feat(web): add nuqs for URL search-param state

[GRACENOBLE]

Summary

• Installs nuqs v2.8.9 and adds NuqsAdapter as the outermost wrapper in app/providers.tsx, enabling useQueryState / useQueryStates hooks across the app
• Updates web/docs/trpc.md to reflect the corrected provider nesting order (NuqsAdapter > SessionProvider > TRPCProvider) and documents the Toaster sibling
• Adds web/docs/url-state.md covering useQueryState, built-in parsers, createSearchParamsCache for Server Components, and a when-to-use guide vs plain useSearchParams

Test plan

• pnpm build passes with no TypeScript errors
• pnpm lint passes
• Dev server starts and existing pages load without errors

Summary by CodeRabbit

• New Features

  • Added support for URL search-param state across the app.
  • Introduced a new documentation guide covering how to use and manage URL-based state.

• Documentation

  • Added a docs index entry for the new URL state guide.
  • Updated provider wiring examples and refreshed the verification date in existing docs.

• Chores

  • Added the required dependency for URL state handling.

[coderabbitai]

📝 Walkthrough

Walkthrough

Adds nuqs (^2.8.9) as a dependency and mounts NuqsAdapter as the outermost wrapper in app/providers.tsx. Adds a new url-state.md documentation page covering nuqs usage, updates the trpc.md provider snippet to reflect the new nesting, and adds an index entry for the new doc.

nuqs integration

Layer / File(s)	Summary
nuqs dependency and provider wiring web/package.json, web/app/providers.tsx	nuqs added to dependencies; NuqsAdapter imported and placed as the outermost wrapper around SessionProvider, TRPCProvider, and Toaster.
Documentation web/docs/url-state.md, web/docs/trpc.md, web/docs/_index.md	New url-state.md documents client usage, parsers, server component access, and when to prefer nuqs. trpc.md provider snippet updated to show NuqsAdapter wrapping, and the docs index gains a new row.

Estimated code review effort

🎯 1 (Trivial) | ⏱️ ~5 minutes

Poem

A NuqsAdapter hops to the top,
Wrapping the tree without a stop.
Search params now live in the URL bar,
Parsed and synced both near and far.
🐇 The rabbit approves — what a clean swap!

🚥 Pre-merge checks | ✅ 5

✅ Passed checks (5 passed)

Check name	Status	Explanation
Title check	✅ Passed	The title accurately summarizes the main change: adding nuqs for URL search-param state in the web app.
Docstring Coverage	✅ Passed	No functions found in the changed files to evaluate docstring coverage. Skipping docstring coverage check.
Linked Issues check	✅ Passed	Check skipped because no linked issues were found for this pull request.
Out of Scope Changes check	✅ Passed	Check skipped because no linked issues were found for this pull request.
Description Check	✅ Passed	Check skipped - CodeRabbit’s high-level summary is enabled.

✨ Finishing Touches

📝 Generate docstrings

• Create stacked PR
• Commit on current branch

🧪 Generate unit tests (beta)

• Create PR with unit tests
• Commit unit tests in branch add-nuqs

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