nearly flat, the classic dark-matter evidence. The halo here is a toy tied to the 3D scene, not a fit to a real galaxy.