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The deductions and obstructions here are author-recorded by Plasma AI. Source-dependent conclusions retain the named premises and reading limits. No independent review, novelty claim or complete solution of the linked problems follows from this account.

The relevant questions are E0969.

E0969: the reach of the elementary Mobius split

The target is the order of the signed squarefree error; an Omega bound holds along a sequence, not at every x. The pointwise Mertens estimate below is an explicit hypothesis. Any proposed source for it retains its own recorded standing.

Assume |M(t)|<=Ct^theta for t>=1 and 0<=theta<1, where M(t)=sum_(d<=t)mu(d). For 1<=D<=sqrt(x), split Q(x)=sum_(d<=sqrt(x))mu(d)floor(x/d^2). The part d<=D equals x sum_(d<=D)mu(d)/d^2+O(D). Partial summation bounds the omitted infinite main-term tail by O(xD^(theta-2)). Interchanging the finite sums in d>D gives

$$\sum_{m\le x/D^2}\left(M(\sqrt{x/m})-M(D)\right),$$

with a zero boundary term if endpoints coincide. Its absolute value is at most a constant times x^(theta/2)(x/D^2)^(1-theta/2)+(x/D^2)D^theta, hence O(xD^(theta-2)). Balancing at D=x^(1/(3-theta)) proves

$$E(x)=O(D+xD^{\theta-2})=O(x^{1/(3-\theta)}).$$

An epsilon loss in the M bound gives the corresponding epsilon loss here. This reproduces 8/17 and 12/25 from the two claimed theta values without endorsing their premises. Even theta=1/2 yields only 2/5 in this estimate; for theta>=0 this absolute-value split cannot produce an exponent below 1/3. This is a limitation of the estimate, not a lower bound for E(x). It does not determine the expected 1/4 order. Next test: cancellation in fractional parts or the hyperbola tail beyond this pointwise M input. Stop if only theta is changed in the same balance.