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/galaxy-client.ts

@@ -15,8 +15,8 @@ import type { Core } from "../platform/core/index";
 import {
 	ExecuteCommandRequestSchema,
 	type ExecuteCommandResponse,
-} from "../proto/galaxy/gateway/v1/edge_gateway_pb";
-import type { EdgeGatewayClient } from "./connect";
+} from "../proto/edge/v1/edge_gateway_pb";
+import type { GatewayClient } from "./connect";
 
 /**
  * Signer produces a raw 64-byte Ed25519 signature over canonicalBytes.
@@ -35,7 +35,7 @@ export type Sha256 = (payload: Uint8Array) => Promise<Uint8Array>;
 
 export interface GalaxyClientOptions {
 	core: Core;
-	edge: EdgeGatewayClient;
+	edge: GatewayClient;
 	signer: Signer;
 	sha256: Sha256;
 	deviceSessionId: string;
@@ -53,7 +53,7 @@ export interface ExecuteCommandResult {
 
 export class GalaxyClient {
 	private readonly core: Core;
-	private readonly edge: EdgeGatewayClient;
+	private readonly edge: GatewayClient;
 	private readonly signer: Signer;
 	private readonly sha256: Sha256;
 	private readonly deviceSessionId: string;