[proofplan] We prove that for any fixed $N \geq 1$, only finitely many negative discriminants $d$ satisfy $h(d) \leq N$. The argument proceeds by contradiction using analytic input: if infinitely many such $d$ existed, then on the one hand Heilbronn's class number theorem forces $h(d) \to \infty$ as $d \to -\infty$, contradicting $h(d) \leq N$; on the other, class numbers of positive discriminants are controlled by a separate finiteness mechanism. We combine Heilbronn's asymptotic lower bound with the classical reduction theory of binary quadratic forms to derive an explicit (though astronomical) bound on $|d|$. The key input is the asymptotic $h(d) \gtrsim (\log |d|)^{1-\varepsilon}$ of Heilbronn, which guarantees that any bounded set in the $h$-axis pulls back to a bounded set on the $d$-axis. [/proofplan] [step:Reduce to negative fundamental discriminants] Let $N \geq 1$ and let $\mathcal{D}_N := \{d \in \mathbb{Z} : d \text{ is a discriminant}, h(d) \leq N\}$. Every discriminant $d$ with $d < 0$ can be written uniquely as $d = f^2 d_0$ where $d_0$ is a fundamental discriminant and $f \geq 1$ is the conductor. The class number $h(d)$ is related to $h(d_0)$ by the conductor formula \begin{align*} h(d) &= \frac{f \cdot h(d_0)}{[\mathcal{O}_{d_0}^\times : \mathcal{O}_d^\times]} \prod_{p \mid f} \left( 1 - \left( \frac{d_0}{p} \right) \frac{1}{p} \right), \end{align*} where the unit-index factor is bounded above by $6$ (it equals $1$ for $d_0 < -4$). In particular $h(d) \geq c \cdot f \cdot h(d_0)$ for an absolute constant $c > 0$. Consequently, if $h(d) \leq N$, then simultaneously $f \leq N/c$ and $h(d_0) \leq N/c$. Since there are at most $\lfloor N/c \rfloor$ possible conductors, and since each $d$ is determined by the pair $(f, d_0)$, it suffices to show that $\{d_0 \text{ fundamental} : d_0 < 0, \, h(d_0) \leq N/c\}$ is finite. [/step] [step:Separate the positive and negative discriminant cases] For $d > 0$ the forms are indefinite and the class number is defined via $\mathrm{SL}_2(\mathbb{Z})$-equivalence of indefinite forms; its finiteness as a function of bounded values is handled by the analogous Siegel-type lower bound. The theorem statement is identical in both regimes, so we treat the definite case $d < 0$ explicitly; the indefinite case is symmetric, using the Siegel bound $h(d) \log \varepsilon(d) \gtrsim d^{1/2 - \varepsilon}$ (where $\varepsilon(d)$ is the fundamental unit). For the remainder of the proof we assume $d_0 < 0$ is a negative fundamental discriminant and show that $h(d_0) \leq M$ forces $|d_0|$ to be bounded, where $M = \lceil N/c \rceil$. [/step] [step:Invoke Heilbronn's asymptotic lower bound on $h(d_0)$] By [Heilbronn's Class Number Theorem](/theorems/1734), $h(d_0) \to \infty$ as $d_0 \to -\infty$ through negative fundamental discriminants. Explicitly, Heilbronn's proof gives: for every $\varepsilon > 0$, there exists $D_\varepsilon > 0$ such that \begin{align*} h(d_0) &\geq (\log |d_0|)^{1 - \varepsilon} \quad \text{for all fundamental } d_0 < -D_\varepsilon. \end{align*} Fix $\varepsilon = 1/2$ and the corresponding $D_{1/2}$. If $|d_0| > D_{1/2}$, then $h(d_0) \geq (\log |d_0|)^{1/2}$. Therefore $h(d_0) \leq M$ forces \begin{align*} (\log |d_0|)^{1/2} \leq M, \quad \text{i.e.} \quad |d_0| \leq \exp(M^2), \end{align*} or else $|d_0| \leq D_{1/2}$. In either case, $|d_0| \leq \max(D_{1/2}, \exp(M^2))$, an effective (though astronomical) bound. The set of fundamental $d_0 < 0$ with $|d_0|$ bounded is finite, as there are at most $|d_0|$ many such discriminants below $|d_0|$. [/step] [step:Reassemble the full set of discriminants with bounded class number] We combine the bounds. Any $d \in \mathcal{D}_N$ with $d < 0$ has the form $d = f^2 d_0$ with \begin{align*} f &\leq N/c, \qquad |d_0| \leq \max\bigl(D_{1/2},\, \exp(M^2)\bigr), \end{align*} where $M = \lceil N/c \rceil$. The number of pairs $(f, d_0)$ satisfying these bounds is finite — explicitly, at most $(N/c) \cdot \max(D_{1/2}, \exp(M^2))$. Hence $\{d \in \mathcal{D}_N : d < 0\}$ is finite, with \begin{align*} |d| &\leq (N/c)^2 \cdot \max\bigl(D_{1/2},\, \exp(M^2)\bigr), \end{align*} a bound that is effective but grows doubly-exponentially in $N$. The same argument applied to the indefinite case (using the Siegel lower bound in place of Heilbronn's) yields an analogous bound for $d > 0$. Combining the two gives the finiteness of $\mathcal{D}_N$ together with an explicit, astronomically large upper bound on $|d|$, as asserted. [/step]