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/payload_hash_integration_test.go

@@ -15,7 +15,7 @@ import (
 func TestExecuteCommandRejectsPayloadHashWithInvalidLength(t *testing.T) {
 	t.Parallel()
 
-	delegate := &recordingEdgeGatewayService{}
+	delegate := &recordingGatewayService{}
 	server, runGateway := newTestGateway(t, ServerDependencies{
 		Service:      delegate,
 		SessionCache: staticSessionCache{lookupFunc: func(context.Context, string) (session.Record, error) { return newActiveSessionRecord(), nil }},
@@ -38,7 +38,7 @@ func TestExecuteCommandRejectsPayloadHashWithInvalidLength(t *testing.T) {
 func TestExecuteCommandRejectsPayloadHashMismatch(t *testing.T) {
 	t.Parallel()
 
-	delegate := &recordingEdgeGatewayService{}
+	delegate := &recordingGatewayService{}
 	server, runGateway := newTestGateway(t, ServerDependencies{
 		Service:      delegate,
 		SessionCache: staticSessionCache{lookupFunc: func(context.Context, string) (session.Record, error) { return newActiveSessionRecord(), nil }},
@@ -62,7 +62,7 @@ func TestExecuteCommandRejectsPayloadHashMismatch(t *testing.T) {
 func TestSubscribeEventsRejectsPayloadHashWithInvalidLength(t *testing.T) {
 	t.Parallel()
 
-	delegate := &recordingEdgeGatewayService{}
+	delegate := &recordingGatewayService{}
 	server, runGateway := newTestGateway(t, ServerDependencies{
 		Service:      delegate,
 		SessionCache: staticSessionCache{lookupFunc: func(context.Context, string) (session.Record, error) { return newActiveSessionRecord(), nil }},
@@ -85,7 +85,7 @@ func TestSubscribeEventsRejectsPayloadHashWithInvalidLength(t *testing.T) {
 func TestSubscribeEventsRejectsPayloadHashMismatch(t *testing.T) {
 	t.Parallel()
 
-	delegate := &recordingEdgeGatewayService{}
+	delegate := &recordingGatewayService{}
 	server, runGateway := newTestGateway(t, ServerDependencies{
 		Service:      delegate,
 		SessionCache: staticSessionCache{lookupFunc: func(context.Context, string) (session.Record, error) { return newActiveSessionRecord(), nil }},