Stage 12: observability & performance (OTel/OTLP, domain metrics, guest GC)

- pkg/telemetry: shared OTel provider bootstrap (none/stdout/otlp + W3C
  propagators + Go runtime metrics); backend/internal/telemetry becomes a thin
  facade keeping its gin middleware.
- Telemetry parity: gateway and the Telegram connector gain telemetry runtimes
  and config (GATEWAY_/TELEGRAM_ SERVICE_NAME + OTEL_*); otelgrpc instruments the
  backend push server, the gateway's backend+connector clients and the connector
  server. Default exporter stays none (collector/dashboards are Stage 14).
- Operational metrics (variant attribute on game-scoped ones): game_replay_duration,
  game_move_validate_duration, games_started_total, games_abandoned_total,
  game_cache_active, chat_messages_total{kind}, gateway edge_request_duration.
  Wired via the SetMetrics setter pattern (default no-op meter).
- TODO-3: account.GuestReaper deletes guests with no game seat past
  BACKEND_GUEST_RETENTION (default 30d, swept every BACKEND_GUEST_REAP_INTERVAL).
- Tests: pkg/telemetry exporter selection; game/social/edge metric recording via
  a manual reader; config (otlp accepted, guest knobs); inttest guest reaper.
- Docs: PLAN.md re-scopes Stage 12 and adds Stage 13 (alphabet-on-wire) + Stage 14
  (CI/deploy) with the agreed dictionary-versioning resolution; ARCHITECTURE 11/13,
  TESTING, the three READMEs and FUNCTIONAL(+ru) updated.

backend/internal/game/cache.go

@@ -63,6 +63,7 @@ type gameCache struct {
 
 type cachedGame struct {
 	game       *engine.Game
+	variant    string
 	lastAccess time.Time
 }
 
@@ -82,11 +83,12 @@ func (c *gameCache) get(id uuid.UUID) (*engine.Game, bool) {
 	return e.game, true
 }
 
-// put stores g as the live game for id.
-func (c *gameCache) put(id uuid.UUID, g *engine.Game) {
+// put stores g as the live game for id. variant labels the entry so the active-
+// games gauge can report counts by variant without inspecting engine internals.
+func (c *gameCache) put(id uuid.UUID, g *engine.Game, variant string) {
 	c.mu.Lock()
 	defer c.mu.Unlock()
-	c.entries[id] = &cachedGame{game: g, lastAccess: c.now()}
+	c.entries[id] = &cachedGame{game: g, variant: variant, lastAccess: c.now()}
 }
 
 // remove drops id from the cache (used on a finished game and after a failed
@@ -119,3 +121,16 @@ func (c *gameCache) size() int {
 	defer c.mu.Unlock()
 	return len(c.entries)
 }
+
+// countByVariant tallies the resident games by their variant label. It backs the
+// game_cache_active observable gauge; the resident set is the bounded number of
+// live (active) games, so the scan under the lock is cheap.
+func (c *gameCache) countByVariant() map[string]int {
+	c.mu.Lock()
+	defer c.mu.Unlock()
+	out := make(map[string]int, len(c.entries))
+	for _, e := range c.entries {
+		out[e.variant]++
+	}
+	return out
+}