[chillwur]

Problem

Formatting wasn't standardized across apps/frontend and apps/backend. Without a shared formatter, small style differences (quote style, line wrapping, trailing commas) show up as noise in diffs and code review, distracting from the actual logic changes.

Solution

• Added a root .prettierrc.json with project conventions: 2-space indent and LF line endings (matching .editorconfig), double quotes, semicolons, trailing commas everywhere, and a 100-character print width.
• Added a .prettierignore covering dist, .next, node_modules, target, and pnpm-lock.yaml.
• Added root format (prettier --write .) and format:check (prettier --check .) scripts.
• Added eslint-config-prettier to both apps/frontend and apps/backend and appended it as the last entry in each workspace's ESLint flat config, so any ESLint stylistic rules that conflict with Prettier are disabled and formatting is Prettier's job alone.
• Ran pnpm format once across the repo and committed the resulting formatting-only diff (mostly Markdown table alignment and collapsing a few lines under the new print width) — no logic changes.
• Verified pnpm format:check, pnpm lint, pnpm typecheck, and pnpm test all pass after the pass.

closes #5