Attach the right witness to the 857.567 record in 87a

constants/87a.md

@@ -16,8 +16,8 @@ Upper bounds on $C_{87}$ come from explicit constructions of infinite tamely ram
 | ----- | --------- | -------- |
 | $1058.565$ | [Mar1978] | Martinet's original construction of infinite $2$-class field towers of totally real number fields. |
 | $954.293$ | [HM2002] | Hajir–Maire, refined Golod–Shafarevich with tame ramification. |
-| $913.493$ | [Mar2006] | Martin, further refinement of the [HM2002] construction. |
-| $857.567$ | [HMR2019] | Hajir–Maire–Ramakrishna, "cutting towers" via the refined Golod–Shafarevich criterion; explicit example is an $8$-th root class-field tower over the totally real field of [HM2002] with $2$-class group of rank $8$, yielding $\mathrm{rd} \le 3^4\cdot 5^4\cdot 7^4\cdot 13^2\cdot 29^4\cdot 53^2\cdot 109^2 \le 857.5662\dots$ Current record. |
+| $913.493$ | [Mar2006] | Martin, further refinement of the [HM2002] construction; the degree-$8$ field of [HMR2019, §3.3.1] has discriminant $3^4\cdot 5^4\cdot 7^4\cdot 13^2\cdot 29^4\cdot 53^2\cdot 109^2$ and root discriminant $< 913.4927$. |
+| $857.567$ | [HMR2019] | Hajir–Maire–Ramakrishna, "cutting towers" via the refined Golod–Shafarevich criterion. The totally real example [HMR2019, §3.3.3] is a *degree-$12$* field $\mathrm{K}$ with $\mathrm{rd}_{\mathrm{K}} < 770.6432$, cut at a single prime above $13$ of norm $13$, giving $\mathrm{rd}_{\mathrm{K}_S^{[1]}} = \mathrm{rd}_{\mathrm{K}}\cdot 13^{\frac{1}{12}(1-\frac{1}{2})} < 857.5662\dots$ — a saving of a factor $13^{1/24}$. Current record. |
 
 ## Known lower bounds