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The following are source statements or pointers, not additional complete proof reconstructions.

Earlier maxima. Published pp.793–796 compare $M(x)$ with the number $W(x)$ of totient values, using Pollack–Pomerance–Treviño and Ford/Maier–Pomerance. Remark 1.3 quotes older bounds for the nonincreasing maximum $M^\downarrow$ and the constant maximum $M^0$, including

$$M^0(x)+x^{0.18}<M^\downarrow(x) \ll x\exp\!\left(-(\tfrac12+o(1)) \sqrt{\log x\log_2x}\right)$$

and

$$x^{0.7156}\ll M^0(x) \ll x\exp\!\left(-(1+o(1)) \frac{\log x\log_3x}{\log_2x}\right).$$

These quoted results, their dependence on the cited shifted-prime theorems, and their later status are not independently compiled here. They are not necessary inputs to Tao's new main argument.

Finer weak-maximum conjectures. The source records $M(x)\le\pi(x)+O(1)$ and the more precise conjecture $M(x)=\pi(x)+64$ for $x\ge31957$, with historical numerical evidence. Remark 4.3 says the latter would imply Legendre's conjecture at all primes, using “a little more computation” (published p.812). That finite baseline or certificate is not supplied here; only the complete eventual implication is compiled. No present-status assertion about these finer conjectures follows from their appearance in the 2024 paper.

RH comparison. Remark 4.4 imports Selberg's 1943 estimate

$$\sum_{n\le x}\frac{(p_{n+1}-p_n)^2}{p_n}\ll\log^3x$$

under the Riemann hypothesis. Its cited application estimates how many prime-square insertions can arise from the specified prime gaps. The Selberg theorem and that ancillary counting consequence are kept as an external pointer here. In particular, this is not a theorem $M(x)-\pi(x)=O(\log^3x)$ under RH.

Prime-tuples domain. The introductory statement on published p.812 claims a positive singular series for prime pairs $p,ap+b$ assuming only $(a,b)=1$ and $a>0$. This omits local admissibility: for example $a=b=1$ admits only $p=2$ with $p+1$ prime. The intended conjectural comparison uses two distinct linear forms $t,at+b$ with $b\ne0$ and no prime dividing their product for every integer $t$. Under $(a,b)=1$, the prime two requires that $a,b$ are not both odd. This qualification concerns the background statement, not the self-contained hypothesis of Proposition 4.5.

Maynard comparison. On published p.813 the source cites Maynard's dense-clusters theorem, Polymath's parameter calculations, and Ford's Lemma 2 to assert that some $1\le k\le49$ has

$$\#\{p\le x:\lceil p/2^k\rceil\text{ prime}\} \gg x/\log^{50}x$$

for infinitely many $x$. The quoted combination of external results is not proved here and is not an input to Proposition 4.5.

Other proposed improvements. Section 4.3 asks about a power-saving bound $M(x)=\pi(x)+O(x^\theta)$, $\theta<1$, and an analogous asymptotic on intervals $(x,x+x^\theta]$. Its proven local observations are the half bound and the composite barrier. The surrounding proposals remain dated questions rather than conclusions of the compiled proof.

Source. Tao, published paper, published pp.793–796 and 811–815, historical and conditional remarks. This page uses that published version.

Bears on. Problem 49.