fix(sling): resolve rig path for cross-rig bead hooking

gt sling failed when hooking rig-level beads from town root because
bd update doesn't support cross-database routing like bd show does.

The fix adds a ResolveHookDir helper that:
1. Extracts the prefix from bead ID (e.g., "ap-xxx" → "ap-")
2. Looks up the rig path from routes.jsonl
3. Falls back to townRoot if prefix not found

Also removes the BEADS_DIR environment override which was preventing
routing from working correctly.

Fixes #148

internal/beads/routes.go

@@ -190,3 +190,60 @@ func FindConflictingPrefixes(beadsDir string) (map[string][]string, error) {
 
 	return conflicts, nil
 }
+
+// ExtractPrefix extracts the prefix from a bead ID.
+// For example, "ap-qtsup.16" returns "ap-", "hq-cv-abc" returns "hq-".
+// Returns empty string if no valid prefix found (empty input, no hyphen,
+// or hyphen at position 0 which would indicate an invalid prefix).
+func ExtractPrefix(beadID string) string {
+	if beadID == "" {
+		return ""
+	}
+
+	idx := strings.Index(beadID, "-")
+	if idx <= 0 {
+		return ""
+	}
+
+	return beadID[:idx+1]
+}
+
+// GetRigPathForPrefix returns the rig path for a given bead ID prefix.
+// The townRoot should be the Gas Town root directory (e.g., ~/gt).
+// Returns the full absolute path to the rig directory, or empty string if not found.
+// For town-level beads (path="."), returns townRoot.
+func GetRigPathForPrefix(townRoot, prefix string) string {
+	beadsDir := filepath.Join(townRoot, ".beads")
+	routes, err := LoadRoutes(beadsDir)
+	if err != nil || routes == nil {
+		return ""
+	}
+
+	for _, r := range routes {
+		if r.Prefix == prefix {
+			if r.Path == "." {
+				return townRoot // Town-level beads
+			}
+			return filepath.Join(townRoot, r.Path)
+		}
+	}
+
+	return ""
+}
+
+// ResolveHookDir determines the directory for running bd update on a bead.
+// Since bd update doesn't support routing or redirects, we must resolve the
+// actual rig directory from the bead's prefix. hookWorkDir is only used as
+// a fallback if prefix resolution fails.
+func ResolveHookDir(townRoot, beadID, hookWorkDir string) string {
+	// Always try prefix resolution first - bd update needs the actual rig dir
+	prefix := ExtractPrefix(beadID)
+	if rigPath := GetRigPathForPrefix(townRoot, prefix); rigPath != "" {
+		return rigPath
+	}
+	// Fallback to hookWorkDir if provided
+	if hookWorkDir != "" {
+		return hookWorkDir
+	}
+	return townRoot
+}

