Correct the Headroom finding: LiteLLM does not know the router's models; install headroom-ai on all three machines

docs/NEXT.md

@@ -113,6 +113,15 @@ The one place it would earn its keep here: a failed run in `Agents>agents` offer
 asks "what does this failure need" with four options and records the answer, the confidence and
 the options on the task. A person still decides, and it never decides a merge.
 
+## Headroom: measured facts
+
+- The **tool** path works: `headroom-ai` is now installed for Pi on `.13`, `.27` and `.51`, and
+  `compress()` answers from `http://192.168.20.13:8787` (measured: 2809 tokens before and after
+  for a single long user message, which headroom protects from compression).
+- The **provider** path fails, and not for the reason first given: `docker logs headroom` shows
+  LiteLLM rejecting the router's model ids, so a completion returns 502. The router answers the
+  same request in 1.1 s. Fixing it means teaching headroom's LiteLLM the router's models.
+
 ## Headroom: a tool, not a provider
 
 The `headroom-bridge` extension gives an agent two tools (`compress_for_agent`,