feat(ui): display current git branch in browser tab, header, and nav tabs

- Add branch indicator in browser tab title ([test-gay] GaC-Srtool...)
- Add dynamic tab switching updates preserving branch prefix in browser title
- Add branch badge in brand header and right-aligned branch pill in navigation tabs
- Add scripts/sync-branch.js and version.json for automated branch detection

css/styles.css

@@ -3001,3 +3001,50 @@ input:checked + .slider:before {
 .slot-picker-inv-card.active-wearer {
   border-color: var(--accent-gold);
 }
+
+
+/* Git Branch Indicators */
+.badge-branch {
+  display: inline-flex;
+  align-items: center;
+  padding: 0.12rem 0.5rem;
+  border-radius: 6px;
+  background: rgba(99, 102, 241, 0.15);
+  border: 1px solid rgba(99, 102, 241, 0.35);
+  color: #a5b4fc;
+  font-size: 0.72rem;
+  font-weight: 700;
+  font-family: ui-monospace, SFMono-Regular, Menlo, Monaco, Consolas, monospace;
+  letter-spacing: 0.02em;
+  user-select: none;
+}
+
+.nav-branch-pill {
+  margin-left: auto;
+  display: inline-flex;
+  align-items: center;
+  gap: 0.35rem;
+  padding: 0.25rem 0.65rem;
+  background: rgba(99, 102, 241, 0.12);
+  border: 1px solid rgba(99, 102, 241, 0.3);
+  border-radius: 9999px;
+  color: #a5b4fc;
+  font-size: 0.75rem;
+  font-family: ui-monospace, SFMono-Regular, Menlo, Monaco, Consolas, monospace;
+  font-weight: 600;
+  letter-spacing: 0.02em;
+  white-space: nowrap;
+  user-select: none;
+  transition: all 0.2s ease;
+}
+
+.nav-branch-pill:hover {
+  background: rgba(99, 102, 241, 0.22);
+  border-color: rgba(99, 102, 241, 0.55);
+  color: #c7d2fe;
+}
+
+.branch-icon {
+  flex-shrink: 0;
+  opacity: 0.85;
+}