Polish web demo tests and refresh notebooks

- add README live-demo badge
- tighten engine validation for jump grids and detector indices
- replace flaky FastAPI TestClient web tests with direct handler tests
- remove unused httpx dev dependency
- clean stale frontend comments
- re-execute notebooks with clean sequential outputs

webapp/static/app.js

@@ -329,7 +329,7 @@ async function runMonteCarlo() {
 }
 
 
-/** Initialize collapsible cards to match the ClimbingBoardGPT interaction. */
+/** Initialize collapsible parameter cards. */
 function initCollapsibleCards() {
   document.querySelectorAll(".card.collapsible > h2").forEach((heading) => {
     const card = heading.parentElement;