feat(tui-go): health-checks pane

Surface the health subsystem (llama/event-store/disk probes) in the TUI,
mirroring the session-stats pane: ClientMessage.GetHealthChecks ->
ServerMessage.HealthChecks (health.checks, reuses HealthReport) ->
StreamQueries.healthChecks via HealthInspectionService; Go OverlayHealth on
key H + palette 'health checks', pull-based fetch, per-subject status with
degraded loud. Empty/disabled -> visible fallback (no blank/lie). Go+Kotlin
golden tests pin the wire contract field-for-field. Observability spec §4/§3.

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

@@ -178,6 +178,8 @@ func (m Model) handleNormalKey(k tea.KeyMsg) (tea.Model, tea.Cmd) {
 		m.openArtifacts()
 	case "I":
 		m.openIdeas()
+	case "H":
+		m.openHealth()
 	case "S":
 		m.openStats()
 	case "g":
@@ -444,6 +446,10 @@ func (m Model) handleOverlayKey(k tea.KeyMsg) (tea.Model, tea.Cmd) {
 		if runeIs(k, "S") {
 			m.overlay = OverlayNone
 		}
+	case OverlayHealth:
+		if runeIs(k, "H") {
+			m.overlay = OverlayNone
+		}
 	case OverlayIdeas:
 		switch {
 		case k.Type == tea.KeyUp || runeIs(k, "k"):
@@ -513,6 +519,17 @@ func (m *Model) openStats() {
 	m.client.Send(protocol.GetSessionStats(m.selectedID))
 }
 
+// openHealth opens the system health-checks pane and fetches the current report.
+// Health is system-scoped (not session-scoped), so it opens from anywhere.
+func (m *Model) openHealth() {
+	m.overlay = OverlayHealth
+	// Always re-fetch and show a loading state: health changes out from under us, so a
+	// cached report would be stale (TUI spec §2 — render from the fetched report).
+	m.health = nil
+	m.healthLoading = true
+	m.client.Send(protocol.GetHealthChecks())
+}
+
 // openModelsOverlay opens the model picker, pre-selecting the resident model.
 func (m *Model) openModelsOverlay() {
 	m.overlay = OverlayModels
@@ -566,6 +583,7 @@ func paletteCommands() []paletteCmd {
 		{"events", "event inspector", "browse the event stream"},
 		{"artifacts", "view artifacts", "browse this session's artifacts"},
 		{"stats", "session stats", "metrics for the selected session"},
+		{"health", "health checks", "system health probe status"},
 		{"config", "edit config", "view / change correx settings"},
 		{"mode", "toggle mode", "switch chat / steering"},
 		{"cancel", "cancel session", "stop the selected session"},
@@ -605,6 +623,8 @@ func (m Model) execPalette(id string) (tea.Model, tea.Cmd) {
 		m.openArtifacts()
 	case "stats":
 		m.openStats()
+	case "health":
+		m.openHealth()
 	case "config":
 		m.openConfig()
 	case "mode":