The science
A pure qubit state is |psi> = alpha|0> + beta|1> with |alpha|^2 + |beta|^2 = 1. Its Bloch vector is
x = 2 Re(alpha* beta) y = 2 Im(alpha* beta) z = |alpha|^2 - |beta|^2
Rn(t) = exp(-i t n.sigma / 2) rotates the Bloch vector by angle t about axis n
H = rotation by pi about (x+z)/sqrt2, S = Rz(pi/2), T = Rz(pi/4) (up to a global phase)
Born rule: P(0 in Z) = |alpha|^2, P(0 in X) = |alpha+beta|^2/2, P(0 in Y) = |alpha - i beta|^2/2
Independent checks. (1) Unitarity: the norm stays 1 after every gate. (2) The state is evolved with 2x2 complex matrices, and separately the Bloch vector is rotated with Rodrigues formula; the two must agree. (3) Measurement shots are sampled from the Born probability and the frequency must lie within 3 standard errors sqrt(p(1-p)/N) of it (one run in about 300 will show a NOTE by chance).
Simplifications: a single ideal qubit, no decoherence or gate error; each batch of shots measures fresh identical copies of the state (preparation is repeated), so the displayed state is not collapsed; global phase is ignored because it is unobservable. The 3D view is a plain orthographic projection with front/back shading.