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

constants/87a.md

@@ -29,6 +29,7 @@ Upper bounds on $C_{87}$ come from explicit constructions of infinite tamely ram
 
 ## Additional comments and links
 
+- **The nearest visible improvement, and why it is out of reach.** The record above cuts by $\tau\_{\mathfrak{p}\_{13}}^2$, because Proposition 1.8(iii) gives only that the generator of tame inertia $\tau\_{\mathfrak{p}\_{13}}$ has depth *at least* two in $G\_S$. [HMR2019, §3.3.4] observes that if that depth were greater than two, one could cut by $\tau\_{\mathfrak{p}\_{13}}$ itself; the Golod–Shafarevich polynomial would then be at most $1 - 9t + 20t^2 + t^3$, which still has a root in $(0,1)$. The same degree-$12$ field $\mathrm{K}$ would then have an infinite $2$-Hilbert class field tower — no ramification at $13$ needed at all — and the totally real record would fall from $857.567$ to $\mathrm{rd}\_{\mathrm{K}} < 770.644$, a $10\%$ improvement with no new field. The authors write that they “do not see how to check the depth of $\tau\_{\mathfrak{p}\_{13}}$ in $G\_S$”, and [HMR2019, §5.1] raises the general obstruction as Question 5.1: given $S = \\{\mathfrak{p}\\}$ with $\mathfrak{p}$ coprime to $p$ and $G\_S$ infinite, how deep can $\tau\_{\mathfrak{p}}$ be? This is a hope rather than a bound, so nothing in the table above changes; but it is the cheapest known route to the next record.
 - **Sibling: totally complex Martinet constant.** There is an entirely analogous constant $C^{-}$ defined by taking the $\liminf$ over *totally complex* (a.k.a. totally imaginary) number fields. For that variant the corresponding history is: Martinet 1978 gives $92.368$; Hajir–Maire 2002 give $82.1004$; and Hajir–Maire–Ramakrishna 2019 give the current record $78.427$ (via a $9$-th root class-field tower, with $\mathrm{rd} \le 78.427$). The GRH lower bound in this signature is $44.763$ and the unconditional lower bound is $22.382$ ([Odl1990]). If the totally complex variant later becomes the subject of separate literature it could be split off as `87b`.
 - Martinet's constant has geometric implications far beyond number theory. Tsfasman–Vlăduț [TV1991] used towers with bounded root discriminant to construct **explicit lattice sphere packings** in high dimension. More recently, the same towers underpin the **large-degree number-field constructions** used in the [OpenAI counterexample to the Erdős unit distance conjecture](https://teorth.github.io/optimizationproblems/constants/84a.html) ([ABGLSSTWW2026]) and in the **disproof of the sum-product conjecture over the reals** ([84b](https://teorth.github.io/optimizationproblems/constants/84b.html); [BSSZ2026]). In both of these applications one needs $\mathrm{rd} \le O(1)^d$, which is exactly the Martinet regime; the [HMR2019] bound $857.567$ is precisely the "Martinet's constant" $C_2 \le 857.57$ quoted in [BSSZ2026, §5].
 - **Conditional vs. unconditional.** All the upper bounds above are *unconditional*: they are explicit constructions of infinite towers. The two [Odl1990] lower bounds have very different status — the unconditional bound $60.8$ can be improved only via genuine analytic progress on zero-density estimates for Dedekind zeta functions, whereas the GRH-conditional $215.3$ bound would immediately follow from a proof of GRH. Closing the gap between $215.3$ and $857.6$ would in particular disprove (a strong form of) GRH.