feat(web): move the chat action cluster to the pinned bottom and align the button sets

- task 01: relocate the .chat-actions row (New chat + Share, comments byte-identical with a Phase 65 note) from the top of the column to the bottom of .chat-shell, directly above the composer
- task 02 (owner-locked A1): wrap the row + #composer in ONE sticky .chat-bottom unit (position: sticky; bottom: env(safe-area-inset-bottom, 0), no z-index) — the pills stay at the bottom of the screen at every scroll position and settle into flow above the footer
- task 03 (owner-locked A2): right-align the bottom row to the column's right edge (justify-content: flex-end), mirroring the right-aligned Save-as-doc corner; the five action pills share one 44px / 999px-pill geometry
- task 04: dedicated Playwright suite tests/e2e/test_bottom_chat_actions.py (resting geometry, the A1 pin across the sticky range, A2 alignment + DOM order + mobile stack + 360px overflow bound + 44px touch targets, New chat / Share click-through) — green in isolation
- task 05: regression matrix green in isolation (pinned_composer 4, save_share_ux 5, chat_persistence 4, share_chat 4, chat_history 5, smoke 3); full gate green — unit + integration pass, app/ coverage 99% (>90%), ruff + pyright clean

tests/unit/test_pinned_composer.py

@@ -21,6 +21,15 @@ block. The pin is therefore TWO rules, and both are pinned here:
   bottom edge at every scroll position, settling back into flow (above
   the footer) once the document bottom is reached;
 
+Phase 65 (task 02, owner-locked A1, 2026-09-01, `TODO.md` L3): the pin
+moved up one level — the `.chat-bottom` wrapper (the row + this form,
+the LAST child of `.chat-shell`) carries `position: sticky; bottom:
+env(safe-area-inset-bottom, 0)` and is the unit that stays glued to the
+viewport bottom; the composer's own sticky pair above and the
+`.messages` grow stay exactly as phase 52 pinned them (the composer's
+sticky is redundant inside the wrapper but kept — and pinned here —
+for the computed-style contract).
+
 The pin is CSS-only:
 
 * `.composer` carries the sticky pair (``position: sticky`` + the
@@ -34,15 +43,11 @@ The pin is CSS-only:
 * the sticky context survives: `.chat-shell` / `.app-main` / `.messages`
   gain no ``overflow`` and `.messages` gains no inner scroller — the page
   keeps scrolling at the document level (the phase-42 model);
-* ``index.html`` needs no change: the composer is already the LAST child
-  of `.chat-shell` (the sticky shift range is that column's box) and the
-  `#message-input` / `#send-btn` / `#send-status` markup is untouched;
-* the sticky context survives: `.chat-shell` / `.app-main` / `.messages`
-  gain no ``overflow`` and `.messages` gains no inner scroller — the page
-  keeps scrolling at the document level (the phase-42 model);
-* ``index.html`` needs no change: the composer is already the LAST child
-  of `.chat-shell` (the sticky shift range is that column's box) and the
-  `#message-input` / `#send-btn` / `#send-status` markup is untouched;
+* ``index.html`` (phase 65 task 02, owner-locked A1): the row + composer
+  are wrapped in the `.chat-bottom` unit — the wrapper (NO id) is now
+  the LAST child of `.chat-shell` (the sticky shift range is still that
+  column's box) and the `#message-input` / `#send-btn` / `#send-status`
+  markup is untouched;
 * ``app.js`` gains NO page-scroll call site — the phase-42 invariant
   (``scrollReveal`` is still the single ``window.scrollTo``; no
   ``scrollIntoView``, no ``window.scrollY``) is re-pinned here so the
@@ -303,25 +308,47 @@ def test_document_level_scroll_is_untouched() -> None:
 # ---------- index.html: no DOM change needed ----------
 
 
-def test_composer_is_last_child_of_the_chat_shell() -> None:
-    """The composer must remain the LAST element child of `.chat-shell`
-    (the sticky containing block): the sticky shift range is that
-    column's box, so a sibling after the form would carve the range away
-    and re-break the pin. The `#message-input` / `#send-btn` /
-    `#send-status` markup is untouched (the pin ships no DOM change)."""
+def test_chat_bottom_unit_is_last_child_of_the_chat_shell() -> None:
+    """Phase 65 (task 02, owner-locked A1): the LAST element child of
+    `.chat-shell` is the `.chat-bottom` wrapper — NO id (nothing in JS
+    binds it; the bindings live on the inner elements, the move is pure
+    HTML/CSS) — holding exactly the `.chat-actions` row and then the
+    `#composer` form, in that order: the row + composer are ONE sticky
+    unit, and the wrapper owns the shell's bottom slot, so the sticky
+    shift range is still that column's box (a sibling after it would
+    carve the range away and re-break the pin). The composer form keeps
+    `novalidate` and its contract ids."""
     shell = _tree().find("chat-shell")
     last = shell["children"][-1]
-    assert last["tag"] == "form", (
-        "the composer <form> must stay the last child of .chat-shell"
+    assert last["tag"] == "div" and (
+        last["attrs"].get("class") or ""
+    ).split() == ["chat-bottom"], (
+        "the .chat-bottom wrapper must be the last child of .chat-shell"
     )
-    assert last["attrs"].get("id") == "composer"
-    assert "novalidate" in last["attrs"], (
+    assert last["attrs"].get("id") is None, (
+        "the wrapper carries no id — the JS bindings live on the inner "
+        "elements"
+    )
+    kids = last["children"]
+    assert len(kids) == 2, (
+        "the unit holds exactly two element children: .chat-actions, then "
+        "#composer"
+    )
+    row, form = kids
+    assert row["tag"] == "div" and (
+        row["attrs"].get("class") or ""
+    ).split() == ["chat-actions"], (
+        "the first child is the .chat-actions row (the New chat → Share "
+        "DOM order is pinned by test_save_chat_ui.py)"
+    )
+    assert form["tag"] == "form" and form["attrs"].get("id") == "composer"
+    assert "novalidate" in form["attrs"], (
         "phase 48: the composer form stays `novalidate` (a `required` "
         "input would swallow the Stop click)"
     )
     ids = {
         node["attrs"].get("id")
-        for node in _walk(last)
+        for node in _walk(form)
         if node["attrs"].get("id")
     }
     assert {"composer", "message-input", "send-btn", "send-status"} <= ids, (