feat(api): add health probes, OpenAPI doc, and router

Co-Authored-By: Claude Sonnet 4.6 <noreply@anthropic.com>

crates/api/src/openapi.rs

@@ -0,0 +1,25 @@
+use axum::{Json, Router, extract::State, routing::get};
+use utoipa::OpenApi;
+
+use crate::{AppState, health};
+
+#[derive(OpenApi)]
+#[openapi(
+    paths(health::live, health::ready),
+    components(schemas(health::Live, health::Ready)),
+    info(title = "Collection Management System", version = "0.0.0")
+)]
+struct ApiDoc;
+
+/// Serve the OpenAPI document, overriding the title from runtime config so the
+/// product name is never hardcoded.
+async fn openapi_json(State(state): State<AppState>) -> Json<utoipa::openapi::OpenApi> {
+    let mut doc = ApiDoc::openapi();
+    doc.info.title = state.app_name.clone();
+    Json(doc)
+}
+
+/// OpenAPI routes, parameterized over [`AppState`].
+pub(crate) fn routes() -> Router<AppState> {
+    Router::new().route("/api-docs/openapi.json", get(openapi_json))
+}