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/api/connect.ts

@@ -1,20 +1,21 @@
-// `createEdgeGatewayClient` builds a typed Connect-Web client for the
+// `createGatewayClient` builds a typed Connect-Web client for the
 // gateway's authenticated edge surface. It speaks the Connect protocol
 // over HTTP/1.1 (or HTTP/2 if the host upgrades the connection) — the
 // gateway listener built in Phase 4 natively serves Connect, gRPC, and
 // gRPC-Web on the same h2c port.
 //
-// The factory is intentionally thin: callers provide the gateway base
-// URL (e.g. https://api.galaxy.test), and receive a typed
-// `EdgeGatewayClient`. Authentication, signing, and response
+// The factory is intentionally thin: callers provide the Connect base
+// URL (the same-origin `/rpc` prefix from `gatewayRpcBaseUrl`), and
+// receive a typed
+// `GatewayClient`. Authentication, signing, and response
 // verification live one layer up, in `GalaxyClient`.
 
 import { createClient, type Client } from "@connectrpc/connect";
 import { createConnectTransport } from "@connectrpc/connect-web";
-import { EdgeGateway } from "../proto/galaxy/gateway/v1/edge_gateway_pb";
+import { Gateway } from "../proto/edge/v1/edge_gateway_pb";
 
-export type EdgeGatewayClient = Client<typeof EdgeGateway>;
+export type GatewayClient = Client<typeof Gateway>;
 
-export function createEdgeGatewayClient(baseUrl: string): EdgeGatewayClient {
-	return createClient(EdgeGateway, createConnectTransport({ baseUrl }));
+export function createGatewayClient(baseUrl: string): GatewayClient {
+	return createClient(Gateway, createConnectTransport({ baseUrl }));
 }