internal/beads/routes_test.go

@@ -52,6 +52,143 @@ func TestGetPrefixForRig_NoRoutesFile(t *testing.T) {
 	}
 }
 
+func TestExtractPrefix(t *testing.T) {
+	tests := []struct {
+		beadID   string
+		expected string
+	}{
+		{"ap-qtsup.16", "ap-"},
+		{"hq-cv-abc", "hq-"},
+		{"gt-mol-xyz", "gt-"},
+		{"bd-123", "bd-"},
+		{"", ""},
+		{"nohyphen", ""},
+		{"-startswithhyphen", ""}, // Leading hyphen = invalid prefix
+		{"-", ""},                 // Just hyphen = invalid
+		{"a-", "a-"},              // Trailing hyphen is valid
+	}
+
+	for _, tc := range tests {
+		t.Run(tc.beadID, func(t *testing.T) {
+			result := ExtractPrefix(tc.beadID)
+			if result != tc.expected {
+				t.Errorf("ExtractPrefix(%q) = %q, want %q", tc.beadID, result, tc.expected)
+			}
+		})
+	}
+}
+
+func TestGetRigPathForPrefix(t *testing.T) {
+	// Create a temporary directory with routes.jsonl
+	tmpDir := t.TempDir()
+	beadsDir := filepath.Join(tmpDir, ".beads")
+	if err := os.MkdirAll(beadsDir, 0755); err != nil {
+		t.Fatal(err)
+	}
+
+	routesContent := `{"prefix": "ap-", "path": "ai_platform/mayor/rig"}
+{"prefix": "gt-", "path": "gastown/mayor/rig"}
+{"prefix": "hq-", "path": "."}
+`
+	if err := os.WriteFile(filepath.Join(beadsDir, "routes.jsonl"), []byte(routesContent), 0644); err != nil {
+		t.Fatal(err)
+	}
+
+	tests := []struct {
+		prefix   string
+		expected string
+	}{
+		{"ap-", filepath.Join(tmpDir, "ai_platform/mayor/rig")},
+		{"gt-", filepath.Join(tmpDir, "gastown/mayor/rig")},
+		{"hq-", tmpDir}, // Town-level beads return townRoot
+		{"unknown-", ""}, // Unknown prefix returns empty
+		{"", ""},         // Empty prefix returns empty
+	}
+
+	for _, tc := range tests {
+		t.Run(tc.prefix, func(t *testing.T) {
+			result := GetRigPathForPrefix(tmpDir, tc.prefix)
+			if result != tc.expected {
+				t.Errorf("GetRigPathForPrefix(%q, %q) = %q, want %q", tmpDir, tc.prefix, result, tc.expected)
+			}
+		})
+	}
+}
+
+func TestGetRigPathForPrefix_NoRoutesFile(t *testing.T) {
+	tmpDir := t.TempDir()
+	// No routes.jsonl file
+
+	result := GetRigPathForPrefix(tmpDir, "ap-")
+	if result != "" {
+		t.Errorf("Expected empty string when no routes file, got %q", result)
+	}
+}
+
+func TestResolveHookDir(t *testing.T) {
+	// Create a temporary directory with routes.jsonl
+	tmpDir := t.TempDir()
+	beadsDir := filepath.Join(tmpDir, ".beads")
+	if err := os.MkdirAll(beadsDir, 0755); err != nil {
+		t.Fatal(err)
+	}
+
+	routesContent := `{"prefix": "ap-", "path": "ai_platform/mayor/rig"}
+{"prefix": "hq-", "path": "."}
+`
+	if err := os.WriteFile(filepath.Join(beadsDir, "routes.jsonl"), []byte(routesContent), 0644); err != nil {
+		t.Fatal(err)
+	}
+
+	tests := []struct {
+		name        string
+		beadID      string
+		hookWorkDir string
+		expected    string
+	}{
+		{
+			name:        "prefix resolution takes precedence over hookWorkDir",
+			beadID:      "ap-test",
+			hookWorkDir: "/custom/path",
+			expected:    filepath.Join(tmpDir, "ai_platform/mayor/rig"),
+		},
+		{
+			name:        "resolves rig path from prefix",
+			beadID:      "ap-test",
+			hookWorkDir: "",
+			expected:    filepath.Join(tmpDir, "ai_platform/mayor/rig"),
+		},
+		{
+			name:        "town-level bead returns townRoot",
+			beadID:      "hq-test",
+			hookWorkDir: "",
+			expected:    tmpDir,
+		},
+		{
+			name:        "unknown prefix uses hookWorkDir as fallback",
+			beadID:      "xx-unknown",
+			hookWorkDir: "/fallback/path",
+			expected:    "/fallback/path",
+		},
+		{
+			name:        "unknown prefix without hookWorkDir falls back to townRoot",
+			beadID:      "xx-unknown",
+			hookWorkDir: "",
+			expected:    tmpDir,
+		},
+	}
+
+	for _, tc := range tests {
+		t.Run(tc.name, func(t *testing.T) {
+			result := ResolveHookDir(tmpDir, tc.beadID, tc.hookWorkDir)
+			if result != tc.expected {
+				t.Errorf("ResolveHookDir(%q, %q, %q) = %q, want %q",
+					tmpDir, tc.beadID, tc.hookWorkDir, result, tc.expected)
+			}
+		})
+	}
+}
+
 func TestAgentBeadIDsWithPrefix(t *testing.T) {
 	tests := []struct {
 		name     string