[P2 exact] Integer structure of the five wrapping-type cells

[LightChainr]

Parent: #650. Census is done (PR #653 L=3,4; PR #657 axis L=5 and diamond L=4). This ticket reads those integers and looks for exact structure. No new enumeration. No Huawei.

What is already known

On every size reached, the 4×4 is supported on five cells only:

neither × both, both × neither, dir0 × dir0, dir1 × dir1, both × both

and a_k = #(both×neither) − #(neither×both). dir0×dir0 = dir1×dir1 at every k. both-two mass is 0. Row sums are C(N,k).

Mertens–Ziff explain the support. They do not give generating functions for the five counts as functions of (L,k).

Questions

1. Closed forms or recurrences for the five sequences in k, at fixed L, and across L=3,4,5 (axis) and diamond L=2,3,4 if the tables exist.
2. OEIS / Catalan / Motzkin / nested-path hits. Quote the match or the gap. Site frontier is already known to be a strict subset of Catalan on another ticket ([DeepSeek / compute] Enumerate the matching-lattice boundary state space — the single biggest unknown in #636's cost model #638) — do not redo [DeepSeek / compute] Enumerate the matching-lattice boundary state space — the single biggest unknown in #636's cost model #638; only ask whether these wrapping-type counts are named sequences.
3. Any exact identity beyond MZ pairing: e.g. generating functions, roots of M_L, factorizations already in exact_matching_polynomial.py --factor.
4. What would L=6 (2^36, Bernstein already on PR Issue #639: exact Bernstein ladder — new rungs axis L=5, diamond L=4, axis L=6 (two-kernel verified); diamond L=5 reported out of reach #649, wrapping-type not enumerated) have to show to kill a proposed closed form. Do not enumerate L=6.

Tripwire

Every identity you claim must hold on the committed tables as exact integers. A float in an exact artifact is a bug. If you guess a formula, test it on all existing L before writing it down.

Deliverable

notes/wrapping-five-cell-integers-YYYYMMDD.md + optional tiny script that reads the JSON tables. Draft PR against main. No STATUS, no merge, do not close #640/#651.

Wait for ASSIGNED_MACHINE.

Related: #640, #651, #638, #649, #653, #657.

[LightChainr]

ASSIGNED_MACHINE: local

Read existing wrapping JSON only (PR #653, #657). No enumeration, no Huawei. Draft PR against main. Do not edit STATUS. Do not merge. Do not close #640/#651.

[LightChainr]

Read JSON only (PR #653 / #657). No new census on the Mac. L=6 wrapping is out of scope and would be Huawei if ever assigned.

[LightChainr]

PR: #669

Verified against the #659 tripwire (read-only JSON, integers, no L=6, no STATUS). Identities hold on all five committed tables. Ticket stays open; not merging; not closing #640/#651.