Store Assistent chats, UI state, and taste in sqlite.

Runtime session data belongs in one DB so History can FTS-search and follow the volume; persona overlays and sidecar cards stay files.

Co-authored-by: Cursor <cursoragent@cursor.com>

Assets/assistent.css

@@ -427,6 +427,23 @@
     letter-spacing: 0.03em;
 }
 
+.sa-chats-search {
+    appearance: none;
+    width: 100%;
+    box-sizing: border-box;
+    border: 1px solid color-mix(in srgb, currentColor 22%, transparent);
+    background: color-mix(in srgb, #000 28%, transparent);
+    color: inherit;
+    border-radius: 0.35rem;
+    padding: 0.35rem 0.5rem;
+    font: inherit;
+    font-size: 0.82rem;
+}
+
+.sa-chats-search:focus {
+    outline: 1px solid color-mix(in srgb, #6cf 55%, currentColor);
+}
+
 .sa-chats-panel-hint {
     font-size: 0.72rem;
     opacity: 0.65;