[Autoplectic]

Problem

The synergy-aware pair search finds parents that inform the target only jointly, as in an XOR. Its pool took the remaining candidates in (lag, process) order and stopped at the first 20. In larger networks, XOR parents at longer lags were therefore never searched. For example, with 8 processes and parents (5, lag 3) and (6, lag 2), the pair sits beyond the 20th candidate and is missed entirely.

Change

• SkeletonSettings.max_pairs (default 5000 pairs, i.e. 100 candidates) replaces the 20-candidate cap. max_pair_candidates now defaults to None and still caps the pool when set. The order is unchanged and does not look at the target, so the maximum statistic stays a valid test.
• TargetSkeleton.pair_pool_dropped lists the candidates left out, and a single warning says so.
• The pair test is curtailed for every estimator unless curtail=False. It reports no p-value, and curtailment never changes a decision.
• Joint codes are built pair by pair instead of all at once. A 100-candidate pool has 4950 pairs, each a column as long as the data.

Results

From validation/pair_pool.py, with 4000 samples, the default permutation counts, and 10 XOR plus 10 null targets per size:

Processes	Pool	XOR recovered	False parents per target	Seconds per target
8	first 20	5/10	0.00	2.3
8	5000 pairs	10/10	0.10	3.8
12	first 20	2/10	0.05	2.7
12	5000 pairs	10/10	0.05	7.8
20	first 20	0/10	0.05	3.7
20	5000 pairs	10/10	0.25	25.6

The pair test keeps its level with the larger pool. On null targets it rejected 5 of 100 with 12 processes (1540 pairs) and 6 of 100 with 20 processes (4560 pairs). At 20 processes, the false parents were one greedy addition and two spurious pairs across 20 targets, which is consistent with the per-search level.

Tests

• test_pair_pool_order_and_budget.
• test_pair_search_reaches_candidates_beyond_twenty, which also checks the warning and pair_pool_dropped.
• test_pair_search_curtailment_keeps_the_decision.
• Full suite: 278 passed, 13 skipped. ruff, ty and sphinx -W are clean.

Motivated by Hourican & Quax, "Operationalising Weaker Faithfulness" (arXiv:2610.31724).