feat(v0.5): add CTASK_PROJECT_ROOT check to ctask doctor

Three-state check matching the seed-dir pattern: INFO when unset
(points at the default discovery location), INFO with user-scope
advisory when set and present, FAIL when set but the directory
doesn't exist. Advisory wording is recommendatory, not prescriptive
(per spec amendment).

Co-Authored-By: Claude Opus 4.7 (1M context) <noreply@anthropic.com>

cmd/doctor.go

@@ -200,6 +200,9 @@ func runDoctor(cmd *cobra.Command, args []string) error {
 		&failed,
 	)
 
+	// Check 8: CTASK_PROJECT_ROOT (v0.5).
+	checkProjectRoot(&passed, &failed)
+
 	// Summary
 	fmt.Println()
 	fmt.Printf("%d checks passed, %d failed\n", passed, failed)
@@ -210,6 +213,28 @@ func runDoctor(cmd *cobra.Command, args []string) error {
 	return nil
 }
 
+// checkProjectRoot prints the three-state CTASK_PROJECT_ROOT doctor line:
+//   - unset           -> INFO pointing at the default location under CTASK_ROOT
+//   - set and exists  -> INFO with custom-root advisory
+//   - set and missing -> FAIL (increments failed counter)
+func checkProjectRoot(passed, failed *int) {
+	_ = passed // reserved for future symmetry with checkSeedDir
+	envValue := os.Getenv("CTASK_PROJECT_ROOT")
+	if envValue == "" {
+		defaultProj := filepath.Join(config.ResolveRoot(), "projects")
+		fmt.Printf("  [INFO] CTASK_PROJECT_ROOT: not set (projects discovered under %s)\n", defaultProj)
+		return
+	}
+	resolved := config.ResolveProjectRoot()
+	if info, err := os.Stat(resolved); err == nil && info.IsDir() {
+		fmt.Printf("  [INFO] CTASK_PROJECT_ROOT: %s (custom -- recommended: set at user scope so all terminals can discover these workspaces)\n", resolved)
+		return
+	}
+	fmt.Printf("  [FAIL] CTASK_PROJECT_ROOT configured but not found: %s\n", resolved)
+	fmt.Printf("         Fix: create the directory or unset CTASK_PROJECT_ROOT.\n")
+	*failed++
+}
+
 // checkSeedDir prints one of the three seed-directory doctor states.
 // envValue is the raw value of the CTASK_SEED_* variable (empty means unset).
 // resolved is the absolute path returned by the config resolver (used only