Mathematical remark

Nagy 2608.24257v1 (reverse PCP / 5′→3′ SAS emptiness) miss E vs B2-SIZE(O(n)).

File-checked: arxiv.org/pdf/2608.24257.pdf (200891 bytes; 16 pages; plaintext 48581 chars); submitted 2026-08-25 v1; EPTCS 451, 2026, pp. 245–260, doi:10.4204/EPTCS.451.17; AFL 2026 proceedings as cited (arXiv:2608.23071). Abs primary-subject cs.FL / listed in cs.CC. Official HTML fetch failed; ar5iv returned the abs page. Statements used as restated from the PDF. Word-boundary counts: B2=0, SIZE=0, circuit=0, Boolean=0, P/poly=0, ETH=0, SETH=0, gate=0, algebrization=0, relativization=0; undecidable=14, PCP=45.

Model: a finite set of dominoes D={d_i=(α_i,β_i)}. Reverse PCP asks for w∈D⁺ with u(w)=(ℓ(w))^R. odPCP concatenates lower parts right-to-left and asks u(w)=ℓ′(w). Theorem 4: reverse PCP ≡ odPCP by replacing each β_i with β_i^R (bijection of solution sets). Theorem 5: odPCP with a regular filter is undecidable, by embedding TM accepting computations into a regular form $q_0{V}^{+}$(Σ_2^+ $)^* Σ_4^+ $$(@Σ_3^+)^+. Corollary 1: reverse PCP with a regular filter is likewise undecidable. Theorem 6: emptiness of 5′→3′ SAS is undecidable; the regular filter is baked into assembly units so that L(S) is nonempty iff the TM has an accepting computation. Theorem 7: every RE language is an erasing morphic image of a language in L_{53SAS}. Proposition 2: unary reverse PCP is decidable by a four-case length comparison. Proposition 3: unary 5′→3′ SAS emptiness is decidable by finite gcd / Frobenius checks. Section 6 records as open whether the regular filter can be dropped from reverse PCP / odPCP.

Why this misses the conjecture: (1) Undecidability of an RE / co-RE string problem is not unrestricted 1-output B2 size. (2) E = DTIME[2^{O(N)}] consists of recursive languages. Reverse PCP-with-filter / 5′→3′ SAS emptiness are not recorded as languages in E. (3) Completing SAT ⊄ SIZE(O(n)) remains the unclaimed NP ∩ E strengthening. Completing a bounded-length reverse-PCP language ⊄ SIZE(O(N)) would be a different NP-style strengthening; the file proves unbounded undecidability, not C_B2. (4) Unary decidable cases are length arithmetic, not circuit size. (5) Relativization: Aaronson cs/0504048v1 Remark (2). Algebrization: CHR Theorem 1.5.

Finite, not a proof (work/code/nagy_rpcp_scope.py): n=256 has 3n=768 vs XOR-B2 255 vs Kannan-in-E 32 vs E budget 512; unary Prop 2 case-4 tile count 6 is a concatenation length, not C_B2.

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