**Abstract** : The Stern-Gerlach experiment is notoriously counter-intuitive. The official theory is that the spin of a fermion remains always aligned with the magnetic field. Its directions
are thus quantized: It can only be spin up or down.
But that theory is based on mathematical errors in the way it (mis)treats spinors and group theory.
We present here a mathematically rigorous theory for a fermion in a magnetic field, which is no longer counter-intuitive.
It is based on an understanding of spinors in SU(2) which is only Euclidean geometry.
Contrary to what Pauli has been reading into the Stern-Gerlach experiment,
the spin directions are not quantized.
The new corrected paradigm, which solves all conceptual problems, is that the fermions precess around the magnetic-field just like
Einstein and Ehrenfest had conjectured.
Surprizingly this leads to only two energy states, which should be qualified as precession-up and precession-down rather than
spin-up and spin down.
Indeed, despite the presence of the many different possible angles $\theta$ between
the spin axis ${\mathbf{s}}$ and the magnetic field ${\mathbf{B}}$, the fermions can only have two possible energies $m_{0}c^{2}\pm\mu B$.
The values $\pm\mu B$ do thus not correspond to the continuum of values $-{\boldsymbol{\mu\cdot}}{\mathbf{B}}$ Einstein and Ehrenfest had conjectured.
The energy term $V= -{\boldsymbol{\mu\cdot}}{\mathbf{B}}$ is a macroscopic quantity. It is a statistical average over a large ensemble of fermions distributed
over the two microscopic energy states $\pm\mu B$,
and as such not valid for individual fermions. The two fermion states $\pm\mu B$ are not potential-energy states.
We also explain the
mathematically rigorous meaning of the up and down spinors. They represent left-handed and right-handed reference frames, such that now everything is intuitively clear
and understandable in simple geometrical terms. The paradigm shift does not affect the Pauli principle.