diff --git a/constants/81a.md b/constants/81a.md
index d1060c3..4d6a3f4 100644
--- a/constants/81a.md
+++ b/constants/81a.md
@@ -8,28 +8,35 @@ $C\_{81a}$, Brun's Constant, is the sum of the reciprocals of the twin primes.
 
 | Bound                  | Reference      | Comments                                      |
 | ---------------------- | -------------- | --------------------------------------------- |
-| 2.175398               | [K2007]        |                                               |
-| 2.288513               | [PT2018]       |                                               |
+| 2.347                  | [CP2005]       | The first rigorous upper bound, from the Crandall--Pomerance argument; attributed as such in [PT2018, §1]. |
+| 2.175398               | [K2007]        | **Conditional on the Generalised Riemann Hypothesis.**  [PT2018, §1] records this as "under the assumption of the Generalised Riemann Hypothesis we have $B < 2.1754$", so it is not comparable with the unconditional rows. |
+| 2.288513               | [PT2018]       | Unconditional, and the value carried in the README table.  Sharpens the Crandall--Pomerance bound by about 13%. |
 
 
 ## Known lower bounds
 
 | Bound                  | Reference      | Comments                                      |
 | ---------------------- | -------------- | --------------------------------------------- |
-| 1.8267324395006        | [N1995]        | Led to the discovery of the Pentium FDIV bug. |
-| 1.83180806343237901198 | [N2010]        |                                               |
-| 1.840503               | [PT2018]       |                                               |
+| 1.8267324395006        | [N1995]        | A lower bound is just a partial sum, since $B$ is a sum of positive terms; this one comes from the enumeration to $10^{14}$ of that reference.  The much-quoted $B \approx 1.902$ from the same computations is an extrapolation, not a bound.  This enumeration led to the discovery of the Pentium FDIV bug. |
+| 1.83180806343237901198 | [N2010]        | Continues the same enumeration through the twin-prime pairs from $10^{16}$ to $2\cdot 10^{16}$. |
+| 1.840503               | [PT2018]       | Unconditional, and the value carried in the README table. |
 
 
 ## Additional comments and links
 
+- The upper bounds here are of two kinds.  An unconditional bound follows from a
+  Crandall--Pomerance style estimate; a smaller bound is available under the
+  Generalised Riemann Hypothesis.  The README table records the best
+  *unconditional* bound, which is why it reads $2.288513$ rather than the
+  smaller conditional $2.175398$ in the table above.
 - Similar constants exist for other prime families, e.g., cousin primes.
 - [Wikipedia page on Brun's theorem](https://en.wikipedia.org/wiki/Brun%27s_theorem)
 
 ## References
 
-- [K2007] Klyve, Dominic. [Explicit Bounds on Twin Primes and Brun’s Constant](https://doi.org/10.1349/ddlp.153).
-- [N1995] Nicely, Thomas. [Enumeration to 1e14 of the twin primes and Brun's constant](https://faculty.lynchburg.edu/~nicely/twins/twins.html).
-- [N2010] Nicely, Thomas. [Enumeration of the twin-prime pairs from 1e16 to 2e16](https://faculty.lynchburg.edu/~nicely/twins/t2_0001.html).
+- [CP2005] Crandall, Richard and Pomerance, Carl. *Prime Numbers: A Computational Perspective*, second edition, Springer, New York, 2005.
+- [K2007] Klyve, Dominic. [Explicit Bounds on Twin Primes and Brun’s Constant](https://doi.org/10.1349/ddlp.153).  PhD thesis, Dartmouth College.
+- [N1995] Nicely, Thomas. [Enumeration to 1e14 of the twin primes and Brun's constant](https://faculty.lynchburg.edu/~nicely/twins/twins.html).  (The Lynchburg pages return 404 as of 17 August 2026; the author died in 2019 and the trnicely.net domain has since been taken over by an unrelated site, so it is not a substitute.)
+- [N2010] Nicely, Thomas. [Enumeration of the twin-prime pairs from 1e16 to 2e16](https://faculty.lynchburg.edu/~nicely/twins/t2_0001.html).  (Also 404 as of 17 August 2026.)
 - [PT2018] Platt, Dave and Trudgian, Tim. [Improved bounds on Brun's constant](https://doi.org/10.48550/arXiv.1803.01925).