Implementation Verification Report (v0.4 SCM)

This report summarises what is implemented in the v0.4 Signed Common Meadows (SCM) codebase and what the current contracts are.

Scope: the v0.4 Python packages zeroproof/ and zeroproofml/ (compatibility shim).

✅ Core domain: SCMValue and ⊥

Code: zeroproof/scm/value.py

  • SCMValue represents either a numeric payload (float/complex) or the absorptive bottom element ⊥ (is_bottom=True).
  • Bottom is absorptive for + and ·; division by zero yields bottom.
  • Factories: scm_real, scm_complex, scm_bottom.

✅ Scalar arithmetic helpers

Code: zeroproof/scm/ops.py

  • Scalar helpers implement totalised arithmetic: scm_add, scm_sub, scm_mul, scm_div, scm_inv, scm_neg, scm_pow.
  • Transcendentals are bottom-aware with domain checks: scm_log, scm_exp, scm_sqrt, scm_sin, scm_cos, scm_tan.

✅ Vectorised SCM + masks (NumPy / Torch / JAX)

Code: zeroproof/scm/ops.py

Vectorised entry points propagate a separate boolean bottom mask: - NumPy: scm_*_numpy - Torch: scm_*_torch - JAX: scm_*_jax

Each returns (payload, mask) and treats: - mask=True as ⊥ - division by zero / inverse of zero as ⊥ (adds to the mask)

✅ IEEE-754 bridge (scalar)

Code: zeroproof/utils/ieee_bridge.py

  • from_ieee: collapses NaN and ±inf to ⊥
  • to_ieee: maps ⊥ to NaN (tooling-friendly sentinel)

✅ Gradient policies (SCM semantics)

Code: zeroproof/autodiff/policies.py

  • Policy enum: CLAMP, PROJECT, REJECT, PASSTHROUGH
  • Context manager: gradient_policy(...)
  • Utilities: apply_policy, apply_policy_vector

✅ Torch SCM rational layer with bottom mask + policy hook

Code: zeroproof/layers/scm_rational.py

  • SCMRationalLayer.forward(x) -> (output, bottom_mask)
  • Singularities are detected via denominator ≈ 0 and surfaced as bottom_mask
  • Policies are applied by registering a backward hook on output gradients

✅ Trainer loop: coverage-aware early stopping

Code: zeroproof/training/trainer.py

  • SCMTrainer.fit() returns per-step logs including loss and coverage
  • Coverage is estimated from NaN on decoded tensors or from projective denominators
  • Supports gradient accumulation and mixed precision (AMP)

Intentional non-goals (v0.4)

  • No Transreal tags (+∞, −∞, Φ) in the v0.4 core; Transreal-era scripts under examples/archive_tr/ are for historical reference and are not part of the v0.4 import path.
  • No “Mask‑REAL”/“Hybrid” transreal modes in the v0.4 public API; use gradient policies and/or projective tuples instead.