We're sharing ZeroProofML, a small framework for scientific ML problems where the target can be genuinely undefined or non-identifiable: poles, assay censoring boundaries, kinematic locks, etc. The underlying issue is division by zero. Not as a numerical bug, but as a semantic event that shows up whenever a learned rational function hits a pole, a normalization denominator vanishes, or a physical quantity becomes non-identifiable.
The motivating issue is semantic, not just numerical. A common fix for denominator pathologies is ε-regularization: replacing N/D with N/(D+ε). That often keeps training stable, but it also changes the meaning. A point that should decode to "undefined" becomes a large finite scalar instead. Our approach builds on Common Meadows, an algebraic framework from theoretical computer science (Bergstra & Tucker) where division is total: dividing by zero returns an absorptive element ⊥ that propagates through all subsequent operations. The specific variant we use is Signed Common Meadows (SCM), which additionally preserves sign/direction information at the singular boundary.
The practical difficulty is that ⊥ annihilates gradients, so you can't train directly in strict mode. Our solution is 'Train on Smooth, Infer on Strict': during training, the model works with smooth projective tuples ⟨N, D⟩ so gradients still flow; at inference, we switch to strict decoding where the denominator crossing the singular boundary emits an explicit state rather than an ε-stabilized large number. Rational neural nets already help with representation: they can model pole-like growth and sharp transitions more naturally than plain MLPs. ZeroProofML builds on that rational inductive bias, but adds a stricter semantic layer: near the singular boundary, the model does not have to return a clipped finite surrogate.
3 domains (10 seeds):
- Dose–response (pharma): strict decoding reduces false finite predictions on censored inputs from 57.3% (rational+ε baseline) to about 1.2×10⁻⁴, with FN_in = 0.
- RF filter extrapolation (electronics): under 33× OOD extrapolation in Q_f, the shared-denominator SCM model improves peak-retention yield from 39.8% to 77.3% and substantially reduces phase error.
- Inverse kinematics (robotics): the projective parameterization reduces seed-to-seed variance by 31.8×
A few limitations:
- In Dose, reconciling censored-direction supervision with high-quality regression is still an open optimization problem.
- In robotics, there is a bias-variance trade-off.
- For ordinary smooth regression problems, this is unnecessary overhead.
We claim that arithmetic design is an inductive bias, and in singular regimes it can matter whether the model represents division-by-zero explicitly.
Blog: https://domezsolt.substack.com/p/from-brahmagupta-to-backpropagation
Paper: https://zenodo.org/records/18944466
Code: https://gitlab.com/domezsolt/zeroproofml
Feedback and cooperation suggestions welcome!
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