feat(server,tui): session stats pane — metrics over the WS bus + Bubble Tea overlay

Surfaces the observability metrics (observability-spec §3-tier-2) in the Go TUI:
a session-summary pane showing duration, token throughput per provider, tool
time, approval latency per tier, failure counts, and the session wall-time
accounting split. Press `S` (or palette → "session stats").

Mirrors the artifacts/config fetch pattern exactly — a WS request/response, not a
new transport. Server-side reuses the already-tested MetricsInspectionService
(one MetricsReport definition, two wires: REST `correx stats` + WS).

Server:
- ClientMessage.GetSessionStats + ServerMessage.SessionStats(@SerialName
  session.stats), reusing MetricsReport as the nested payload.
- StreamQueries.sessionStats replays via MetricsInspectionService (pure read,
  replay-neutral); routed in GlobalStreamHandler.
- ServerMessageSerializationTest golden pins the session.stats wire format.

TUI (Go/Bubble Tea):
- protocol: TypeSessionStats + StatsDto/nested structs + GetSessionStats encoder
  + golden decode test (cross-language contract).
- OverlayStats pane (statsModal), `S` key + palette entry + footer hint,
  loading/cache-by-session state, bounded breakdown rows.
- demo `stats` preview case for serverless visual verification.

apps/tui-go/internal/app/update.go

@@ -169,6 +169,8 @@ func (m Model) handleNormalKey(k tea.KeyMsg) (tea.Model, tea.Cmd) {
 		m.overlay = OverlayToolPalette
 	case "v":
 		m.openArtifacts()
+	case "S":
+		m.openStats()
 	case "g":
 		m.openConfig()
 	case "m":
@@ -429,6 +431,10 @@ func (m Model) handleOverlayKey(k tea.KeyMsg) (tea.Model, tea.Cmd) {
 		case runeIs(k, "m"):
 			m.overlay = OverlayNone
 		}
+	case OverlayStats:
+		if runeIs(k, "S") {
+			m.overlay = OverlayNone
+		}
 	}
 	return m, nil
 }
@@ -450,6 +456,21 @@ func (m *Model) openArtifacts() {
 	m.client.Send(protocol.ListArtifacts(m.selectedID))
 }
 
+// openStats opens the session-stats pane for the selected session and requests its
+// metrics from the server. No-op when no session is selected.
+func (m *Model) openStats() {
+	if m.selectedID == "" {
+		return
+	}
+	m.overlay = OverlayStats
+	// Reuse a cached report only if it's for this session; otherwise show a loading state.
+	if m.statsFor != m.selectedID {
+		m.stats = nil
+		m.statsLoading = true
+	}
+	m.client.Send(protocol.GetSessionStats(m.selectedID))
+}
+
 // openModelsOverlay opens the model picker, pre-selecting the resident model.
 func (m *Model) openModelsOverlay() {
 	m.overlay = OverlayModels
@@ -502,6 +523,7 @@ func paletteCommands() []paletteCmd {
 		{"models", "swap model", "pick / pin the local model"},
 		{"events", "event inspector", "browse the event stream"},
 		{"artifacts", "view artifacts", "browse this session's artifacts"},
+		{"stats", "session stats", "metrics for the selected session"},
 		{"config", "edit config", "view / change correx settings"},
 		{"mode", "toggle mode", "switch chat / steering"},
 		{"cancel", "cancel session", "stop the selected session"},
@@ -539,6 +561,8 @@ func (m Model) execPalette(id string) (tea.Model, tea.Cmd) {
 		m.overlayEventIdx = 0
 	case "artifacts":
 		m.openArtifacts()
+	case "stats":
+		m.openStats()
 	case "config":
 		m.openConfig()
 	case "mode":