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

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

crates/api/src/lib.rs

@@ -1 +1,24 @@
 //! HTTP API: router, handlers, and OpenAPI document.
+
+mod health;
+mod openapi;
+
+use axum::Router;
+use db::Db;
+
+/// Shared application state passed to handlers.
+#[derive(Clone)]
+pub struct AppState {
+    /// Database handle for this organization.
+    pub db: Db,
+    /// User-facing product name (from config). Never hardcoded.
+    pub app_name: String,
+}
+
+/// Build the application router from shared state.
+pub fn build_app(state: AppState) -> Router {
+    Router::new()
+        .merge(health::routes())
+        .merge(openapi::routes())
+        .with_state(state)
+}