Mobile app: sync /tabs/projects with recursive-projects redesign + table renames

Problem

apps/mobile/app/(tabs)/projects/index.tsx still references the pre-rename schema (trip_items, trip_id) and the pre-redesign ProjectWithCount shape. CI Lint & Typecheck fails on this file with:

Type '...' is missing the following properties from type 'ProjectWithCount': slug, dueDate, parentProjectId, kind

Same failure has been on main for at least the last several commits — confirmed by running pnpm -r exec tsc --noEmit on a clean main checkout. CI is therefore globally red on main, blocking auto-merge for all PRs.

Scope

• Rename queries: trip_items → cards → options (the rename has gone two rounds since this item was filed; today's mobile code should target options + project_id).
• Update ProjectWithCount mapping to include slug, kind, parentProjectId, dueDate (added by the recursive-projects redesign on 2026-05-02).
• Decide whether the mobile project list should still be flat (kind=plan only?) or kind-aware like the web app — needs a UX decision.
• Update any related mobile screens that read the old shape (likely the project detail screen too).

Acceptance

• pnpm -r exec tsc --noEmit green from repo root.
• Mobile project list builds + runs in Expo with the new schema.
• CI Lint & Typecheck green on main.

Surfaced from: in-conversation review while closing YOPSYS-78 — pre-existing main-CI failure unrelated to this PR's diff.

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Plane: YOPSYS-114