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/static/manifest.webmanifest

@@ -2,28 +2,28 @@
 	"name": "Galaxy",
 	"short_name": "Galaxy",
 	"description": "Galaxy — a turn-based space strategy game.",
-	"id": "/",
-	"start_url": "/",
-	"scope": "/",
+	"id": "./",
+	"start_url": "./",
+	"scope": "./",
 	"display": "standalone",
 	"orientation": "any",
 	"background_color": "#0a0e1a",
 	"theme_color": "#0a0e1a",
 	"icons": [
 		{
-			"src": "/icons/icon-192.png",
+			"src": "icons/icon-192.png",
 			"sizes": "192x192",
 			"type": "image/png",
 			"purpose": "any"
 		},
 		{
-			"src": "/icons/icon-512.png",
+			"src": "icons/icon-512.png",
 			"sizes": "512x512",
 			"type": "image/png",
 			"purpose": "any"
 		},
 		{
-			"src": "/icons/icon-maskable-512.png",
+			"src": "icons/icon-maskable-512.png",
 			"sizes": "512x512",
 			"type": "image/png",
 			"purpose": "maskable"