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	1	# Graded ("fractal") connectors — pre-registration
	2
	3	2026-10-18. Written and committed before any code or run.
	4
	5	## Question
	6
	7	Linas (diary ten-J:12245-12251): connectors "asking for mating at multiple scales". Our reading: a
	8	connector carries a chain of classes (word → fine class → coarser classes), and two pieces mate at the
	9	deepest level where both have evidence. Two claims:
	10
	11	1. Better structure: scoring links by the finest level that has evidence ("backoff") parses better than
	12	any single level, because it uses word detail where data allow and class detail where they don't.
	13	2. Constraint without deletion: ranking level-2 candidates by how well they assemble under graded
	14	connectors — never deleting any — keeps the true phrases that the hard rule of `experiments/level2`
	15	Test A lost (it kept 34%), while still finding them with far fewer candidates than plain frequency order.
	16
	17	## Corpus and hierarchy
	18
	19	The planted grammar of `across-cluster/ac.c` unchanged. Gold hierarchy per token, from
	20	`level2/l2.c` `merge_class`: W word → G gold class (30) → S1 number merged → S2 bare parts of
	21	speech. Gold level-2 units: noun- and prepositional-phrase spans (as in level2). Tiers 3K, 10K, 30K training
	22	sentences; 2K held-out; 10 seeds; common random numbers.
	23
	24	This tests the mechanism with a known hierarchy (as Test A used gold classes); a learned hierarchy comes
	25	later.
	26
	27	## Part 1 — level-1 parse with graded connectors
	28
	29	Per level L, gated window PMI over L's alphabet (W = 6; PMI > 0 and G > 10.83; as `ac.c`). Edge scorers for
	30	a token pair:
	31	- single level W, G, S1, S2: that level's gated PMI, else no edge;
	32	- BACKOFF: the gated PMI at the finest level (W, then G, S1, S2) that passes the gate.
	33
	34	Parse: Eisner max-weight projective forest (as `ac.c`). Metric: held-out link F1 against gold trees.
	35
	36	## Part 2 — level-2 candidates ranked, never deleted
	37
	38	Candidate types: contiguous spans of length 2–6, typed by their G-class sequence (the same alphabet for all
	39	arms). Arms differ only in order:
	40	- C-G: all types by training count (frequency order; nothing deleted);
	41	- HARD: only types that occur as complete assemblies in the G-level parse, by count; all other types
	42	deleted (Test A's rule);
	43	- GRADED: all types, tiered by the best structure any training instance achieved in the BACKOFF parse
	44	(complete assembly > connected > neither), by count within a tier; nothing deleted;
	45	- GRADED-G: as GRADED but tiered by the G-level parse (isolates the multi-scale part);
	46	- GRADED-R: as GRADED with a random hierarchy (each level a random relabelling with the same class
	47	sizes) — control.
	48
	49	Metric: type recall at budget B = share of held-out gold spans (length ≤ 6) whose G-type is among the
	50	first B types of the arm's ranking; B ∈ {300, 1K, 3K, 10K, all}. For HARD, "all" = its whole list.
	51
	52	## Predictions (bars fixed now; paired over 10 seeds; "beats" = 95% CI of the paired difference excludes 0)
	53
	54	- F1 (graded structure): at tiers 3K and 10K, BACKOFF beats every single level on link F1; at 30K it is
	55	within 0.01 of the best single level or better.
	56	- F2 (no deletion): GRADED's type recall at "all" ≥ 0.95 at every tier (HARD stays at its Test-A level).
	57	- F3 (still a constraint): at B = 1K, GRADED beats C-G at every tier; and at B = \|HARD list\| (per seed),
	58	GRADED ≥ HARD in mean.
	59	- F4 (the multi-scale part matters): at B = 1K, GRADED beats GRADED-G at tiers 3K and 10K.
	60	- F5 (control): at B = 1K, GRADED-R does not beat C-G at any tier.
	61
	62	## Not tested here
	63
	64	A learned hierarchy, recursion past level 2, the Linas port.

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