When we published our inverse kinematics results, we led with the most impressive number: 47% error reduction in bucket B1, the region where the Jacobian determinant falls between 10⁻⁵ and 10⁻⁴. It's a real result, validated across three random seeds with near-zero variance. But here's the full picture: our overall test MSE was 0.141 versus 0.142 for the best ε-ensemble baseline. On most of the workspace, we're essentially tied.

So did we just cherry-pick a metric? Not quite, but it's worth understanding what's actually happening. We divided the test set into buckets by |det(J)|—a proxy for distance to singularity. Bucket B0 contains the most extreme cases (|det(J)| ≤ 10⁻⁵), B1 the next tier, and so on through B4 for the well-conditioned regions far from singularities. In B0, we achieved 0.0022 MSE versus 0.0032 for ε-ensemble (29.7% reduction). In B1, it was 0.0013 versus 0.0024 (46.6% reduction). But B2 through B4? Roughly identical performance. These buckets also represent very different amounts of workspace—B0 and B1 combined might be 5–10% of configurations, while B3–B4 dominate the dataset.

This matters because singularities aren't uniformly distributed problems. A robot working in the middle of its workspace will never see these cases. For those applications, ZeroProofML offers no advantage over standard methods—you're just adding complexity for no benefit. But if your application specifically operates near workspace boundaries (think assembly tasks at maximum reach, or surgical robots requiring extreme articulation), those "rare" configurations become your entire operating envelope. A 47% error reduction in the region where you actually work is enormous. A 0.7% overall improvement when you never go there is meaningless.

We also tested against several baselines beyond simple ε-regularization: learnable ε (where the network learns the regularization parameter), smooth surrogates like P/√(Q² + α²), and a 5‑member ensemble with different ε values. The ensemble came closest to our performance but took 12× longer to train and doesn't provide deterministic outputs. Damped Least Squares from classical robotics handles singularities but with O(n³) complexity per iteration and systematic bias proportional to the damping factor. Our approach trains in 182 seconds on a modern CPU, runs in O(n²) per forward pass, and produces bit‑identical results across runs given the same seed—useful for safety certification.

What doesn't work well: ZeroProofML provides no benefit when singularities are removable (you can rewrite the math to avoid division), when they're artifacts of poor problem formulation, or when the function you're learning is fundamentally smooth. We also haven't validated on problems where singularities form complex sets rather than isolated points or manifolds. The theoretical guarantees assume the poles are well‑separated and the training data provides sufficient coverage—violate those assumptions and convergence behavior is unknown.

The honest takeaway: we solved a specific problem (learning rational functions with poles) in a specific domain (robot inverse kinematics) with measurable improvements in the exact regions that matter for that problem. Whether this translates to your application depends entirely on whether you have the same mathematical structure and whether near‑singularity performance is your bottleneck. If you're hitting NaN errors, training instabilities, or dead zones in learned representations near mathematical singularities, this might help. If your models work fine already, stick with what works. We're still figuring out where else the approach applies—and where it doesn't.