feat(deploy): single-origin path-based deployment + project site

Serve the whole stack behind one host: site at /, game UI at /game/,
gateway REST at /api + /healthz, Connect at /rpc (prefix stripped by the
edge Caddy). The built artifact is domain-agnostic — the UI talks to the
gateway same-origin via relative URLs, so the same bundle runs under any
host with no rebuild and with CORS disabled.

- Rename the Connect proto service galaxy.gateway.v1.EdgeGateway ->
  edge.v1.Gateway; regenerate Go + TS; public path /rpc/edge.v1.Gateway.
- Move the game UI under base path /game (env BASE_PATH); make the
  manifest, service-worker scope, WASM loader, and all navigation
  base-aware via a withBase helper.
- Relative API + /rpc Connect prefix; Vite dev proxy mirrors the strip.
- Rewrite the edge Caddy (dev + prod) for path-based routing; empty CORS
  allow-lists (same-origin); single host.
- New VitePress project site (site/): i18n en/ru with switcher, LaTeX
  math, minimal monospace theme; built and served at /.
- dev-deploy compose/Makefile + CI (dev-deploy, prod-build, new
  site-build) build and seed the site; probes hit /, /game/, /healthz.
- Sync docs (ARCHITECTURE, gateway README/openapi, dev-deploy &
  local-dev READMEs, CLAUDE.md, ui/PLAN).

Co-Authored-By: Claude Opus 4.7 (1M context) <noreply@anthropic.com>

ui/frontend/src/service-worker.ts

@@ -9,14 +9,16 @@
 //
 // SvelteKit registers this worker automatically in the production build.
 
-import { build, files, version } from "$service-worker";
+import { base, build, files, version } from "$service-worker";
 
 const sw = self as unknown as ServiceWorkerGlobalScope;
 
 const CACHE = `galaxy-cache-${version}`;
-// "/" is the SPA shell (adapter-static fallback); precaching it makes the
-// start_url load offline.
-const PRECACHE = ["/", ...build, ...files];
+// `${base}/` is the SPA shell (adapter-static fallback); precaching it
+// makes the start_url load offline. `base` is empty at the root and
+// `/game` under the single-origin deployment, and `$service-worker`
+// derives it from `location.pathname` so it stays correct in a subdir.
+const PRECACHE = [`${base}/`, ...build, ...files];
 
 sw.addEventListener("install", (event) => {
 	event.waitUntil(
@@ -65,7 +67,7 @@ sw.addEventListener("fetch", (event) => {
 				const cached = await cache.match(request);
 				if (cached) return cached;
 				if (request.mode === "navigate") {
-					const shell = await cache.match("/");
+					const shell = await cache.match(`${base}/`);
 					if (shell) return shell;
 				}
 				throw err;