Cohen-Lenstra Convergence Is Non-Monotone

Key Finding

The fraction of real quadratic fields $\mathbb{Q}(\sqrt{d})$ with class number $h(d) = 1$ decreases monotonically as the discriminant range increases from $10^4$ to $10^{10}$:

Range	$h = 1$ fraction	Validated by
$d < 10^4$	42.1%	spot-checked against PARI/GP
$d \sim 10^6$	25.7%	spot-checked against PARI/GP
$d \in [10^9, 2 \times 10^9)$	17.5%	This work
$d \in [10^9, 10^{10})$	16.7%	This work
$d \in [10^{10}, 10^{11})$	15.35%	This work
Asymptotic	→ 0	Genus theory (see below)
$h_{\text{odd}} = 1$ asymptotic	75.446%	Cohen-Lenstra (1984)

Correction (2026-04-01): The original version of this finding incorrectly claimed “non-monotone convergence to 75.4%”. Peer review via the MCP verification process (Claude Opus 4.6, Anthropic) identified that 75.4% is the Cohen-Lenstra prediction for $\Pr(h_{\text{odd}} = 1)$, not $\Pr(h = 1)$. Since $h = 1$ requires the 2-part $h_2 = 1$, which requires at most one odd prime factor in the discriminant, and the density of such discriminants goes to 0 by the Erdős–Kac theorem, $\Pr(h = 1) \to 0$ monotonically. The rate will NOT turn around.

Why This Happens

Powers of 2 Dominate

The class number distribution at moderate discriminants is dominated by genus theory — the 2-rank of the class group is determined by the number of prime factors of $d$. For discriminants with many prime factors, $h(d)$ tends to be a large power of 2.

$h$	Fraction at $d \sim 10^{10}$	Structure
1	16.7%	Trivial class group
2	22.2%	Genus theory ($d$ has 2 prime factors)
4	19.8%	Genus theory ($d$ has 3 prime factors)
8	10.9%	Genus theory ($d$ has 4 prime factors)
16	4.5%	Genus theory ($d$ has 5 prime factors)

These five values account for 74.1% of all discriminants — the entire distribution is shaped by genus theory at this scale.

The Odd Part Is Where Cohen-Lenstra Applies

Cohen-Lenstra heuristics predict the distribution of the odd part of the class group, not the full class number. The 2-part is deterministic (from genus theory). As $d \to \infty$, the average number of prime factors of $d$ grows as $\log \log d$, so the genus-theoretic 2-part grows, making $h = 1$ less likely even as the odd part concentrates at 1.

This explains the monotonic decrease: as $d$ grows, discriminants have more prime factors (Erdős–Kac: the number of prime factors of $d$ concentrates around $\log \log d$), so the 2-part grows, making $h = 1$ less likely. The rate $\Pr(h = 1)$ goes to 0 — it does NOT converge to 75.4%.

The Cohen-Lenstra prediction of 75.446% applies to the odd part: $\Pr(h_{\text{odd}} = 1) \to 75.4\%$. This convergence is extremely slow — at $d \sim 10^{10}$, the 3-divisibility rate is 15.3% vs the predicted 44%, still far from asymptotic.

3-Divisibility

Statistic	Observed ($d \sim 10^{10}$)	Cohen-Lenstra prediction
$3 \mid h$	15.3%	$\approx 43.99\%$
$5 \mid h$	4.9%	$\approx 23.84\%$
$7 \mid h$	2.3%	$\approx 16.33\%$

The p-divisibility rates are also far from the asymptotic predictions, again due to the dominance of the 2-part at moderate discriminants.

Computational Details

• 2,735,671,820 fundamental discriminants processed in the $[10^9, 10^{10})$ range
• 30 minutes on 8× NVIDIA B200 DGX cluster (Blackwell, 192 GB HBM3e each, CUDA 12.8) for the first 2.7 billion
• Total wall-clock for the full 30 billion discriminants (including $[10^{10}, 10^{11})$): approximately 5 hours
• 1.5M discriminants/sec throughput
• Method: GPU sieve + CF regulator (log-space, spot-checked against PARI/GP for $d < 10^6$) + Euler product (9,592 primes). Note: spot-checking a small prefix does not guarantee absence of silent errors over the full 30 billion inputs. A randomized cross-check on a $\geq 0.1\%$ subsample using an independent algorithm is planned.
• Raw data: every (d, h) pair preserved in binary format, to be uploaded to Hugging Face

Full paper: experiment page Source code: github.com/cahlen/idontknow

References

• Cohen, H. and Lenstra, H.W. Jr. (1984). “Heuristics on class groups of number fields.” Number Theory Noordwijkerhout 1983, Lecture Notes in Mathematics 1068, pp. 33–62.
• Jacobson, M.J. Jr., Ramachandran, S., and Williams, H.C. (2006). “Numerical results on class groups of imaginary quadratic fields.” Mathematics of Computation, 75(254), pp. 1003–1024.
• Stevenhagen, P. (1993). “The number of real quadratic fields having units of negative norm.” Experimental Mathematics, 2(2), pp. 121–136.

---

Computed 2026-03-30 on 8× NVIDIA B200. All code and data at github.com/cahlen/idontknow. Published at bigcompute.science.

This work was produced through human–AI collaboration (Cahlen Humphreys + Claude). Not independently peer-reviewed. All code and data open for verification at github.com/cahlen/idontknow.

Update: d ∈ [10^10, 10^11] — 27.4 Billion Discriminants (2026-03-31)

Statistic	[10^9, 10^10]	[10^10, 10^11]
Discriminants	2,735,671,820	27,356,719,769
h=1 rate	16.70%	15.35%
h=2	22.17%	21.27%
h=4	19.77%	19.92%
h=8	10.90%	11.67%
h=16	4.52%	5.15%
3 divides h	15.28%	15.48%

The h=1 rate continues to fall (16.7% → 15.4%). Powers of 2 are increasing (h=8: 10.9% → 11.7%, h=16: 4.5% → 5.2%), consistent with genus theory dominating at larger discriminants. Total: 30 billion discriminants across both ranges.

Computational details for the 27B block: Wall-clock time was approximately 4.5 hours on 8× NVIDIA B200 (Blackwell) GPUs at ~82% average GPU utilization (measured via nvidia-smi). Throughput sustained at ~1.7M discriminants/sec across the cluster. Validation: a random 0.01% subsample (~2.7M discriminants) from the $[10^{10}, 10^{11})$ range was spot-checked against PARI/GP qfbclassno(), with exact agreement on all sampled values.

Genus Theory Shift: Powers of 2 Redistributing

The class number distribution is dominated by powers of 2 (genus theory), and this dominance is evolving:

h	d ~ 10^9	d ~ 10^10	Change
1	16.71%	15.35%	-1.35%
2	22.17%	21.27%	-0.90%
4	19.77%	19.92%	+0.15%
8	10.90%	11.67%	+0.77%
16	4.52%	5.15%	+0.63%

The h=1 and h=2 “drain” is flowing into h=8 and h=16. This is consistent with discriminants at larger d having more prime factors on average (grows as log log d), which increases the 2-rank of the class group via genus theory. The total power-of-2 share is slowly decreasing (74.1% to 73.4%), meaning odd class numbers are very gradually gaining ground.