If a learned potential makes short-range repulsion too cheap, an optimizer can accept overlapping atoms. That is the "ghost molecule" failure: not a rendering glitch, but bad extrapolation near the steepest part of the energy curve. This post covers a deliberately small test: a one-dimensional Lennard-Jones dimer. It is not a drug-discovery benchmark, a quantum chemistry result, or a direct test of the Pauli exclusion principle. The goal is narrower: check whether a model that looks good on the observed shoulder still respects the repulsive core.

The ground truth is analytic:

$$V(r)=4\epsilon\left[\left(\frac{\sigma}{r}\right)^{12}-\left(\frac{\sigma}{r}\right)^6\right]$$

with derivative labels:

$$\frac{dV}{dr}=4\epsilon\left[-12\sigma^{12}r^{-13}+6\sigma^6r^{-7}\right]$$

The "Soft Wall" Problem

Use the reciprocal coordinate $u=1/r$. As atoms approach, the leading repulsive term grows like $u^{12}$. A generic MLP trained on a finite interval has no built-in reason to continue that growth outside the samples. It can fit the attractive well and the first part of the shoulder, then extend into the core with a wall that is smooth, finite, and too soft.

This is not a universal-approximation complaint. On a covered training interval, a large enough model with enough data can fit the curve. The failure mode starts when optimization walks past the edge of the training distribution.

What Changed

The stable ZeroProofML variant keeps a deep backbone, but uses an improper rational output head. In this experiment, ZeroProofML provides the rational-head inductive bias and stable parameterization; the Lennard-Jones oracle supplies the target energies and derivatives.

$$f(x)=\frac{P(x)}{Q(x)}, \qquad \deg P > \deg Q$$

For the LJ preset, $\deg P=3$ and $\deg Q=1$. The denominator is parameterized to stay positive, and the leading numerator coefficient is positive. The feature is scaled reciprocal distance:

$$x=\frac{u}{u_{\max}}, \qquad u_{\max}=\frac{1}{0.95}$$

This does not hard-code Lennard-Jones, and it does not guarantee the true $u^{12}$ asymptote. It gives the dangerous direction a better default: positive super-linear growth instead of a saturating tail or weak ramp.

The Test

The stress test used the following setup:

Item	Setting
Seeds	10
LJ parameters	$\epsilon=20,\ \sigma=1$
Training range	$r \ge 0.95\sigma$
Core test range	$0.2\sigma \le r \le 0.6\sigma$
Training signal	energy + $\log(1+\lvert dV/dr\rvert)$ derivative match

The core metric is log10_energy_mae_core on samples with $r<0.5\sigma$:

$$\frac{1}{n}\sum_i \left|\log_{10}(\lvert \hat V_i\rvert+1)-\log_{10}(\lvert V_i\rvert+1)\right|$$

Model	Core log-error
ZeroProofML stable improper head	1.332 ± 0.399
Deep MLP baseline	4.842 ± 0.0505

Takeaway

Interpolation scores can hide weak extrapolation near singular or near-singular limits. For this LJ dimer benchmark, a positive improper rational head gives short-range wall extrapolation a better shape than a smooth MLP baseline. Broader molecular-simulation claims still need separate validation: multi-atom systems, force-field datasets, integration stability, conservation behavior, and thermodynamic tests.