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/protocol/protocol.go

@@ -51,6 +51,7 @@ const (
 	TypeRouterNarration  = "router.narration"
 	TypeArtifactList     = "artifact.list"
 	TypeConfigSnapshot   = "config.snapshot"
+	TypeSessionStats     = "session.stats"
 )
 
 // ServerMessage is a flat decode of every server->client variant. Field names
@@ -131,6 +132,68 @@ type ServerMessage struct {
 	ConfigFields          []ConfigFieldDto `json:"fields"`
 	ConfigRestartRequired []string         `json:"restartRequired"`
 	ConfigError           *string          `json:"error"`
+
+	// session.stats — derived metrics for a session (reply to GetSessionStats)
+	Stats *StatsDto `json:"stats"`
+}
+
+// StatsDto mirrors the server's MetricsReport: a session's derived metrics
+// (observability-spec §2/§3). Raw sums plus the derived ratios the pane renders.
+type StatsDto struct {
+	SessionID          string                 `json:"sessionId"`
+	EventCount         int64                  `json:"eventCount"`
+	SessionDurationMs  int64                  `json:"sessionDurationMs"`
+	InferenceCount     int64                  `json:"inferenceCount"`
+	InferenceMs        int64                  `json:"inferenceMs"`
+	PromptTokens       int64                  `json:"promptTokens"`
+	CompletionTokens   int64                  `json:"completionTokens"`
+	TokensPerSecond    float64                `json:"tokensPerSecond"`
+	PerProvider        []ProviderStatsDto     `json:"perProvider"`
+	ToolCount          int64                  `json:"toolCount"`
+	ToolMs             int64                  `json:"toolMs"`
+	PerTool            []ToolStatsDto         `json:"perTool"`
+	ApprovalsRequested int64                  `json:"approvalsRequested"`
+	ApprovalsResolved  int64                  `json:"approvalsResolved"`
+	ApprovalsPending   int64                  `json:"approvalsPending"`
+	ApprovalWaitMs     int64                  `json:"approvalWaitMs"`
+	AvgApprovalWaitMs  int64                  `json:"avgApprovalWaitMs"`
+	PerTier            []TierApprovalStatsDto `json:"perTier"`
+	Failures           FailureMetricsDto      `json:"failures"`
+	InferencePct       float64                `json:"inferencePct"`
+	ToolPct            float64                `json:"toolPct"`
+	ApprovalWaitPct    float64                `json:"approvalWaitPct"`
+}
+
+type ProviderStatsDto struct {
+	Provider         string  `json:"provider"`
+	CompletedCount   int64   `json:"completedCount"`
+	TotalLatencyMs   int64   `json:"totalLatencyMs"`
+	PromptTokens     int64   `json:"promptTokens"`
+	CompletionTokens int64   `json:"completionTokens"`
+	TokensPerSecond  float64 `json:"tokensPerSecond"`
+}
+
+type ToolStatsDto struct {
+	ToolName        string `json:"toolName"`
+	CompletedCount  int64  `json:"completedCount"`
+	TotalDurationMs int64  `json:"totalDurationMs"`
+}
+
+type TierApprovalStatsDto struct {
+	Tier           string `json:"tier"`
+	RequestedCount int64  `json:"requestedCount"`
+	ResolvedCount  int64  `json:"resolvedCount"`
+	TotalWaitMs    int64  `json:"totalWaitMs"`
+	AvgWaitMs      int64  `json:"avgWaitMs"`
+}
+
+type FailureMetricsDto struct {
+	InferenceFailures int64 `json:"inferenceFailures"`
+	InferenceTimeouts int64 `json:"inferenceTimeouts"`
+	ToolFailures      int64 `json:"toolFailures"`
+	ToolRejections    int64 `json:"toolRejections"`
+	StageFailures     int64 `json:"stageFailures"`
+	WorkflowFailures  int64 `json:"workflowFailures"`
 }
 
 type ConfigFieldDto struct {
@@ -275,6 +338,11 @@ func ListArtifacts(sessionID string) []byte {
 	return encode("ListArtifacts", map[string]any{"sessionId": sessionID})
 }
 
+// GetSessionStats asks the server for a session's derived metrics (replied to with session.stats).
+func GetSessionStats(sessionID string) []byte {
+	return encode("GetSessionStats", map[string]any{"sessionId": sessionID})
+}
+
 // GetConfig requests the current editable config (replied to with a config.snapshot).
 func GetConfig() []byte {
 	return encode("GetConfig", map[string]any{})