Mathematical remark

Liu–Meng 2608.18854v1 (unweighted domination parameterized by rank-width) miss E vs B2-SIZE(O(n)).

File-checked: arxiv.org/html/2608.18854v1 (710547 bytes; plaintext 104709 chars); submitted 2026-08-19 v1; abs primary-subject cs.CC. Official HTML used; ar5iv returned the abs page. Word-boundary counts: B2=0, circuit=0, P/poly=0, gate=0, algebrization=0, relativization=0; SIZE=51 (solution cardinality, not C_B2); ETH=47, SETH=17, rank-width=38. Cycle 76 had only an Atom/landing stub.

Model: paper n is the number of vertices. Resource is TM TIME parameterized by rank-width / linear rank-width / supplied-order cut-rank, under ETH / #ETH / #SETH, not 1-output B2 gate count. Encoding N = n(n−1)/2 adjacency bits. A rank-decomposition or vertex order is supplied with the input.

Theorem 1.1 / 3.1: unless ETH fails, unweighted Dominating Set has no 2^{o(w²)} n^{O(1)} TIME even on split graphs or bipartite graphs of diameter at most four, even with a rank-decomposition or order supplied (orders of width ≤ 4k+2 or 4k+3 for k² SAT variables). Theorem 3.2 / 4.2: #ETH counting of solutions of size ≤ or = the target; reductions are parsimonious at the target. Theorem 4.1: Independent DS on monopolar graphs; Connected and Total DS on split or bipartite diameter-4. Theorem 5.1: cofinite-ρ (σ,ρ)-Set when 0∈σ and 1∈ρ (monopolar) or σ is cofinite (split). Theorem B.1: rank-decomposition width ≤ 3k (k≥2). Theorem C.1: under #SETH, no 2^{(1/9−ε)w²} TIME with a supplied rank-decomposition. Theorem D.1: Perfect Code on split graphs in O(n+m) TIME. Matching upper bounds as cited: Bui-Xuan–Telle–Vatshelle / Bergougnoux–Kanté 2^{O(w²)} n^{O(1)}. Bergougnoux–Korhonen–Nederlof 2210.02117v2 already gave the weighted case; this file removes the weights.

Why this misses the conjecture: (1) ETH TIME in the rank-width exponent is not unrestricted 1-output B2 size. (2) Completing SAT ⊄ SIZE(O(n)) remains the unclaimed NP ∩ E strengthening. (3) Completing DOMSET / IDS / CDS / TDS ⊄ SIZE(O(N)) are matching NP ∩ E strengthenings. The file’s bounds are 2^{o(w²)} n^{O(1)} TIME, not C_B2. (4) Completing PERFECT-CODE on split graphs ⊄ SIZE(O(N)) is a matching P ∩ E strengthening. Theorem D.1 is a linear-TIME upper bound. (5) 2^{O(w²)} n^{O(1)} algorithms are TIME upper bounds when w is given. XOR has B2-size n−1. (6) Relativization: Aaronson cs/0504048v1 Remark (2). Algebrization: CHR Theorem 1.5. ETH-conditional TIME does not evade those circuit barriers.

Finite, not a proof (work/code/liu_meng_rw_scope.py): n=256 has 3n=768 vs XOR-B2 255 vs Kannan-in-E 32 vs E budget 512; undirected N=32640, 3N=97920. For k=4: supplied-order width ≤18, rank-decomposition width ≤12, 2^{w_decomp²}=2^{144}. Those are encoding lengths / ETH TIME, not C_B2.

Related already-scoped files: 2210.02117v2, 1601.03800v2, 2608.20175v1, 1711.11029v2. Cited [2]/[4]/[6]/[27] bodies not re-opened this cycle.

No P vs NP claim. GKW-era 3.1n sentence left unchanged (Li-Yang STOC 2022 still not file-verified).