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>

gateway/internal/grpcapi/push_heartbeat_test.go

@@ -7,7 +7,7 @@ import (
 	"testing"
 	"time"
 
-	gatewayv1 "galaxy/gateway/proto/galaxy/gateway/v1"
+	edgev1 "galaxy/gateway/proto/edge/v1"
 
 	"github.com/stretchr/testify/assert"
 	"github.com/stretchr/testify/require"
@@ -66,7 +66,7 @@ func TestHeartbeatingStreamRealSendResetsSilenceTimer(t *testing.T) {
 		defer ticker.Stop()
 		for range 6 {
 			<-ticker.C
-			if err := hb.Send(&gatewayv1.GatewayEvent{EventType: "real.event"}); err != nil {
+			if err := hb.Send(&edgev1.GatewayEvent{EventType: "real.event"}); err != nil {
 				t.Errorf("real Send failed: %v", err)
 				return
 			}
@@ -134,16 +134,16 @@ func TestHeartbeatingStreamSendErrorPropagates(t *testing.T) {
 	require.NotNil(t, hb)
 	defer hb.Stop()
 
-	err := hb.Send(&gatewayv1.GatewayEvent{EventType: "real.event"})
+	err := hb.Send(&edgev1.GatewayEvent{EventType: "real.event"})
 	require.ErrorIs(t, err, wantErr)
 }
 
 // capturingStream is a minimal grpc.ServerStreamingServer that pushes
 // every Send into a channel so tests can assert on the wire frame.
 type capturingStream struct {
-	grpc.ServerStreamingServer[gatewayv1.GatewayEvent]
+	grpc.ServerStreamingServer[edgev1.GatewayEvent]
 
-	events  chan *gatewayv1.GatewayEvent
+	events  chan *edgev1.GatewayEvent
 	sendErr atomic.Pointer[errorBox]
 }
 
@@ -152,10 +152,10 @@ type errorBox struct{ err error }
 func newCapturingStream(t *testing.T) *capturingStream {
 	t.Helper()
 
-	return &capturingStream{events: make(chan *gatewayv1.GatewayEvent, 16)}
+	return &capturingStream{events: make(chan *edgev1.GatewayEvent, 16)}
 }
 
-func (s *capturingStream) Send(event *gatewayv1.GatewayEvent) error {
+func (s *capturingStream) Send(event *edgev1.GatewayEvent) error {
 	if box := s.sendErr.Load(); box != nil {
 		return box.err
 	}
@@ -172,7 +172,7 @@ func (s *capturingStream) SetTrailer(metadata.MD)       {}
 func (s *capturingStream) SendMsg(any) error            { return errors.New("capturingStream.SendMsg: unused") }
 func (s *capturingStream) RecvMsg(any) error            { return errors.New("capturingStream.RecvMsg: unused") }
 
-func (s *capturingStream) recv(t *testing.T, timeout time.Duration) *gatewayv1.GatewayEvent {
+func (s *capturingStream) recv(t *testing.T, timeout time.Duration) *edgev1.GatewayEvent {
 	t.Helper()
 
 	select {