[P2 proof] #680: NN+NNN strip frontier partitions are noncrossing — proof + small-width control + convention reconciliation

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p680 — #680 delivery: Lemma A (NN+NNN strip frontier partitions are noncrossing)

Outcome: proof for the noncrossing claim (all widths and depths, open strip); class-set equality (NN vs NN+NNN) re-verified at w ≤ 6, R ≤ 4 in the honest convention; #675's 28/66 counts reconciled as a different convention.

Contents

• notes/p680-nn-nnn-noncrossing-lemma-20260913.md — the deliverable note:
  • Theorem 1 (proved, all w, all R): every frontier occupancy partition is noncrossing. The PR [Superseded by #702] Finite-sector obstruction — unrestricted no-go withdrawn #675 §4 sketch is completed in full rigour: (§2.1) planarization lemma — the occupied king graph has the same connectivity as a planar graph G* (all NN bonds + at most one diagonal per face; both diagonals of a face active ⇒ all four corners occupied ⇒ the face's NN cycle already merges them), drawn in the closed half-plane with the frontier sites exactly on the boundary; (§2.2) a joint-minimality choice of the crossing tuple makes the two connecting paths avoid the frontier (explicit case analysis, all nine cases written out); (§2.3) Jordan-curve contradiction (ray-parity identifies the bounded side; the boundary arc through ∞ puts d in the unbounded component).
  • Claim 2 (verified w ≤ 6, R ≤ 4 — status unchanged from [Superseded by #702] Finite-sector obstruction — unrestricted no-go withdrawn #675): the reachable class sets of NN and NN+NNN coincide at every tested (w,R).
  • Convention correction: [Superseded by #702] Finite-sector obstruction — unrestricted no-go withdrawn #675's "28 states at w=5 over 4 rows and 66 at w=6 over 3 rows, identical sets; strictly inside 42/132" mixes conventions. 28 and 66 are reproduced exactly as the NN side of the delayed-closing / fully-occupied-frontier convention (28 at (5,R=4); 66 at (6,R=4); the "3 rows" reading gives 65 — off-by-one suspected). Under that convention the two rules' class sets are not equal (13 vs 28; 24 vs 65/66), though all classes stay noncrossing. The equality Lemma A claims is true in the honest convention (all bonds, free frontier occupancy), where the ambient is not C_w (51 > 42 at w=5 saturation, because vacancy-leap blocks exist).
  • The converse stays false at the cluster level (witness re-verified: footprint {0,2} over a vacant middle position via one diagonal relay; the partition {0,2} is still NN-reachable via a longer relay).
• scripts/p680_frontier_partition_enumeration.py — exact integer enumeration (transfer BFS with union-find transitions + independent brute-force per-configuration cross-check wherever w·R ≤ 16; both conventions; witness function). Stdlib only.
• results/p680-noncrossing-lemma-20260913/ — REPORT.md, metadata.json, commands.txt, raw class lists per (w,R).

Class-count sequences (honest convention, w = 2..6)

• R = 1: NN [4, 8, 16, 32, 64] = NN+NNN
• R = 2: NN [4, 9, 21, 50, 120] = NN+NNN
• R = 3: NN [4, 9, 21, 51, 127] = NN+NNN
• R = 4: NN [4, 9, 21, 51, 127] = NN+NNN (saturated from R = 3)

Boundaries honored

No width above 6 enumerated; no Huawei budget; no #636 started; no STATUS/ROADMAP/RESEARCH-FRONTIER edits; nothing merged or closed; only new files added. The cyclic (periodic-y) strip is explicitly out of scope (Remark 6 of the note).

Full Matching-One repository CI has not been run for this commit.