Generated and pasted board backgrounds move into .backgrounds/ — the board root stops collecting the app's own pictures

New app-written background images (Theme tab ▸ Pattern, and Edit ▸ Paste
as Board Background) now land in a hidden `.backgrounds/` folder at
board root instead of beside index.md, matching the `.trash/` app-managed
pattern. `background.image` stores the qualified relative reference
(`.backgrounds/facets.png`); the resolver needed no change at all, since
it already accepted any relative path inside the board root — the same
mechanism that already resolved `art/backdrops/sunset.png` resolves the
new location for free. `BoardWriter.writeBoardImage` now creates its
destination folder if missing, since `.backgrounds/` won't exist until a
board's first generated or pasted background.

The Finder collision-ladder (`BoardStore.boardImageName`) is rescoped to
`.backgrounds/`'s own contents, and its overwrite-in-place check now
recognizes only the qualified form as "ours" — a legacy bare
`background.image: facets.png` from before this change is read as a
foreign reference rather than migrated, so a regeneration writes a fresh
`.backgrounds/` file and orphans the old one in place, per the no-migration
ruling. The Theme tab's Pattern/Solid mode-detection was updated to
recognize both the legacy and current spellings as the generator's own
output.

The board loader needed no change: `.backgrounds/` is a hidden,
non-UUID-shaped name, and `.skipsHiddenFiles` already keeps every hidden
entry off the lane walk before any name-based exclusion is consulted —
pinned with a new loader test. Deliberately did not add `.backgrounds` to
IntegrityRules' claimed-name/squatter-displacement table: that table
mirrors a specific existing DESIGN ruling this card doesn't amend.

Claude-Session: https://claude.ai/code/session_014PtZdPwqZuqEDLc6wZMtEy

Kanban/Storage/BackgroundField.swift

@@ -67,9 +67,11 @@ extension FrontmatterDocument {
     ///
     /// The doc comment above says "There is no image picker and none is planned"; that sentence held
     /// until generated backgrounds (DESIGN/explorations/board-backgrounds.md), which do not make one
-    /// either. What the generator writes is a *file it just created in the board folder* and the name
-    /// it wrote it under — the app is not browsing the user's pictures, it is naming its own output —
-    /// so the hand-written path stays the escape hatch it always was, and this write preserves it the
+    /// either. What the generator writes is a *file it just created* — in `.backgrounds/`
+    /// (`BoardBackdrop.backgroundsFolderName`, ruled 2026-08-09), never the board root, for every new
+    /// write — and the relative reference to it, which is still simply a path the reader was already
+    /// able to resolve; the app is not browsing the user's pictures, it is naming its own output — so
+    /// the hand-written path stays the escape hatch it always was, and this write preserves it the
     /// same way the colour write preserves an image: by subkey.
     ///
     /// Everything else is `setStyleValue`'s, deliberately shared rather than restated: the same