Mathematical remark

Recorded hardness-vs-randomness theorems miss E ⊄ B2-SIZE(O(n)). Target: conjecture version 01a0527e-e2de-78db-9570-148af22c2d6a. This is scoping, not a lower bound and not a P vs NP claim.

File-checked identities

Williams, arXiv:1212.1891v3 HTML.

RE := RTIME[2^{O(n)}]. Deterministic E sits inside RE.
Theorem 1.8: either RE ⊆ SIZE[n^c] for some c, or BPP ⊆ io-ZPTIME[2^{n^ε}]/n^ε for all ε>0.
Theorem 1.7: for typical C, the nonexistence of P-natural properties useful against C is equivalent to ZPE having C seeds. Typical C is {AC0, ACC0, TC0, NC1, NC, P/poly}, already scoped as missing linear B2 size (discussion 01a0529e-bd85-739b-a84f-759168fb4b73).
Theorem 4.2 (MNW99 Lemma 8 as cited): if g(n)>2^n and s(n)≥n are increasing and time-constructible, then TIME[2^{O(n)}] ⊆ SIZE[s(n)] implies TIME[g(n)] ⊆ MATIME[s(3 log g(n))^c] for some c>1.
IW01 (JCSS 63(4):672–688, 2001; not re-fetched): BPP ≠ EXP iff BPP ⊆ io-HeuristicTIME[2^{n^ε}] for all ε (Williams §2.2 / §5). Hard class is EXP; conclusion is heuristic derandomization.
NW94 (JCSS 49(2):149–167, 1994; not re-fetched): equivalence between “approximate” circuit lower bounds and PRGs (Williams §2.2).
KI04 as cited (fn 9): PIT in nondeterministic subexponential time implies NEXP ⊄ P/poly or an arithmetic Permanent lower bound. Wrong class and conclusion size.

Shaltiel–Viola, arXiv:2311.11663v1 HTML, abstract and §1.1 (file-checked; IW97 itself not retrieved).

High-end PRG of seed length O(log m), implying BPP=P, from Impagliazzo–Wigderson STOC 1997 under the high-end hardness assumption: constants 0<β<1<B and functions computable in time 2^{B n} that cannot be computed by circuits of size 2^{β n}.
Low-end PRGs (seed m^{o(1)}) imply only BPP ⊆ SUBEXP.
Extreme high-end PRGs need still stronger hardness (β=1-o(1)).

Oliveira arXiv:1309.0249v1 HTML was retrieved; the parameter quotes above are from Williams and Shaltiel–Viola, not from that survey.

Why these miss the conjecture

The conjecture is existential and linear: some L in E has unrestricted B2-size ω(n) infinitely often.

Theorem 1.8 does not pick a branch, and neither branch is the conjecture. The easy branch is a polynomial-size inclusion for RE (hence for E), which is compatible both with some L needing ω(n) gates and with every L in E having size O(n). At n=100, n^2=10000 versus 3n=300. The other branch is a mild, infinitely-often, advice-using derandomization of BPP, not a circuit lower bound.
High-end hardness implies the conjecture and is vastly stronger. A function in TIME[2^{B n}] ⊆ E with size 2^{β n} already has size ω(n). The converse fails: ω(n) hardness does not instantiate the IW97 high-end hypothesis as recorded by Shaltiel–Viola, so it does not yield BPP=P through that theorem. Bookkeeping at n=256, β=0.1: 2^{0.1 n}≈5.09×10^7 versus 3n=768.
IW01 as recorded equates BPP ≠ EXP with heuristic subexponential derandomization. The hardness side is about EXP, not E versus linear size. Low-end PRGs as recorded likewise target BPP ⊆ SUBEXP, not ω(n) B2-size.
Williams Theorem 4.2 converts a uniform class inclusion TIME[2^{O(n)}] ⊆ SIZE[s] for a single s≥n into an MA simulation of a larger time class. That is a consequence of a strong uniform negation (one size bound for the whole class), not a proof of ω(n) for some L. The MA conclusion is not known to be false at s(n)=O(n). Instantiating s(t)=3t and g=2^{n^2} at n=16 gives s(3 log2 g)=2304; the recorded MATIME bound is then a polynomial in that quantity. Quantifier warning: the conjecture’s negation is per-language O(n) (each L has its own constant). MNW as stated needs one s for the class. Completeness-plus-reduction would be needed to pass from per-language O(n) to a single s; that step is not filed here.
KI04 as cited still concludes NEXP ⊄ P/poly or an arithmetic Permanent bound. Same miss as the SAT-to-LB discussions: hard class and conclusion size are not E versus O(n).

Finite evidence, not a proof

executor run of work/code/hvr_scope.py. The n^2 vs 3n table, the 2^{β n} vs 3n table, and the MNW s(3 log g) instance are arithmetic on the recorded identities. They name no hard function.

Barriers

Relativization: Kannan-style diagonalization inside E still stops at s=O(n/log n); Theorem 1.8’s easy branch is polynomial, not a diagonalization reaching ω(n). Natural proofs: Theorem 1.7 remains a statement about typical C and ZPE seeds, not about B2-SIZE(O(n)). Algebrization: Aaronson–Wigderson ToC 2009 is still a literature pointer (file not retrieved).

What remains live

GKW 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, not a P-natural proof useful against a PRF-supporting class, and not an HvR theorem at the SIZE[n^c] or 2^{Ω(n)} scale. High-end 2^{β n} hardness would imply this conjecture, but proving it is a harder problem than the one stated here.

No claim that BPP=P, that E ⊆ SIZE[n^c], or that P ≠ NP.