feat(ui): move in-game status to the board — highlight, score badge, full-width rack

Remove the under-board status strip and relocate its signals:
- bag count -> a badge on the exchange/pass control + the foot of the move table
- whose-turn / win-lose -> a thin strip above the score plaques
- the tentative-move caption -> the board itself: staged tiles tint green (legal) or
  pink (illegal), the board tiles a formed word runs through go a shade darker, and an
  orange score badge sits on the main word (digit sized like a tile value, clamped on-board)

The word geometry (covered cells + badge anchor) is a new pure client-side helper
(ui/src/lib/formed.ts), independent of the move evaluator, so it works on the local or
network preview path alike; the badge's number still comes from the preview score.

Rack: a seven-column grid filling the tray width in both layouts — square, full-width
tiles — with the confirm control in the fixed 7th slot.

Settings: a touch-only "Zoom the board" toggle (default on, device-local) gates the
tile-placement auto-zoom; taking a hint while zoomed in now zooms out so the highlighted
hint word is never left off-screen.

Docs (FUNCTIONAL +_ru, UI_DESIGN, ARCHITECTURE) and e2e/unit tests updated.

ui/src/lib/formed.ts

@@ -0,0 +1,164 @@
+// Client-side geometry of the play being composed: which cells the formed word(s) cover, and
+// where to anchor the move-score badge. This is deliberately independent of the move evaluator
+// (lib/dict/eval.ts) — the network `evaluate` returns legality/score/word strings but no cell
+// coordinates, and the board highlight plus the score badge must work even while the local
+// dictionary is still warming up. Legality and the score come from the preview; the geometry is
+// derived here purely from the board and the staged tiles. Both functions are pure (testable in
+// lib/formed.test.ts); the board stays the source of truth for legality.
+
+import type { Board } from './board';
+import { BOARD_SIZE } from './premiums';
+
+/** Play axis: horizontal or vertical. */
+export type Dir = 'H' | 'V';
+
+/** MainWord locates the play's main word: its start cell, axis and length in cells. */
+export interface MainWord {
+  row: number;
+  col: number;
+  dir: Dir;
+  len: number;
+}
+
+/** FormedGeometry is the composed play's shape: the main word plus the set of every cell (as a
+ *  "row,col" key) that the main word and any cross words cover — committed tiles included. */
+export interface FormedGeometry {
+  main: MainWord;
+  cells: Set<string>;
+}
+
+/** A minimal cell reference; PendingTile satisfies it structurally. */
+interface Cell {
+  row: number;
+  col: number;
+}
+
+const key = (r: number, c: number): string => `${r},${c}`;
+
+/**
+ * filledPredicate returns a test for whether a cell holds a tile once the staged play is on the
+ * board: either a committed tile sits there or a pending tile is being placed there.
+ */
+function filledPredicate(board: Board, pending: readonly Cell[]): (r: number, c: number) => boolean {
+  const pend = new Set(pending.map((p) => key(p.row, p.col)));
+  return (r, c) =>
+    r >= 0 &&
+    r < BOARD_SIZE &&
+    c >= 0 &&
+    c < BOARD_SIZE &&
+    (board[r]?.[c] != null || pend.has(key(r, c)));
+}
+
+/**
+ * span walks the maximal contiguous filled run through cell (row, col) along dir and returns the
+ * run's start index and length on the moving axis (columns for H, rows for V).
+ */
+function span(
+  filled: (r: number, c: number) => boolean,
+  dir: Dir,
+  row: number,
+  col: number,
+): { start: number; len: number } {
+  if (dir === 'H') {
+    let s = col;
+    while (filled(row, s - 1)) s--;
+    let e = col;
+    while (filled(row, e + 1)) e++;
+    return { start: s, len: e - s + 1 };
+  }
+  let s = row;
+  while (filled(s - 1, col)) s--;
+  let e = row;
+  while (filled(e + 1, col)) e++;
+  return { start: s, len: e - s + 1 };
+}
+
+/**
