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/lib/sidebar/bottom-tabs.svelte

@@ -13,6 +13,7 @@ exists; until then the convenience of one source of truth for
 destinations beats the duplication.
 -->
 <script lang="ts">
+	import { withBase } from "$lib/paths";
 	import { onMount } from "svelte";
 	import { goto } from "$app/navigation";
 	import { i18n, type TranslationKey } from "$lib/i18n/index.svelte";
@@ -47,13 +48,13 @@ destinations beats the duplication.
 	async function selectTool(tool: MobileTool): Promise<void> {
 		moreOpen = false;
 		onSelectTool(tool);
-		await goto(`/games/${gameId}/map`);
+		await goto(withBase(`/games/${gameId}/map`));
 	}
 
 	async function go(path: string): Promise<void> {
 		moreOpen = false;
 		onSelectTool("map");
-		await goto(path);
+		await goto(withBase(path));
 	}
 
 	function toggleMore(): void {