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.css

@@ -1,4 +1,4 @@
-/* Theme tokens and page structure intentionally mirror ClimbingBoardGPT. */
+/* Theme tokens and page structure for the Adaptive Barrier Monitor demo. */
 :root {
   --base00: #1A1B26;
   --base01: #16161E;

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;

webapp/static/index.html

@@ -159,7 +159,7 @@
       </div>
     </section>
 
-    <!-- Left column, row 2: supporting information, mirroring ClimbingBoardGPT. -->
+    <!-- Left column, row 2: supporting information. -->
     <section class="controls" id="col-info">
       <div class="card explain">
         <h2>How to read it</h2>