+ * formedGeometry derives the composed play's word geometry from the board and the staged tiles.
+ * It returns the main word (the maximal run along the play axis through the staged tiles) and the
+ * set of every cell the main word and each cross word (a perpendicular run of two or more cells
+ * through a staged tile) cover. It returns null when nothing is staged, or when the staged tiles
+ * are not on a single line — a shape only an illegal play produces, which the caller never asks
+ * about (it is invoked only for a legal preview). The play axis is fixed by the staged tiles when
+ * two or more are colinear; a lone tile takes whichever axis forms the longer word.
+ */
+export function formedGeometry(board: Board, pending: readonly Cell[]): FormedGeometry | null {
+  if (pending.length === 0) return null;
+  const filled = filledPredicate(board, pending);
+  const first = pending[0];
+
+  let dir: Dir;
+  if (pending.length === 1) {
+    const h = span(filled, 'H', first.row, first.col);
+    const v = span(filled, 'V', first.row, first.col);
+    dir = h.len >= v.len ? 'H' : 'V';
+  } else if (pending.every((p) => p.row === first.row)) {
+    dir = 'H';
+  } else if (pending.every((p) => p.col === first.col)) {
+    dir = 'V';
+  } else {
+    return null;
+  }
+
+  const cells = new Set<string>();
+
+  // Main word: the maximal run along the play axis through the staged tiles (contiguous via any
+  // committed tiles between them), collected from any one staged tile.
+  const ms = span(filled, dir, first.row, first.col);
+  let main: MainWord;
+  if (dir === 'H') {
+    main = { row: first.row, col: ms.start, dir, len: ms.len };
+    for (let c = ms.start; c < ms.start + ms.len; c++) cells.add(key(first.row, c));
+  } else {
+    main = { row: ms.start, col: first.col, dir, len: ms.len };
+    for (let r = ms.start; r < ms.start + ms.len; r++) cells.add(key(r, first.col));
+  }
+
+  // Cross words: the perpendicular run through each staged tile, kept only when it is a real word
+  // (two or more cells).
+  const cross: Dir = dir === 'H' ? 'V' : 'H';
+  for (const p of pending) {
+    const cs = span(filled, cross, p.row, p.col);
+    if (cs.len < 2) continue;
+    if (cross === 'H') {
+      for (let c = cs.start; c < cs.start + cs.len; c++) cells.add(key(p.row, c));
+    } else {
+      for (let r = cs.start; r < cs.start + cs.len; r++) cells.add(key(r, p.col));
+    }
+  }
+
+  return { main, cells };
+}
+
+/** Where the move-score badge is anchored: a main-word cell and the tile corner it sits on. */
+export interface BadgeAnchor {
+  row: number;
+  col: number;
+  corner: 'tr' | 'bl';
+}
+
+/**
+ * badgePlacement chooses the main-word tile and corner for the move-score badge, so the badge
+ * points into the board and away from the nearest edge (the board clamps the final pill fully
+ * on-board). The rules: a word touching the right or top edge anchors at a tile's bottom-left
+ * corner (horizontal → its first letter, vertical → its last letter); a word touching the left or
+ * bottom edge — and any mid-board word — anchors at the top-right corner (horizontal → its last
+ * letter, vertical → its first letter). A full-width horizontal word (it touches both side edges)
+ * anchors at the last letter's top-right, which the board then clamps leftward.
+ */
+export function badgePlacement(main: MainWord): BadgeAnchor {
+  const last = BOARD_SIZE - 1;
+  const h = main.dir === 'H';
+  const minRow = main.row;
+  const maxRow = h ? main.row : main.row + main.len - 1;
+  const minCol = main.col;
+  const maxCol = h ? main.col + main.len - 1 : main.col;
+
+  if (h && main.len === BOARD_SIZE) {
+    return { row: main.row, col: maxCol, corner: 'tr' };
+  }
+  if (maxCol === last || minRow === 0) {
+    return h ? { row: main.row, col: minCol, corner: 'bl' } : { row: maxRow, col: main.col, corner: 'bl' };
+  }
+  return h ? { row: main.row, col: maxCol, corner: 'tr' } : { row: minRow, col: main.col, corner: 'tr' };
+}