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>

tools/local-dev/docker-compose.yml

@@ -180,7 +180,7 @@ services:
       GATEWAY_PUBLIC_HTTP_ANTI_ABUSE_SEND_EMAIL_CODE_IDENTITY_RATE_LIMIT_BURST: "1000"
       GATEWAY_PUBLIC_HTTP_ANTI_ABUSE_CONFIRM_EMAIL_CODE_IDENTITY_RATE_LIMIT_REQUESTS: "10000"
       GATEWAY_PUBLIC_HTTP_ANTI_ABUSE_CONFIRM_EMAIL_CODE_IDENTITY_RATE_LIMIT_BURST: "1000"
-      # public_misc class wraps the authenticated EdgeGateway gRPC
+      # public_misc class wraps the authenticated edge.v1.Gateway gRPC
       # endpoints (ExecuteCommand, SubscribeEvents). The gateway's
       # default for this class is 0 bytes, which rejects every
       # non-empty body with HTTP 413; override with a generous limit
@@ -200,7 +200,7 @@ services:
       GATEWAY_AUTHENTICATED_GRPC_ANTI_ABUSE_MESSAGE_CLASS_RATE_LIMIT_BURST: "1000"
     ports:
       - "${LOCAL_DEV_GATEWAY_REST_PORT:-8080}:8080"
-      # Authenticated EdgeGateway connect-web/gRPC listener. The
+      # Authenticated edge.v1.Gateway connect-web/gRPC listener. The
       # browser reaches it via the Vite dev proxy in
       # ui/frontend/vite.config.ts.
       - "${LOCAL_DEV_GATEWAY_GRPC_PORT:-9090}:9090"