feat(tui): show session workspace (cwd) in the status bar

SessionWorkspaceBoundEvent already recorded the bound workspace root but
it was never surfaced. Map it to a new session.workspace_bound frame
(sealed ServerMessage variant, auto-registered; re-emitted on reconnect
via the snapshot replay), carry it on Session.WorkspaceRoot, and show it
home-abbreviated in the status bar (⌂ ~/Programs/correx). Golden test
pins the decode.

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

@@ -73,6 +73,7 @@ func PreviewFrame(kind string, w, h int) string {
 		m.sessionEntered = true
 		if s := m.session("04a546aa"); s != nil {
 			s.CurrentStage = "write_script"
+			s.WorkspaceRoot = "/home/kami/Programs/correx"
 			s.Events = sampleEvents()
 			s.Pending = &Approval{
 				RequestID: "req-1", SessionID: "04a546aa", Tier: "T3", Risk: "MEDIUM",

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

@@ -98,19 +98,20 @@ type Approval struct {
 
 // Session is the UI's view of one server session.
 type Session struct {
-	ID           string
-	Status       string
-	WorkflowID   string
-	Name         string
-	LastEventAt  int64
-	CurrentStage string
-	LastOutput   string
-	LastResponse string
-	Tools        []ToolRecord
-	ToolsByStage map[string][]ManifestTool
-	Events       []EventEntry
-	Pending      *Approval
-	Active       bool // an inference/tool is in flight (drives the spinner)
+	ID            string
+	Status        string
+	WorkflowID    string
+	Name          string
+	LastEventAt   int64
+	CurrentStage  string
+	WorkspaceRoot string // bound cwd, from session.workspace_bound
+	LastOutput    string
+	LastResponse  string
+	Tools         []ToolRecord
+	ToolsByStage  map[string][]ManifestTool
+	Events        []EventEntry
+	Pending       *Approval
+	Active        bool // an inference/tool is in flight (drives the spinner)
 }
 
 // Workflow is a launchable workflow advertised by the server.

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

@@ -175,6 +175,10 @@ func (m *Model) applyServer(msg protocol.ServerMessage) {
 		if s := m.session(msg.SessionID); s != nil {
 			s.addEvent(nowMillis(), "ArtifactCreated", msg.ArtifactID)
 		}
+	case protocol.TypeWorkspaceBound:
+		if s := m.session(msg.SessionID); s != nil {
+			s.WorkspaceRoot = msg.WorkspaceRoot
+		}
 	case protocol.TypeApprovalRequired:
 		m.onApprovalRequired(msg)
 	case protocol.TypeApprovalResolved:

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

@@ -1,6 +1,7 @@
 package app
 
 import (
+	"os"
 	"strings"
 
 	"github.com/charmbracelet/lipgloss"
@@ -74,6 +75,9 @@ func (m Model) renderStatus() string {
 		if s.Status != "" {
 			parts = append(parts, lipgloss.NewStyle().Foreground(statusColor(t, s.Status)).Background(bg).Render(statusLabel(s.Status)))
 		}
+		if s.WorkspaceRoot != "" {
+			parts = append(parts, span("⌂ "+shortPath(s.WorkspaceRoot), t.P.Faint))
+		}
 	}
 
 	model := m.currentModel
@@ -391,6 +395,14 @@ func (m Model) eventRows(w, h int) []string {
 
 // --- width helpers ---
 
+// shortPath abbreviates the user's home dir to ~ for a compact cwd in the status bar.
+func shortPath(p string) string {
+	if home := os.Getenv("HOME"); home != "" && strings.HasPrefix(p, home) {
+		return "~" + p[len(home):]
+	}
+	return p
+}
+
 // fill left-justifies a styled line and pads with bg to width w.
 func (m Model) fill(s string, w int, bg lipgloss.Color) string {
 	return " " + padTo(s, w-1, bg)