Mathematical remark

Osipov 2608.22662v1 (step-recursion resource profiles) miss E vs B2-SIZE(O(n)).

File-checked: arxiv.org/html/2608.22662v1 (411009 bytes; plaintext 61788 chars); submitted 2026-08-23 v1; abs primary-subject cs.CC, also cs.LO. Official HTML used; ar5iv returned the abs page. Companion arXiv:2608.04871 Atom-listed, body not opened. Word-boundary counts: B2=3 are not the binary Boolean basis; SIZE=3 are encoded machine-state width. circuit=0, P/poly=0, ETH=0, SETH=0, gate=0, algebrization=0, relativization=0. Boolean=16 are D/∃/∀/A computation-tree aggregators.

Model: multi-tape TMs, N:=|x|+2, branch modes Q∈{D,∃,∀,A}. TISP_Q(t,s) is path time O(t) and work space O(s). Step-recursion width u is mutable-state bits; descent ρ supplies depth δ_ρ(u). Definition 2.8 is bounded-state dynamics with Boolean semantics BSR^Q. Lemma 2.9 realizes every deterministic instance by one ordinary bounded step recursion over a fixed finite numerical basis. Lemma 2.10 compiles one-step TM transitions into those local maps.

Theorem 3.1: Q-semantics of a width-u dynamics is in TISP_Q(δ_ρ(u) u^{O(1)}, O(u)); conversely a total Q-machine compiles when u stores configurations and δ_ρ(u) dominates branch length. Theorem 3.2: if W is ρ-profile-closed, SR^Q_ρ[W] = Prof_Q(ρ,W) after union over widths. Corollary 3.3: generator growth of φ determines the TISP family. Theorem 3.6: on polynomial widths, δ_{τ_r}(u)=2^{Θ((log_2 u)^r)} and an infinite strict descent-quotient chain between the divisive and predecessor rows; the file states this does not assert language-class separation. Theorem 3.9: divisive row = TIME_Q(W); predecessor row = SPACE_Q(W). Section 4 calibrates P/NP/PSPACE/EXP on W_0 and E/NE/ESPACE on W_E as TM classes. Corollary 4.1: counting aggregator = #P.

Why this misses the conjecture: (1) TISP / recursion-algebra equality is not unrestricted B2 size. (2) “Boolean” is computation-tree aggregation, not the 16-element basis B2. (3) Calibrating E = DTIME(2^{O(N)}) restates the definition of E; it does not prove E ⊄ B2-SIZE(O(n)). (4) Completing SAT ⊄ SIZE(O(n)) remains the unclaimed NP ∩ E strengthening. (5) Relativization: Aaronson cs/0504048v1 Remark (2). Algebrization: CHR Theorem 1.5.

Finite, not a proof (work/code/osipov_sr_scope.py): n=256 has 3n=768 vs XOR-B2 255 vs Kannan-in-E 32 vs E budget 512; divisive depth stand-in 256 and Thm 3.6 r=2 stand-in 2^{64} are profile depths, not C_B2.

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