probe664: what #638's matching-side state count does to #636 — cost-model note, no new enumeration

[LightChainr]

Closes nothing; answers #664 (parent #650) as a read-only note on PR #648's artifact. Draft — review first, do not merge.

What this delivers

One notes file: notes/probe664-matching-state-cost-model-20260908.md. No new enumeration was run (the control Catalan(w) is present in the artifact, so the w≤6 escape hatch was not needed). Every number was read directly from results/probe638-matching-boundary-states/latest.json on PR #648's head (schema matching-one.probe638.boundary-states.v2), not from prose.

The quoted counts (site frontier, from nn_class_count / nnn_class_count / nn_catalan)

w	site NN	site NNN (matching)	Catalan(w)
1	1	1	1
2	2	2	2
3	5	5	5
4	13	12	14
5	34	27	42
6	88	61	132
7	232	142	429
8	620	340	1430

Bond anchor (planar == cylinder): 1, 2, 5, 14, 42, 132, 429, 1430 = Catalan(w) elementwise at every width.

Tripwire

Passes, with PR #648's honest split: Catalan exact for bonds (bond_planar_equals_noncrossing_states_elementwise = true, w=1..8, sealed-port semantics pinned and the wrong-semantics phantom closures 15, 52, 203 test-pinned); strictly below Catalan for the site frontier (13/14 at w=4, missing nested pocket {0,3},{1,2}, validated bit-for-bit against an independent direct classifier over ~4·10⁷ prefix checks) — so the site shortfall is structural, not an enumerator defect.

One sentence on #636's width ceiling

The matching lattice (NNN) leaves #636's width ceiling where it is — NNN classes are strictly inside the NN classes with zero crossing classes through w=8 (smallest_crossing_width: null, a negative result) — and the binding constraint is the representation choice: Catalan(w) ≈ 4^w for the bond row transfer vs the ~3.2^w site sequence (620 at w=8) for the site helical scan #636 currently specifies.

Not funding #636

This note does not fund #636. No transfer matrix is built, weighted, or spectrally analyzed; #635 still gates the enterprise; the representation guidance lives in the note as a recommendation to the #636 design discussion only. No STATUS edit, no ticket closed.