Completely rewritten. (9/26)
It seems indeed that nothing like Theorem 1.10 from Mochizuki's IUTT-IV could hold.
Here is an infinite set of counterexamples, assuming for convenience two standard conjectures (the first being in fact a consequence of ABC), that contradict Thm. 1.10 very badly.
Assumptions:
A (Consequence of ABC) For all but finitely many elliptic curves over $\mathbb{Q}$, the conductor $N$ and the minimal discriminant $\Delta$ satisfy $\log{|\Delta|} < (\log{N})^2$.
B (Uniform Serre Open Image conjecture) For each $d \in \mathbb{N}$, there is a constant $c(d) < \infty$ such that for every number field $F/\mathbb{Q}$ with $[F:\mathbb{Q}] \leq d$, and every non-CM elliptic curve $E$ over $F$, and every prime $\ell \geq c(d)$, the Galois representation of $G_F$ on $E[\ell]$ has full image $\mathrm{GL}_2(\mathbb{Z}/{\ell})$. (In fact, it is sufficient to take the weaker version in which $F$ is held fixed. )
Further, as far as I can tell from the proof of Theorem 1.10 of IUTTIV, the only reason for taking $F := F_{\mathrm{tpd}}\big( \sqrt{-1}, E_{F_{\mathrm{tpd}}}[3\cdot 5] \big)$ --- rather than simply $F := F_{\mathrm{tpd}}(\sqrt{-1})$ --- was to ensure that $E$ has semistable reduction over $F$. Since I will only work in what follows with semistable elliptic curves over $\mathbb{Q}$, I will assume, for a mild technical convenience in the examples below, that for elliptic curves already semistable over $F_{\mathrm{tpd}}$, we may actually take $F := F_{\mathrm{tpd}}(\sqrt{-1})$ in Theorem 1.10.
The infinite set of counterexamples. They come from Masser's paper [Masser: Note on a conjecture of Szpiro, Asterisque 1990], as follows. Masser has produced an infinite set of Frey-Hellougarch (i.e., semistable and with rational 2-torsion) elliptic curves over $\mathbb{Q}$ whose conductor $N$ and minimal discriminant $\Delta$ satisfy $$ (1) \hspace{3cm} \frac{1}{6}\log{|\Delta|} \geq \log{N} + \frac{\sqrt{\log{N}}}{\log{\log{N}}}. $$ (Thus, $N$ in these examples may be taken arbitrarily large. ) By (A) above, taking $N$ big enough will ensure that $$ (2) \hspace{3cm} \log{|\Delta|} < (\log{N})^2. $$ Next, the sum of the logarithms of the primes in the interval $\big( (\log{N})^2, 3(\log{N})^2 \big)$ is $2(\log{N})^2 + o((\log{N})^2)$, so it is certainly $> (\log{N})^2$ for $N \gg 0$ big enough. Thus, by (2), it is easy to see that the interval $\big( (\log{N})^2, 3(\log{N})^2 \big)$ contains a prime $\ell$ which divides neither $|\Delta|$ nor any of the exponents $\alpha = \mathrm{ord}_p(\Delta)$ in the prime factorization $|\Delta| = \prod p^{\alpha}$ of $|\Delta|$.
Consider now the pair $(E,\ell)$: it has $F_{\mathrm{mod}} = \mathbb{Q}$, and since $E$ has rational $2$-torsion, $F_{\mathrm{tpd}} = \mathbb{Q}$ as well. Let $F := \mathbb{Q} \big( \sqrt{-1}\big)$. I claim that, upon taking $N$ big enough, the pair $(E_F,\ell)$ arises from an initial $\Theta$-datum as in IUTT-I, Definition 3.1. Indeed:
- Certainly (a), (e), (f) of IUTT-I, Def. 3.1 are satisfied (with appropriate $\underline{\mathbb{V}}, \, \underline{\epsilon}$);
- (b) of IUTT-I, Def. 3.1 is satisfied since by construction $E$ is semistable over $\mathbb{Q}$;
- (c) of IUTT-I, Def. 3.1 is satisfied, in view of (B) above and the choice of $\ell$, as soon as $N \gg 0$ is big enough (recall that $\ell > (\log{N})^2$ by construction!), and by the observation that, for $v$ a place of $F = \mathbb{Q}(\sqrt{-1})$, the order of the $v$-adic $q$-parameter of $E$ equals $\mathrm{ord}_v (\Delta)$, which equals $\mathrm{ord}_p(\Delta)$ for $v \mid p > 2$, and $2\cdot\mathrm{ord}_2(\Delta)$ for $v \mid 2$;
while $\mathbb{V}_{\mathrm{mod}}^{\mathrm{bad}}$ consists of the primes dividing $\Delta$;
- Finally, (d) of IUTT-I, Def. 3.1 is satisfied upon excluding at most four of Masser's examples $E$. (See page 37 of IUTT-IV).
