diff --git a/constants/45a.md b/constants/45a.md
index f9f7316..052f2b2 100644
--- a/constants/45a.md
+++ b/constants/45a.md
@@ -27,7 +27,9 @@ $C_{45}$ is the asymptotic density (if it exists) of the set of odd integers tha
 | $0.107648$ | [CE2018] |  |
 
 
-## Certificate for the $0.490249407811155$ upper-density bound
+## Certificate for the earlier $0.490249407811155$ upper-density bound
+
+The current recorded upper bound is $0.490180063290061$ (see the table above); this section preserves the earlier finite certificate for reference.
 
 Let
 $$
diff --git a/constants/47a.md b/constants/47a.md
index abf49d8..26a7386 100644
--- a/constants/47a.md
+++ b/constants/47a.md
@@ -39,8 +39,8 @@ the optimal weak-type $(1,1)$ constant of the centered Hardy–Littlewood maxima
 
 | Bound | Reference | Comments |
 | ----- | --------- | -------- |
-| $\dfrac{3}{4}-\dfrac{\sqrt{2}}{4}+\dfrac{\sqrt{6}}{2}\approx 1.6211915$ | [Ald2000] | Aldaz's Proposition 1.4 gives a lower bound in every dimension $n\ge 2$. Specializing the formula to $n=2$ gives the displayed value. [Ald2000-prop1.4] |
 | $\dfrac{11+\sqrt{61}}{12}\approx 1.5675208$ | <a href="#Mel2003">[Mel2003]</a>, <a href="#Ald2011">[Ald2011]</a> | Melas proved $c_1=\dfrac{11+\sqrt{61}}{12}$. Since $c_{d+1}\ge c_d$, we get $c_2\ge c_1$. <a href="#Mel2003-c1-formula">[Mel2003-c1-formula]</a> <a href="#Ald2011-monotone">[Ald2011-monotone]</a> |
+| $\dfrac{3}{4}-\dfrac{\sqrt{2}}{4}+\dfrac{\sqrt{6}}{2}\approx 1.6211915$ | [Ald2000] | Aldaz's Proposition 1.4 gives a lower bound in every dimension $n\ge 2$. Specializing the formula to $n=2$ gives the displayed value. [Ald2000-prop1.4] |
 
 ## Additional comments and links