C_87: note the depth-of-inertia route to 770.644 as the nearest visible improvement

[Chessing234]

The follow-up you suggested when merging #151: a line in "Additional comments and links" for [HMR2019, §3.3.4].

The record row cuts by $\tau_{\mathfrak{p}_{13}}^2$ because Proposition 1.8(iii) only gives depth at least two. §3.3.4 says that if the depth were greater than two, cutting by $\tau_{\mathfrak{p}_{13}}$ itself would give Golod–Shafarevich polynomial at most $1 - 9t + 20t^2 + t^3$, which still has a root in $(0,1)$ — so the same degree-12 field would have an infinite 2-Hilbert class field tower with no ramification at 13 at all, and the record would fall to $\mathrm{rd}_{\mathrm{K}} < 770.644$.

Quoted verbatim from the preprint:

In the example above, a hope would be that $\tau_{p_{13}}$ has depth greater than two in $G_S$. In that case we could cut by the relation $\tau_{p_{13}}$ and the corresponding Golod-Shafarevich polynomial would be at most $1 - 9t + 20t^2 + t^3$ which has a root in $(0,1)$. One would then have that $K$ has infinite 2-Hilbert Class Field Tower and the totally real root discriminant record would be $< 770.644$. We do not see how to check the depth of $\tau_{p_{13}}$ in $G_S$. See also the beginning of §5.

I added the pointer to §5.1, since that is where the obstruction is stated in general — Question 5.1 asks how deep the generator of tame inertia at a single prime coprime to $p$ can be when $G_S$ is infinite. As you said, this is a hope rather than a bound, so no table row and no README change: the bullet says explicitly that nothing above it changes.

Rendering: the bullet is inline math only, so underscores are escaped and the one literal brace pair is written \\{ \\}; there are no pipes in it, and the two tables still have four | per row.

Prepared with AI assistance; the §3.3.4 and §5.1 text was read from the arXiv PDF via pdftotext and the quotation above is transcribed from it.