We treat alignment as a family of measurable scaling relations. For each risk category $r$, the alignment burden required to maintain a fixed safety target is modeled as $B_r(N)=a_r N^{\alpha_r}$, where $N$ is a capability proxy. When the budget scales linearly with $N$, scaling is beneficial if $\alpha_r<1$. We define three operationalizations of burden—observed, audited, and true alignment—and clarify their distinctions. A toy model assumes that corrections consume capability headroom, making the relations explicit. We prove that:
- The largest exponent among corrected risks determines the long‑run regime;
- If $\alpha>1$, any policy that keeps headroom above a floor must grow super‑exponentially;
- For burdens that are positive mixtures of power laws, fits on small models underestimate large‑scale exponents;
- An audit that uncovers hidden failures without false positives never underestimates true alignment. Based on these insights we propose a preregistrable protocol and apply reduced versions twice. A preregistered reanalysis of public adversarial‑training data for Pythia classifiers finds that the compute needed to bring attack success below 10% scales as $N^{0.60}$. A preregistered pilot on Qwen2.5 models (0.5B‑72B) yields exponents of -0.05 for truthfulness and 0.48 for stated dispositions (both benefit from scaling under the reduced rule, with local slopes approaching 1 at the top); sycophancy shows 0.89 (or 0.83 when two additional 72B seeds are added), while a planted backdoor remains undetermined: it is removed quickly once its trigger is known but survives blind safety training at four of five sizes. We release four browser games illustrating these laws at www.aisafety.fun. We make no claim about which regime holds for current frontier models.
Review