Now, take $\epsilon := \big( \log{N} \big)^{-2}$ in Theorem 1.10 of IUTT-IV; this is certainly permissible for $N \gg 0$ large enough. I claim that the conclusion of Theorem 1.10 contradicts (1) as soon as $N \gg 0$ is large enough.
For note that Mochizuki's quantity $\log(\mathfrak{q})$ is precisely $\log{|\Delta|}$ (reference: see e.g. Szpiro's article in the Grothendieck Festschrift, vol. 3); his $\log{(\mathfrak{d}^{\mathrm{tpd}})}$ is zero; his $d_{\mathrm{mod}}$ is $1$; and his $\log{(\mathfrak{f}^{\mathrm{tpd}})}$ is our $\log{N}$. By construction, our choice $\epsilon := \big( \log{N} \big)^{-2}$ then makes $1/\ell < \epsilon$ and $\ell < 3/\epsilon$, whence the finaly display of Theorem 1.10 would yield $$ \frac{1}{6} \log{|\Delta|} \leq (1+29\epsilon) \cdot \log{N} + 2\log{(3\epsilon^{-8})} < \log{N} + 16\log{\log{N}} + 32, $$ where we have used $\epsilon \log{N} = (\log{N})^{-1} < 1$ for $N > 3$, and $2\log{3} < 3$.
The last display contradicts (1) as soon as $N \gg 0$ is big enough.
Thus Masser's examples yield infinitely many counterexamples to Theorem 1.10 of IUTT-IV (as presently written).
Added on 10/15. Mochizuki has commented on the apparent contradiction between Masser's examples and Theorem 1.10:
He writes that he will revise portions of IUTT-III and IUTT-IV, and will make them available in the near future. He seems (to me) to suggest the revision (3): $$ \frac{1}{6} \log{|\Delta|} < \log{N} + \omega(N)\cdot \log{\log{N}} + O\big( (\log{\log{N}}) \big), $$ of the preceding display, which is certainly plausible, being in tune with Baker's conjecture: it would be a terrific, explicit version of ABC! Here, $\omega(N)$ is the number of prime factors of $N$. (Technically speaking, Mochizuki's $\log{\mathfrak{q}}$ is almost our $\log{|\Delta|}$, except that he doesn't count the contribution at the prime $2$. But this makes no difference if we restrict to $abc$-triples of bounded $2$-adic valuation - or elliptic curves whose minimal discriminant has bounded $2$-adic valuation).
However, to really get this plausible version (3), I believe he ought to have written simply "$\omega(N)\cdot \log{\log{N}}$" for the adjustment term at the beginning of p. 3 of the above Comments: that is, without the "$-\log{N}$" term! (The inequality in (3) without the $\log{N}$-term on the right is certainly false). Another seemingly problematic point from his Comments, which left me even more perplexed, is the second sentence of (4.): if "ramification index" and "moderately ramified" are taken in the standard, obvious sense (as they are in Definition 1.9), then nothing would change at all in the case of the Masser examples alluded to above! (Those examples are over $\mathbb{Q}$, and are already semistable and with rational $2$-torsion, hence one may simply take $F := \mathbb{Q}(\sqrt{-1})$ - a field that is absolutely unramified at all the odd places! - without having to adjoin the coordinates of the points of order $15$. )
Thus I think the problem might lie deeper than "the fact that the radius of convergence of the $p$-adic log/exp series is $p^{-1/(p-1)}$."