= Solution
For <adiabatic process>[adiabatic] collapse, $T\rho^{1-\gamma}$ is constant. With $\gamma=5/3$,
$$
T\propto\rho^{2/3},
\qquad
M_J\propto T^{3/2}\rho^{-1/2}\propto\rho^{1/2}.
$$
The rising <Jeans mass> produces <adiabatic suppression of fragmentation>: smaller subregions become more pressure-supported as density increases.
For <isothermal fragmentation>, $T$ stays approximately constant, and
$$
M_J\propto\rho^{-1/2}.
$$
The instability scale then falls during collapse, allowing hierarchical fragmentation until cooling fails, opacity rises, or another source of support intervenes.
Primordial metal-free gas cools inefficiently, principally through molecular hydrogen, and remains relatively hot. It therefore has a larger <Jeans mass> and tends toward a top-heavy <initial mass function> of massive <Population III stars>. Metal lines and dust let enriched gas remain cool to higher density, so <Population II stars> extend to much lower birth masses.
These alternatives map directly onto <black-hole seed> channels. Massive Population III remnants produce light <Population III remnant black-hole seeds>. If cooling and fragmentation are strongly suppressed while a primordial halo supplies rapid inflow, near-monolithic collapse can produce a heavy <direct-collapse black-hole seed>. Intermediate cooling and fragmentation in a dense cluster can instead permit a <runaway stellar-collision black-hole seed>. The Jeans argument selects plausible mass scales; angular momentum, feedback, chemistry, and accretion determine which channel actually operates.
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