Fracterm Flattening in ZeroProofML

Common meadow theory guarantees that any term over the signature can be flattened into a single rational function P(x)/Q(x). ZeroProofML adopts this result for Fused Rational Units (FRUs) to make singularity detection and policy selection predictable without expanding the entire network.

Engineering Scope

  • Local flattening only. To avoid degree explosion, flattening is restricted to small rational heads instead of whole models; the rest of the network stays in standard SCM or projective form.
  • Constant-time checks. By keeping the rational part flattened, we can decide whether a forward pass hits the ⊥-producing denominator with an O(1) check on Q_head, rather than walking each intermediate layer.
  • Projective compatibility. FRU outputs can be decoded either strictly into SCM values or as projective tuples (N, D) during training, keeping gradients smooth around poles.

Implementation Status

  • zeroproof.scm.fracterm provides symbolic utilities for representing and simplifying small P/Q terms.
  • zeroproof.layers.fru.FractermRationalUnit currently enforces degree-growth and depth constraints and provides a lightweight placeholder flatten(...) helper; it is not yet a full automatic “flatten an arbitrary PyTorch subgraph” pass.

Worked Example

Consider a two-layer FRU ((x + 1) / x) / (x - 1). Flattening produces

P(x) = x + 1
Q(x) = x(x - 1)

so the singularities live at x ∈ {0, 1} and can be detected by checking Q(x) once, instead of monitoring every intermediate op.

Authoring Guidelines

  1. Represent rational heads explicitly as (numerator, denominator) pairs or helper classes so the final denominator is easy to inspect.
  2. Prefer shallow FRUs (depth ≤ 5) to keep polynomial degrees manageable, matching the hard limit in the migration plan.
  3. Document where flattening occurs in layer docstrings so Sphinx/MkDocs can surface the singularity story alongside the API.

References

  • J.A. Bergstra and A. Ponse, Common Meadows (2019 revision)
  • J.A. Bergstra and J.V. Tucker, Fracterm calculus and totalised fields