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/session_lookup.go

@@ -5,7 +5,7 @@ import (
 	"errors"
 
 	"galaxy/gateway/internal/session"
-	gatewayv1 "galaxy/gateway/proto/galaxy/gateway/v1"
+	edgev1 "galaxy/gateway/proto/edge/v1"
 
 	"google.golang.org/grpc"
 	"google.golang.org/grpc/codes"
@@ -30,15 +30,15 @@ func resolvedSessionFromContext(ctx context.Context) (session.Record, bool) {
 // sessionLookupService resolves the authenticated session from SessionCache
 // after envelope parsing succeeds and before later auth steps run.
 type sessionLookupService struct {
-	gatewayv1.UnimplementedEdgeGatewayServer
+	edgev1.UnimplementedGatewayServer
 
-	delegate gatewayv1.EdgeGatewayServer
+	delegate edgev1.GatewayServer
 	cache    session.Cache
 }
 
 // ExecuteCommand resolves the cached session for req and only then forwards it
 // to the configured delegate with the resolved session attached to ctx.
-func (s sessionLookupService) ExecuteCommand(ctx context.Context, req *gatewayv1.ExecuteCommandRequest) (*gatewayv1.ExecuteCommandResponse, error) {
+func (s sessionLookupService) ExecuteCommand(ctx context.Context, req *edgev1.ExecuteCommandRequest) (*edgev1.ExecuteCommandResponse, error) {
 	record, err := s.lookupSession(ctx)
 	if err != nil {
 		return nil, err
@@ -50,7 +50,7 @@ func (s sessionLookupService) ExecuteCommand(ctx context.Context, req *gatewayv1
 // SubscribeEvents resolves the cached session for req and only then forwards it
 // to the configured delegate with the resolved session attached to the stream
 // context.
-func (s sessionLookupService) SubscribeEvents(req *gatewayv1.SubscribeEventsRequest, stream grpc.ServerStreamingServer[gatewayv1.GatewayEvent]) error {
+func (s sessionLookupService) SubscribeEvents(req *edgev1.SubscribeEventsRequest, stream grpc.ServerStreamingServer[edgev1.GatewayEvent]) error {
 	record, err := s.lookupSession(stream.Context())
 	if err != nil {
 		return err
@@ -63,7 +63,7 @@ func (s sessionLookupService) SubscribeEvents(req *gatewayv1.SubscribeEventsRequ
 }
 
 // newSessionLookupService wraps delegate with the session-cache lookup gate.
-func newSessionLookupService(delegate gatewayv1.EdgeGatewayServer, cache session.Cache) gatewayv1.EdgeGatewayServer {
+func newSessionLookupService(delegate edgev1.GatewayServer, cache session.Cache) edgev1.GatewayServer {
 	return sessionLookupService{
 		delegate: delegate,
 		cache:    cache,
@@ -105,7 +105,7 @@ func cloneSessionRecord(record session.Record) session.Record {
 type resolvedSessionContextKey struct{}
 
 type resolvedSessionContextStream struct {
-	grpc.ServerStreamingServer[gatewayv1.GatewayEvent]
+	grpc.ServerStreamingServer[edgev1.GatewayEvent]
 	ctx context.Context
 }
 
@@ -126,4 +126,4 @@ func (unavailableSessionCache) Lookup(context.Context, string) (session.Record,
 func (unavailableSessionCache) MarkRevoked(string)           {}
 func (unavailableSessionCache) MarkAllRevokedForUser(string) {}
 
-var _ gatewayv1.EdgeGatewayServer = sessionLookupService{}
+var _ edgev1.GatewayServer = sessionLookupService{}