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/PLAN.md

@@ -474,11 +474,11 @@ documents the historical labelling in its package doc.
 Artifacts (delivered):
 
 - `gateway/buf.gen.yaml` extended with `buf.build/connectrpc/go`,
-  generating `gateway/proto/galaxy/gateway/v1/gatewayv1connect/edge_gateway.connect.go`
+  generating `gateway/proto/edge/v1/edgev1connect/edge_gateway.connect.go`
 - `gateway/internal/grpcapi/server.go` rewritten around `http.Server`
-  + `h2c.NewHandler` + `gatewayv1connect.NewEdgeGatewayHandler`
+  + `h2c.NewHandler` + `edgev1connect.NewGatewayHandler`
 - new `gateway/internal/grpcapi/connect_handler.go` adapting the
-  existing `gatewayv1.EdgeGatewayServer` decorator stack to the
+  existing `edgev1.GatewayServer` decorator stack to the
   Connect handler interface, including a `grpc.ServerStreamingServer`
   shim around `*connect.ServerStream[GatewayEvent]` and a gRPC
   `status.Error` → `*connect.Error` translation helper
@@ -492,7 +492,7 @@ Artifacts (delivered):
   `gateway/docs/runbook.md`, and `docs/ARCHITECTURE.md` §15
 - migrated tests: `gateway/internal/grpcapi/server_test.go`,
   `test_fixtures_test.go`, and every `*_integration_test.go` in that
-  package now drive a `gatewayv1connect.EdgeGatewayClient` over
+  package now drive a `edgev1connect.GatewayClient` over
   HTTP/2 cleartext loopback
 - migrated harness: `integration/testenv/grpc_client.go` →
   `connect_client.go`. `SignedGatewayClient` keeps the same public
@@ -580,10 +580,10 @@ Artifacts (delivered):
   browsers; the JSDOM test path lives next to it in
   `ui/frontend/tests/setup-wasm.ts`.
 - `ui/frontend/src/api/connect.ts` — typed Connect-Web transport +
-  `EdgeGatewayClient` factory.
+  `GatewayClient` factory.
 - `ui/frontend/src/api/galaxy-client.ts` — `GalaxyClient` skeleton
   with injected `Signer` and `Sha256` dependencies.
-- `ui/frontend/src/proto/galaxy/gateway/v1/edge_gateway_pb.ts`
+- `ui/frontend/src/proto/edge/v1/edge_gateway_pb.ts`
   (generated) and `ui/frontend/src/proto/buf/validate/validate_pb.ts`
   (generated as a transitive import via `--include-imports`).
 - `ui/frontend/static/core.wasm` (903 KB) + `wasm_exec.js` (TinyGo