SCM Foundations

Signed Common Meadows (SCM) provide a totalised arithmetic with a single absorptive bottom value (⊥). This library implements SCM semantics for machine learning workflows.

Algebraic Rules

  • Total inverse: every element has an inverse; 0^{-1} = ⊥.
  • Absorption: x + ⊥ = ⊥ and x · ⊥ = ⊥ for all x.
  • Weak sign: s(x) = x/|x| for non-zero inputs, 0 at the origin, ⊥ when the argument is ⊥; for complex numbers the sign lies on the unit circle.
  • History-aware sign: when |z| falls below ε_sign, the last valid orientation is kept to prevent oscillation.

Practical Implications

  • Scalar SCM ops (zeroproof.scm.value, zeroproof.scm.ops) return SCMValue(⊥) on division-by-zero / domain errors instead of IEEE NaN/Inf.
  • For arrays/tensors, SCM values are represented as a numeric payload + bottom mask. The mask is authoritative; payload values on masked entries are undefined and must be ignored (use strict decoding to map them to NaN if you need an IEEE sentinel).
  • No layer-by-layer “guard mode” is required: singularity handling is pushed to explicit masks and a single decode at the output boundary.
  • Numerical stability is handled by gradient policies and (optionally) projective tuples, rather than ad-hoc forward clamps.

Fracterm Flattening

For rational heads we exploit fracterm flattening: small rational subgraphs are rewritten as P(x) / Q(x) to reduce singularity checks to the final denominator. Depth is capped (L ≤ 5) to avoid polynomial blow-up.

Terminology

  • Bottom (⊥): absorptive error element; any operation involving ⊥ yields ⊥.
  • Projective tuple: pair (N, D) representing the same value as N/D with (1, 0) denoting ⊥.
  • Coverage: fraction of predictions that remain finite (non-⊥) under SCM semantics.