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/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>