Add category filters and live search to parts page

- Filter chips for in-use categories with OR semantics; "All" chip shown
  when nothing is selected.
- Search input filters as the user types (150 ms debounce, replaceState
  so back-button stays useful).
- Fix sort indicators: the `arrow()` helper read `sort`/`dir` from a
  plain function, which Svelte's static dep tracking doesn't trace —
  the ▲/▼ never updated on client-side sorts. Made it a reactive `$:`
  declaration.

Co-Authored-By: Claude Opus 4.7 (1M context) <noreply@anthropic.com>

prompts.txt

@@ -165,3 +165,11 @@ Short: what it is, prerequisites (Docker), quickstart
 2. Print the resulting file tree.
 3. Print the exact command sequence to bring it up from a fresh clone.
 4. Call out anything you guessed at that I should review before we move on.
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+When I select Record Movement in the Parts page, can't we prepopulate the movement since we know the part?
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+ANd shouldn't some of the fields be defaulted to our best guess based on what we know about the part?
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