Mathematical remark

Recorded algebrization theorems do not prove some L in E outside B2-SIZE(O(n)), and they also forbid an algebrizing proof of that statement. Source: Chen–Hu–Ren, arXiv:2511.14038v1 HTML (ITCS 2026). Aaronson–Wigderson ToC 2009 was not file-retrieved; AW sentences below are CHR’s account. No P vs NP claim.

Model (CHR Def. 2.4–2.5 and footnote 11)

A separation C ⊄ D does not algebrize if there exist an oracle A and a low-degree extension à with C^à ⊂ D^A. This paper uses only multilinear extensions. E = DTIME[2^{O(n)}]; EXP = DTIME[2^{n^{O(1)}}]. SIZE^A[s] is A-oracle circuit size. The conjecture is oracle-free B2-SIZE; the barrier is the standard one: a technique that would still prove the separation in every such algebraic-oracle world cannot succeed if a counterexample world exists.

BPE is the exponential-time analogue of BPP used in CHR §1.2. Standard inclusion, not a CHR theorem: E ⊆ BPE, and this lifts to E^Ã ⊆ BPE^Ã.

Matching barrier: Theorem 1.5

CHR Theorem 1.5: there exist A2 and its multilinear extension Ã_2 such that

BPE^{Ã_2} ⊆ SIZE^{A2}[O(n)]

(all input lengths). CHR state this as an algebrization barrier to proving BPE ⊄ SIZE[O(n)], “i.e., an infinitely-often circuit lower bound for BPE.”

That is the same quantifier as the conjecture (C_n = ω(n) for infinitely many n), for a class containing E. In that world E^{Ã_2} ⊆ BPE^{Ã_2} ⊆ SIZE^{A2}[O(n)]. By Def. 2.4, E ⊄ SIZE[O(n)] does not algebrize.

Nearby statements that are not the conjecture

1. CHR’s account of AW: an oracle A and a multiquadratic extension à with BPEXP^à ⊆ P^A/poly. Wrong time class (BPEXP, not E), wrong size (polynomial, not linear), and a weaker extension. That sentence does not by itself block E ⊄ SIZE[O(n)].

2. Theorem 1.3: pr-PostBPE^{Ã_1} ⊆ i.o.-SIZE^{A1}[O(n)]. This blocks an almost-everywhere linear-size lower bound for a larger class. The conjecture is only infinitely-often, for E. CHR contrast this with the algebrizing MA_E ⊄ P/poly bound (AW Theorem 3.17 as cited).

3. Theorem 1.7 / 5.4: E^{Ã_3} and a robust MA_E subclass have super-half-exponential A3-oracle circuits. That blocks proving C_n ≥ h(n) for super-half-exponential h, not ω(n). Bookkeeping, not a proof (work/code/algeb_linear_scope.py): at n=256, 2^{√n}=65536 vs 3n=768.

What is not forbidden

CHR write that all super-polynomial lower bounds against general circuits they are aware of are algebrizing. Gate elimination, GKW Theorem 1.1 / Open Problem 1.1, and other combinatorial linear-size methods are not recorded there as algebrizing. An algebrizing proof of the conjecture is ruled out; a non-algebrizing proof is not.

The remaining GKW unrestricted-size route is unchanged: Open Problem 1.1 with δ© > 1−1/γ©, plus a matching depth-3 bound for some L in E, or a method that is not a GKW inversion and does not algebrize.

Finite checks are evidence only. Li–Yang 3.1n is still not file-verified; the conjecture significance is left unchanged.