feat(kernel): finish phase3 bridge cutover and phase4 tree projection

- add generic mnote-web query transport and bridge routes for workspace/request/trace

- route sidebar compat traffic through mnote-web and tighten fixture fallback to dev/test

- unify sidebar/page tree/file tree/picker consumers on page_tree projection

- sync phase3/phase4 checklist, breakdown docs, and harness progress state

rust/crates/mnote-web/src/routes/mod.rs

@@ -1,7 +1,9 @@
+mod bridge;
 mod compat;
 mod health;
 mod hermes;
 mod kernel;
+mod query_support;
 mod sse;
 mod ws;
 
@@ -15,10 +17,16 @@ pub fn build_router(state: AppState) -> Router {
 
     Router::new()
         .route("/health", get(health::health))
-        .route("/api/kernel/projections/sidebar", get(kernel::project_sidebar))
+        .route(
+            "/api/kernel/projections/sidebar",
+            get(kernel::project_sidebar),
+        )
         .route("/api/kernel/subtree", get(kernel::subtree))
         .route("/api/kernel/edges", get(kernel::edges))
         .route("/api/kernel/graph", get(kernel::graph))
+        .route("/api/bridge/workspace", get(bridge::workspace))
+        .route("/api/bridge/request", get(bridge::request))
+        .route("/api/bridge/trace", get(bridge::trace))
         .route("/api/stream/events", get(sse::events))
         .route("/api/realtime/ws", get(ws::socket))
         .nest(
@@ -29,7 +37,9 @@ pub fn build_router(state: AppState) -> Router {
         )
         .nest(
             &compat_next_base_path,
-            Router::new().route("/ai-agent/run", post(compat::next_ai_agent_run)),
+            Router::new()
+                .route("/ai-agent/run", post(compat::next_ai_agent_run))
+                .route("/sidebar", get(compat::next_sidebar)),
         )
         .with_state(state)
 }