Signed Common Meadows (SCM) Primer¶

ZeroProofML v0.4 grounds its semantics in signed common meadows: a field of characteristic 0 extended with a total inverse and a single absorptive bottom element (⊥). Division by zero yields ⊥, which then absorbs addition and multiplication, allowing singularities to propagate without ad-hoc guard branches.

Weak Sign Structure¶

SCM augments the meadow core with a sign operator that preserves orientation even when magnitudes blow up. For real inputs the operator is 4-signed; for complex inputs it projects onto the unit circle, returning ⊥ unchanged and locking the last valid orientation when approaching the origin. This weak-sign construction makes the library usable for higher-dimensional robotics where ordered-field assumptions break down.

Bottom-Aware Training and Inference¶

The SCM core already ensures total arithmetic, but training benefits from explicit handling of singular paths. Gradient policies clamp, reject, or project gradients that traverse ⊥ during the backward pass, while the forward graph stays faithful to the algebra. The optional projective extension lifts select subgraphs to homogeneous tuples (N, D), giving the optimizer a smooth manifold and delaying any ⊥ instantiation to the boundary decode step. The guiding rule is:

Train on smooth (policy- or projective-regularized) objects; infer on strict SCM semantics.

Signed Common Meadows (SCM) Primer¶

Weak Sign Structure¶

Bottom-Aware Training and Inference¶

Further Reading¶