How ZeroProofML Works

The 3-minute story

Physical systems often contain singularities: mathematical limits where forces approach infinity, resonance peaks sharpen, or geometric determinants vanish.

• Pharma: Atoms approaching overlap experience infinite repulsion forces ($1/r^{12}$).
• Electronics: Resonant filters have poles ($1/Q$) that create sharp spectral peaks.
• Robotics: Manipulators near full extension hit kinematic locks where the Jacobian determinant vanishes.

Standard neural networks (MLPs) rely on a "Smoothness Inductive Bias" (Lipschitz continuity). When they encounter these sharp discontinuities, they smooth them away. They create "Soft Walls" where physics demands "Hard Walls," "Clipped Peaks" instead of resonances, or dangerous "Panic Spikes" in control. Traditional fixes like clipping or $\epsilon$-regularization treat singularities as numerical bugs to be patched. This introduces position-dependent bias that ruins high-fidelity modeling.

ZeroProofML treats singularities as structural features to be learned. By baking Rational Inductive Biases directly into the network architecture via the Signed Common Meadows (SCM) operator, ZeroProofML allows models to learn the exact differential equations driving the singularity. It safely traverses poles during training without gradient explosion.

The approach: Signed Common Meadows (SCM)

Earlier versions of ZeroProofML experimented with Transreal arithmetic. In v0.4.0 we replaced that with a mathematically grounded, differentiable framework: Signed Common Meadows (SCM).

In an SCM, division is total. Every value lives in $\mathbb{F}_{\perp} = \mathbb{F} \cup \{\bot\}$, and $\frac{1}{0} = \bot$ propagates algebraically instead of yielding exceptions. We additionally track a weak sign so orientation near a pole is explicit (⊥ with history-aware sign cues) just as described by Bergstra & Tucker.

To reconcile this strict algebra with gradient-based learning we split execution into tiers:

1. Projective Backbone (Training): Values are lifted to homogeneous tuples $\langle N, D\rangle$. Layers operate in projective space, using detached renormalization so gradients focus on direction rather than magnitude (``ghost gradients''). This keeps optimization smooth even when $D \to 0$.
2. Target-Tuple Loss Stack: Instead of plain MSE we use an implicit fit loss (projective cross product), a margin loss to keep denominators away from the singular band, and a sign/orientation loss for true poles. These objectives guarantee the smooth-training / strict-inference gap is bounded.
3. Strict SCM Inference: At deployment, fracterm flattening collapses rational heads into $P(\mathbf{x})/Q(\mathbf{x})$. We apply configurable thresholds $(\tau_{\text{train}}, \tau_{\text{infer}})$ to surface bottom_mask and gap_mask so every singular decode is auditable.

This “Train on Smooth, Infer on Strict” protocol is the core of ZeroProofML: smooth, stable gradients for optimization, and provable SCM semantics for deployment.

Why this helps: "Architecture is Destiny"

A ReLU MLP extrapolates linearly ($O(x)$). No matter how much data you feed it, it cannot represent a $1/r^{12}$ potential or a $1/Q$ resonance. It will always predict a "Soft Wall" or a "Clipped Peak" out-of-distribution (OOD).

ZeroProofML changes the Inductive Bias of the layer. By configuring the polynomial degrees of the SCM head, you guarantee the physics:

• Improper SCM ($\deg P > \deg Q$): Guarantees super-linear growth ($O(x^k)$) for forces and potentials.
• Shared-Complex SCM: Guarantees phase coherence for spectral resonances.
• Bounded SCM: Guarantees bell-shaped damping for control systems.

Validated Results (The "Physics Trinity")

We benchmarked ZeroProofML against "Steel-Man" MLP baselines (capacity-matched, 4x256) across three distinct physical singularities (N=11 seeds).

1. Pharma: The "Hard Wall" ($1/r^{12}$)

• Problem: Discovering atomic repulsion from low-energy data to prevent "Ghost Molecules."
• Result: >3,000x reduction in core extrapolation error compared to MLPs.
• Impact: While MLPs created soft walls that collapsed under optimization pressure, ZPML matched the Analytic Oracle's stiffness perfectly ($W=3000$).

2. Electronics: Spectral Extrapolation ($1/Q$)

• Problem: Predicting sharp 5G resonance peaks from low-Q training data (33x OOD).
• Result: 70% Yield (Success Rate) vs 40% for MLPs.
• Impact: ZPML physically moves the poles to the stability boundary. In worst-case failures, it retains 50% of the signal energy, whereas MLPs lose 98% (catastrophic signal erasure).

3. Robotics: Geometric Consistency ($\det J \to 0$)

• Problem: Learning safe Inverse Kinematics near a mechanical lock.
• Result: 31.8x lower variance across training seeds and 60x safer worst-case control spikes.
• Impact: While MLPs achieved slightly lower mean error by overfitting the smooth workspace, they exhibited "Peak Attenuation" (under-reacting to the singularity by ~2.4%). ZPML acts as a deterministic safety constraint, trading average-case precision for structural consistency.

When this matters / doesn't matter

Good candidates:

• Scientific ML: Problems governed by differential equations with poles ($1/r$, $1/x$).
• Extrapolation Tasks: When you need the model to hold up outside the training distribution (OOD).
• Safety-Critical Control: When consistency and bounded behavior are more important than average-case error.
• Generative Modeling: Where physical validity (e.g. no atomic overlaps) is non-negotiable.

Poor candidates:

• Interpolation Tasks: If you stay inside the training distribution, a big MLP is often simpler and slightly more precise.
• Standard CV/NLP: ResNets and Transformers don't need rational biases.
• Smooth Manifolds: If your problem has no singularities, SCM adds compute cost (1.1x) for no benefit.

What this means practically

If your physics involves poles, stop trying to force a ReLU network to learn them. Replace your output head with a Rational SCM layer.

Open questions

We have validated this on geometric, asymptotic, and spectral singularities. We suspect strong applications in:

• Power Systems: Voltage-collapse modeling ($V \to 0$) where Jacobians become singular.
• Quantitative Finance: Correlation matrix rank drops and $1/\det$ risk metrics during stress.
• Fluid Dynamics: Shock waves / turbulence boundaries that defeat smooth approximators.

If standard $\epsilon$-smoothing is biasing your science, try SCM. We are actively looking for collaborators with domain expertise to test the limits of these rational inductive biases.