en.wikipedia.org/wiki/Negative_temperature

This sentence misdescribes the key step of the Purcell–Pound experiment.

Credible summaries of the 1951 experiment say the magnetic field was reversed very rapidly, not merely removed:

• A National Academy of Sciences memoir on Robert V. Pound says the LiF crystal achieved a negative spin temperature by “switching the magnetic field instantaneously … from +10 gauss to -10 gauss” and explains that after this sudden reversal the magnetization became antiparallel to the new field.
• University of Maryland statistical-mechanics notes likewise summarize the classic experiment by saying “The field was then reversed so rapidly that the spins remained in their original orientation and found themselves aligned antiparallel with the magnetic field.”

That distinction matters physically: rapid field reversal is what produces the population inversion/antiparallel magnetization associated with the negative-temperature state. Saying the crystal was simply “removed from this field” describes a different procedure.

This sentence is incorrect for the very two-level canonical example being discussed.

For bounded two-level systems, standard canonical-ensemble treatments explicitly include a negative-temperature branch once the population is inverted. An Oregon State thermal-physics text gives the canonical internal energy of a two-level system as

[
U = \frac{N\Delta}{1+e^{\beta\Delta}}
]

and states that the region with energy fraction above 0.5 corresponds to negative temperature. University of Maryland notes say the same thing in words: as a two-level system is heated past the equal-population point, the temperature goes from +∞ to -∞ and then remains negative while the upper level becomes more populated.

That directly contradicts the claim that the canonical description "never need[s] to enter a negative temperature regime."

You can also see the problem from the article’s own formula for its 2-particle example:

[
E(T)=\frac{2e^{-\beta}+2e^{-2\beta}}{1+2e^{-\beta}+e^{-2\beta}}=\frac{2}{1+e^{\beta}}.
]

At positive temperature ((\beta>0)), this only gives energies between 0 and 1. As (T\to +\infty) ((\beta\to 0)), the energy approaches 1, the equal-population midpoint. To reach energies above 1—the upper half of the allowed energy range for this bounded system—you need (\beta<0), i.e. negative temperature.

So for a canonical two-level system, negative temperatures are not unnecessary; they are precisely what describe the inverted-population/high-energy half of the state space.

This overstates the required flip angle.

Standard NMR references say magnetization inversion is achieved with a 180° pulse:

• The IUPAC Gold Book defines an inversion recovery sequence as a spin-echo sequence preceded by a 180° inversion pulse.
• A University of Washington NMR text says plainly: “if we wish to invert magnetization we must use a 180° pulse.” The same text explains that the signal goes through a null at 180° and only after that becomes negative, which means angles greater than 180° are already past the inversion point rather than the defining condition for it.

So the article’s wording is inaccurate: NMR spin inversion is associated with a 180° pulse (and populations can already be inverted before exceeding 180°), not specifically with pulse widths over 180°.