arXiv:2605.08541v4 Announce Type: replace Abstract: Neural scaling laws approximate a language model's loss as a power-law function of parameter count $N$ and token count $D$. Following Chinchilla-style compute-optimal training, many studies fit scaling laws from runs at one or a few fixed tokens-per-parameter (TPP) ratios $k$, with $D = kN$. We show that this collinear design, combined with the empirically common near-equality of the exponents governing $N$ and $D$, induces an inherent ill-conditioning in the Gauss-Newton least-squares problem: the condition number of the design grows as the inverse square of the gap between the $N$- and $D$-exponents. The scale coefficients become practically unidentifiable, with confidence intervals inflating by orders of magnitude, yielding a "sloppy" model whose extrapolations degrade sharply off the training ray. We prove this for four scaling-law formalisms and derive a closed-form TPP-diversity threshold that characterizes well-conditioned estimation to leading order. Empirically, in controlled experiments with 5M-77M-parameter models, non-collinear designs outperform collinear ones on held-out data with a 97.3% win rate across four laws and five corpora, and 98.7% in bfloat16. We further show the degeneracy is rooted in Jacobian geometry and is not an artifact of the loss function: any smooth objective whose Gauss-Newton weights are bounded above and away from zero inherits the same ill-conditioning.