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.go

@@ -6,7 +6,7 @@ import (
 	"time"
 
 	"galaxy/gateway/internal/telemetry"
-	gatewayv1 "galaxy/gateway/proto/galaxy/gateway/v1"
+	edgev1 "galaxy/gateway/proto/edge/v1"
 
 	"go.opentelemetry.io/otel/attribute"
 	"google.golang.org/grpc"
@@ -26,7 +26,7 @@ import (
 // 15-second default a fully-idle stream costs ~840 KB/day per client;
 // see `docs/ARCHITECTURE.md` for the per-scale projection.
 type heartbeatingStream struct {
-	grpc.ServerStreamingServer[gatewayv1.GatewayEvent]
+	grpc.ServerStreamingServer[edgev1.GatewayEvent]
 
 	interval time.Duration
 	metrics  *telemetry.Runtime
@@ -43,7 +43,7 @@ type heartbeatingStream struct {
 // the wrapping entirely; non-nil returns must have `Stop()` called once
 // the stream lifecycle ends.
 func newHeartbeatingStream(
-	inner grpc.ServerStreamingServer[gatewayv1.GatewayEvent],
+	inner grpc.ServerStreamingServer[edgev1.GatewayEvent],
 	interval time.Duration,
 	metrics *telemetry.Runtime,
 ) *heartbeatingStream {
@@ -64,7 +64,7 @@ func newHeartbeatingStream(
 // so the heartbeat goroutine waits a fresh interval before firing
 // again. A Send that succeeds means the transport just delivered real
 // bytes; the silence window restarts from "now".
-func (s *heartbeatingStream) Send(event *gatewayv1.GatewayEvent) error {
+func (s *heartbeatingStream) Send(event *edgev1.GatewayEvent) error {
 	s.sendMu.Lock()
 	defer s.sendMu.Unlock()
 	if err := s.ServerStreamingServer.Send(event); err != nil {
@@ -158,6 +158,6 @@ func (s *heartbeatingStream) resetTimerLocked() {
 // EventType is left at its proto3 default so the wire frame stays as
 // small as Connect framing allows. See `gatewayHeartbeatEventType` for
 // the security rationale of leaving the event unsigned.
-func buildHeartbeatEvent() *gatewayv1.GatewayEvent {
-	return &gatewayv1.GatewayEvent{EventType: gatewayHeartbeatEventType}
+func buildHeartbeatEvent() *edgev1.GatewayEvent {
+	return &edgev1.GatewayEvent{EventType: gatewayHeartbeatEventType}
 }