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/routes/+layout.svelte

@@ -5,6 +5,7 @@
 	import { goto } from "$app/navigation";
 	import { page } from "$app/state";
 	import { dev } from "$app/environment";
+	import { appBase, withBase } from "$lib/paths";
 	import { i18n } from "$lib/i18n/index.svelte";
 	import { session } from "$lib/session-store.svelte";
 	import { eventStream } from "../api/events.svelte";
@@ -26,7 +27,9 @@
 		// in svelte.config.js) so `vite dev` and the dev-server e2e suite
 		// run without the worker intercepting requests.
 		if (!dev && "serviceWorker" in navigator) {
-			void navigator.serviceWorker.register("/service-worker.js");
+			void navigator.serviceWorker.register(withBase("/service-worker.js"), {
+				scope: withBase("/"),
+			});
 		}
 		return () => {
 			eventStream.stop();
@@ -75,7 +78,9 @@
 			streamSessionId = null;
 		}
 
-		const pathname = page.url.pathname;
+		// page.url.pathname includes the configured base path; strip it so
+		// the route comparisons below stay base-agnostic.
+		const pathname = page.url.pathname.slice(appBase.length);
 		// Debug-only routes under /__debug/* run their own bootstrap
 		// path against the storage primitives and must bypass the
 		// auth guard so Phase 6's Playwright spec can drive the
@@ -84,9 +89,9 @@
 			return;
 		}
 		if (session.status === "anonymous" && pathname !== "/login") {
-			void goto("/login", { replaceState: true });
+			void goto(withBase("/login"), { replaceState: true });
 		} else if (session.status === "authenticated" && pathname === "/login") {
-			void goto("/lobby", { replaceState: true });
+			void goto(withBase("/lobby"), { replaceState: true });
 		}
 	});
 </script>