Model part face in tabletop state

Replace the path->part-list store with a per-part map keyed by part id, so each placed part carries its path, stack index, and face. Derive each path's ordered children for stacking, and flip face-down parts in PartPlacement.

docs/bgm-tabletop-plan.md

@@ -80,15 +80,22 @@ Source-of-truth game state per `bgm-tabletop.md` §2:
 ```ts
 interface GameState {
   surfaces: Record<string, boolean>; // enabled per surface id
-  paths: Record<string, string[]>;   // path -> part list
+  parts: Record<string, PartState>;  // part id -> placement state
+}
+
+interface PartState {
+  path: string;  // the path key this part is on
+  index: number; // the part's position in its path's stack
+  face: boolean; // whether the part's face is up
 }
 ```
 
 - A zustand store holding `GameState`.
 - **Derived render state**: `game state + surface routes => map of piece id to
-  `{ surface, route, candidate, index, stackSize }``, per enabled surface.
-  Computed with a selector/memo so the render list is stable.
-- **Assumption**: each piece id is unique within a path (documented in
+  `{ surface, route, candidate, index, stackSize, face }``, per enabled surface.
+  Computed with a selector/memo so the render list is stable. A path's ordered
+  children (for stacking) are derived from the parts map by sorting on `index`.
+- **Assumption**: each piece id is unique on the board (documented in
   `bgm-tabletop.md`); the render map is keyed by piece id.
 
 ### 4. Setup seeding (`setup.ts`) ✅