Add a Dockerfile so the sync server is one command to deploy

Self-hosting meant building the binary by hand and wiring Postgres up
yourself. The image is a static binary on distroless with a built-in
healthcheck flag, and docker-compose brings up Postgres alongside it.

The image itself is unbuilt here: no container runtime is available in
this environment, so only the static Linux build and the healthcheck
flag have been exercised.

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Assisted-by: Crush:kimi-k3

Server/README.md

@@ -28,6 +28,30 @@ The schema is created on startup. Every request is logged as
 `METHOD PATH STATUS DURATION`; SIGTERM/SIGINT shut down gracefully so an
 in-flight push is not cut off mid-write.
 
+## Run with Docker
+
+`docker-compose.yml` starts Postgres and the server together:
+
+```bash
+cd Server
+POSTGRES_PASSWORD=$(openssl rand -hex 16) docker compose up -d
+docker compose logs -f server      # request log
+```
+
+Or build the image on its own and point it at an existing Postgres:
+
+```bash
+docker build -t prompt-sync Server/
+docker run -d --name prompt-sync -p 8080:8080 \
+  -e PROMPT_SYNC_DATABASE_URL="postgres://user:pass@host:5432/prompt_sync?sslmode=disable" \
+  prompt-sync
+```
+
+The image is a static binary on `distroless` (non-root, no shell, ~6 MB binary),
+so its `HEALTHCHECK` calls `prompt-sync -healthcheck`, which pings the local
+`/healthz` and exits non-zero on failure — no curl needed in the image. That is
+what `docker ps` reports as the container health.
+
 ## Deploy to a Linux host
 
 ```bash