Restore the two gating rules the redesign dropped, and put the dot back in its corner

A1 — hiding the assistant from the dot. The old build kept a chevron button
beside the dot wired to _hide_to_edge; the redesign moved that action into the
panel's ⋯ menu, so hiding went from one click to open-panel → ⋯ → Ẩn. The
drawing has no button there (26px, no text, no chevron), so the reach comes
back as a right-click on the dot, with the tooltip saying so.

A2 — per-part cost. Folding Input/Output/Cache into the total tile's sub-line
carried their token counts but not their prices, which live only on those three
cards, so the cost breakdown left the screen entirely. The wireframe asks for
the tiles anyway — "$0.31 Tổng chi phí · 57 lượt / 395.4K Tổng token / 292.8K
Input / 102.7K Output / 108.9K Cache" — so they are back as tiles, each with its
price, and the turn count rides on the cost tile's label as drawn.

The dot on Cowork. _update_dock_guard lifted it whenever a composer existed,
testing only the vertical axis. On a wide window the composer ends at the chat
column's right edge — x=1688 against a dot at x=1892 — so the dot rose 156px to
clear something that was never under it, and Cowork became the one screen where
it was not in the corner. It now lifts only when the two actually overlap:
18px from the bottom at 1936px wide, still lifted at 1200 and 900 where the
composer does reach under it.

Also: open_tooltip and show_tooltip still said "App Assistant" after the rename.

check_dock_corner covers the corner rule at three widths and fails if the lift
comes back unconditionally.

18/18 checkers pass.

Co-Authored-By: Claude Opus 5 (1M context) <noreply@anthropic.com>

i18n.py

@@ -498,13 +498,17 @@ STRINGS: Dict[str, Dict[str, str]] = {
         "en": "Ask how to use the app…", "ja": "アプリの使い方を質問…",
         "vi": "Hỏi cách sử dụng app…"},
     "help_agent.open_tooltip": {
-        "en": "App Assistant — help using the app",
-        "ja": "アプリアシスタント — アプリの使い方をサポート",
-        "vi": "Trợ lý App — hỗ trợ sử dụng app"},
+        "en": "AI Assistant — help using the app",
+        "ja": "AI Assistant — アプリの使い方をサポート",
+        "vi": "AI Assistant — hỗ trợ sử dụng app"},
     "help_agent.collapse_tooltip": {
         "en": "Minimize", "ja": "最小化", "vi": "Thu nhỏ"},
     "help_agent.hide_tooltip": {
         "en": "Hide to the edge", "ja": "端に隠す", "vi": "Ẩn vào cạnh phải"},
+    "help_agent.dot_hint": {
+        "en": "right-click to hide",
+        "ja": "右クリックで非表示",
+        "vi": "chuột phải để ẩn"},
     # The name on the launcher pill. Deliberately the same in every language —
     # it is a product name, and it only shows on hover, so length is not a
     # constraint the way it was on a permanently visible badge.
@@ -513,8 +517,8 @@ STRINGS: Dict[str, Dict[str, str]] = {
     "help_agent.more_tooltip": {
         "en": "More", "ja": "その他", "vi": "Thêm"},
     "help_agent.show_tooltip": {
-        "en": "Show the App Assistant", "ja": "アプリアシスタントを表示",
-        "vi": "Hiện App Assistant"},
+        "en": "Show the AI Assistant", "ja": "AI Assistant を表示",
+        "vi": "Hiện AI Assistant"},
     "help_agent.empty_reply": {
         "en": "(no answer)", "ja": "(回答なし)", "vi": "(không có phản hồi)"},
     "help_agent.error": {