Wire the hero search to a /search results page

[posthog-eu]

Problem

• Every visitor sees a headline search box ("Search your learning in plain English") that does nothing — the platform's marquee feature is a dead text field.
• components/home/hero.tsx rendered a bare SearchInput: no form, no submit handler, no target. There was no /search route, and nothing linked to one.
• The search backend in app/api/search and lib/search is complete but was unreachable from the UI. The results page was deferred in prompts/intelligent-search.md; this PR finishes that wiring.

Changes

• Navigate on submit. A new SearchForm — a native <form action="/search" method="get"> around SearchInput (name="q") — so the box works without client JavaScript. The hero now uses it.
• Render the results. A new /search page reads q and hands off to a SearchResults client component that posts to /api/search and renders the ranked video and lesson cards, reusing the existing LessonVideoCard/LessonCard.
• Per AGENTS.md §11: a "Found N results across M courses" count, a sort control defaulting to Most Relevant, and loading / error / empty states (the empty state links to the catalog).

Boundaries per AGENTS.md §5 are kept: the browser holds no token and only calls the server route; sorting re-runs the search because the route owns ordering.

Not included: the PostHog localhost capture guard

The report also flags instrumentation-client.ts capturing local dev exceptions into the shared project. Draft #6 already changes that file to capture_exceptions: process.env.NODE_ENV === "production" — exactly the development-host guard the report asks for. Duplicating it here would conflict with #6, so this PR leaves that file to #6 and is scoped to the search wiring, the part that actually costs users.

Why

The reported exception (ReferenceError: SearchForm is not defined) was one-off local dev noise from this feature's aborted work-in-progress. The lasting cost is that search — the product's headline feature — could not be reached from the UI at all.

Test

1. npm run lint — passes.
2. npm run typecheck — passes.
3. npm run build — compiles and generates route types; page-data collection needs real SANITY_API_READ_TOKEN (unrelated to this change, same limit noted in Filter ClerkJS network-noise exceptions before PostHog capture #6).
4. Dev server: /search returns 200 and renders the form, the empty prompt, and the loading state; /search?q=... prefills the box. /'s only failure is the placeholder Sanity token.

Not verified live

Rendering real result cards needs a real Sanity dataset, an OpenAI key, and a deployed Studio for the Context MCP — none available in this environment. The client fetch, states, and card mapping are wired against the existing SearchResponse contract but were not exercised against a live search.

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Created with PostHog Desktop from this inbox report.

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