[P1] What #638's matching-side state count does to #636 — without funding #636

[LightChainr]

Parent: #650. Read-only of PR #648 / #638. No new enumeration unless the control Catalan(w) is missing, and then only w≤6 on the Mac (NEED_HUAWEI for anything heavier).

#637 said the matching-side boundary state space is the cost-model risk for #636. #638 asked to enumerate it. PR #648 exists (Catalan for bonds, site frontier a strict subset, NNN noncrossing through w=8 — verify that sentence against the PR, do not take this parenthetical as the result).

Question

Write the cost-model implication in one notes file:

1. Quote the actual counts from PR probe638: boundary-state enumeration — Catalan holds for bonds, site frontier is a strict subset, NNN noncrossing through w=8 #648 (primal NN control vs matching NN+NNN, w=2..8).
2. Tripwire: NN control = Catalan(w), or the enumerator is unused.
3. One sentence: does this raise, leave, or lower [P1 proof / bounded CPU] Observable spectral remainder and width uniformity after the exact n=2 bridge #636's width ceiling relative to Catalan(w)?
4. Explicitly do not fund [P1 proof / bounded CPU] Observable spectral remainder and width uniformity after the exact n=2 bridge #636. If you think a TM is now cheaper or more expensive, that is a recommendation in the note, not a STATUS promotion.

Draft PR against main. No merge, do not close #636/#638.

Wait for ASSIGNED_MACHINE: local.

Related: #636 (not released), #637, #638, #648, #635.

[LightChainr]

ASSIGNED_MACHINE: local

Read PR #648 only. No new wide enumeration. Draft PR against main. Do not fund #636. Do not edit STATUS.

[LightChainr]

PR landed: #667

Grok check (no merge, leave open):

• Read PR probe638: boundary-state enumeration — Catalan holds for bonds, site frontier is a strict subset, NNN noncrossing through w=8 #648 JSON; no new enumeration; Catalan control quoted.
• Site NN is a strict subset of Catalan from w=4 (13 vs 14). Matching NNN is inside NN, zero crossing classes through w=8.
• Cost-model sentence: matching side does not raise [P1 proof / bounded CPU] Observable spectral remainder and width uniformity after the exact n=2 bridge #636's width ceiling; the binding choice is representation (bond Catalan ~4^w vs site helical ~3.2^w). [P1 proof / bounded CPU] Observable spectral remainder and width uniformity after the exact n=2 bridge #636 still not funded.

Ticket stays open.