[Autoplectic]

Summary

This adds benchmarks for the mechanisms of Hourican & Quax, "Operationalising Weaker Faithfulness" (arXiv:2610.31724). It checks #16 (the pair-search budget) and #17 (lag-0 explanation families) against them.

• datasets: xor_descendant, asymmetric_interaction, or_gate and xor_among, plus contemporaneous_motif(kind, law=) with CONTEMPORANEOUS_MOTIFS. The lag-0 motifs are sampled from their exact laws. law=True returns the dit distribution, so reference families come from explanation_family(decisions_from_distribution(...)).
• benchmarks.synergy_validation:
  • Lagged motifs are scored on whether the parents are recovered.
  • Lag-0 motifs are scored on whether contemporaneous_families returns exactly the population family.
  • Rates come with Clopper–Pearson intervals.
• benchmarks.planted_module_validation: synthetic modules (XOR, XOR with a descendant, OR, independent bits) are planted beside a fixed background, following the paper's neural application. It reports recovered parents, exact parents and false parents.
• validation/synergy.py, a new section in docs/validation.rst, and a literature-table entry.

Results

From validation/synergy.py --planted, with 4000 samples, default settings and 10 replicates:

Check	Rate
XOR, flip 0.1 / 0.25 / 0.45: both parents	10/10 each
XOR with a descendant: y from x1 and x2	10/10
XOR with a descendant: w from y only	9/10
Asymmetric interaction; OR; XOR among 12 processes	10/10 each
Lag-0 XOR: population family	9/10
Lag-0 XOR with a descendant, OR, chain, dependent baseline (empty family)	10/10 each
Lag-0 independent baseline	9/10

For modules planted beside a noisy rule-90 automaton of six cells:

• Every true parent was found in 10/10 replicates of every module.
• Every module variable got exactly its parents in 8, 7, 8 and 9 of 10 replicates.
• The remaining replicates had 0.2–0.4 wrong extra parents each, which is consistent with the per-stage levels.

The XOR at flip 0.45 (0.007 bits of whole-minus-sum information) is found in every run. That signal is below the paper's 0.3-bit protection threshold.

Tests

• tests/test_synergy.py: generators, motif laws, and small runs of both validations.
• Full suite: 423 passed, 13 skipped. ruff, ty and sphinx -W are clean.