Add "Search in bottom bar" to Settings

New Settings > Appearance toggle moves Search into the bottom bar and
out of the top bar, so there's only one search entry point on screen.
Tapping it pushes SearchView same as before - it's never highlighted,
since it isn't a real destination.

- Attached: Search becomes the row's last item, styled like any other
  tab but permanently idle.
- Floating: Search gets its own separate, always fully round satellite
  circle beside the bar instead.

Enabling it costs one regular tab: SettingsController.maxVisibleTabs
is now a dynamic getter (defaultMaxVisibleTabs - 1 while the toggle is
on) instead of the old flat constant, re-enforced immediately if the
user already has all 5 slots filled - the same cleanup path used for
a fresh install or an app update that added a new tab.

Co-Authored-By: Claude Sonnet 5 <noreply@anthropic.com>

.claude/context/design-system/DESIGN_SYSTEM.md

@@ -274,6 +274,7 @@ Mobile always has **5 pinned tabs in the bottom bar**. The rest go in the drawer
 | Top bar | 44px row: `menu` on the left; `plus` and avatar on the right (all accent-text). Then a 34px large title, then the search field. | 64px: `menu`, 22px title, `search`, avatar |
 | Bottom bar | Surface fill, 1px top line. Icon 24 above a 10px label. Active: accent-text. Idle: fg-3. 34px home indicator below. | Surface fill, 80px. Icon 24 above a 12px label (label space is always reserved, just invisible when idle, so nothing shifts on selection). The active icon sits inside a 56×32 accent-soft pill that slides between tabs as one shared indicator, rather than popping in/out per tab. 20px gesture bar below. |
 | Bottom bar - floating (Settings → Appearance → "Bottom bar", both platforms) | Same content, inset 16px from both side edges and clear of the bottom safe area instead of edge to edge, radius 28 (the sheet-top/drawer-edge radius) rather than square corners. No shadow (product UI stays flat - see 1.4); a 1px `line` border stands in for elevation instead. Row height drops slightly (64/72 vs 50/80) to suit. Android's idle tabs drop their reserved label space in this mode - the icon just centers in the button and renders a touch bigger (27 vs 24px) - rather than sitting high with a gap held open under it. `Scaffold.extendBody: true` while floating, so the body scrolls behind the bar's transparent margin instead of stopping short of it. |
+| Bottom bar - "Search in bottom bar" (Settings → Appearance, either bar style) | Adds a Search entry that's never highlighted (tapping it pushes Search, same as the top bar's own search action, which this replaces so there's only one entry point on screen) - the row's last item when attached (same styling as a real tab, just permanently idle), or its own separate, always fully round satellite circle beside the bar when floating. Costs one regular tab: `SettingsController.maxVisibleTabs` drops from 5 to 4 while this is on. |
 | Upload | `plus` in the nav bar | Extended FAB |
 | Overflow icon | `ellipsis` | `ellipsis-vertical` |
 | Settings | Pushed screen with a "‹ Files" back button and a large title. No bottom bar. | Pushed screen with a back arrow and a title. No bottom bar. |

.claude/context/styling.md

@@ -197,6 +197,12 @@ blocks are noted where they matter:
   with a `line` border instead of edge to edge (no opacity/blur knob
   either). The host `Scaffold` needs `extendBody: true` while it's active
   (`main.dart` already wires this off `SettingsController.bottomBarStyle`).
+  Settings' separate "Search in bottom bar" toggle
+  (`SettingsController.searchInBottomBar`) adds a never-highlighted Search
+  entry to either bar style (`NooBottomBar`'s `searchDestination`/
+  `onSearchTap`) and lowers `SettingsController.maxVisibleTabs` by one -
+  use that getter, not `defaultMaxVisibleTabs` from `models/app_tab.dart`,
+  anywhere that needs the *current* cap on regular tabs.
 - [`SyncedHeaderScaffold`](../../lib/widgets/synced_header_scaffold.dart) —
   the pull-to-sync `CustomScrollView` header with the persistent sync-status
   chip and pull-to-refresh gesture/spinner. Every screen (including