feat(btw): keep the inherited thread as reference, not an active plan

`/btw` forks the session, so the model receives the parent's whole
conversation — including whatever plan was in flight when the user typed
the command. Nothing tells it that this history is context rather than
its own task, so the fork frequently carries on with the parent's work
instead of answering the side question, which is the opposite of what
`/btw` is for.

Send a boundary instruction as a synthetic part with every message in a
btw session: with the first question in `startBtwSession`, and with each
later send while the panel is expanded and the composer is talking to the
fork.

The wording is deliberately position-independent — it names the history
inherited from the parent thread rather than "everything before this
boundary". The instruction rides along with each send instead of being
pinned once at fork time, so a positional phrasing would be re-anchored
every turn and would end up telling the model to disregard the btw
session's own earlier turns.

The part is synthetic, so it is filtered out of the rendered transcript
whenever the message also carries user text — which is always the case
here. No visual change.

packages/ui/src/components/chat/ChatInput.tsx

@@ -35,7 +35,7 @@ import {
 import { ReviewFlowDialog, type ReviewFlowExecution } from '@/components/session/ReviewFlowDialog';
 import { BtwPanel } from './btw/BtwPanel';
 import { useBtwPanelState } from './btw/useBtwPanelState';
-import { destroyBtwSession, startBtwSession, type BtwSessionRef } from '@/lib/btw';
+import { BTW_BOUNDARY_INSTRUCTION, destroyBtwSession, startBtwSession, type BtwSessionRef } from '@/lib/btw';
 import { AttachedFilesList, AttachedVSCodeFileChips, ActiveEditorFileSuggestion } from './FileAttachment';
 import { lazyWithChunkRecovery } from '@/lib/chunkLoadRecovery';
 import type { ToolPopupContent } from './message/types';
@@ -1126,7 +1126,13 @@ const ChatInputComponent: React.FC<ChatInputProps> = ({
             composerText: !queuedOnly && inputSnapshot.hasContent ? inputSnapshot.message : null,
             composerAttachments: attachedFiles,
             inlineComments: drafts,
-            syntheticTexts: syntheticParts?.map((part) => part.text) ?? [],
+            // btw mode: the boundary rides with every send, not just the
+            // first one, so the inherited transcript stays reference material
+            // for the whole side conversation.
+            syntheticTexts: [
+                ...(isBtwActive ? [BTW_BOUNDARY_INSTRUCTION] : []),
+                ...(syntheticParts?.map((part) => part.text) ?? []),
+            ],
             linkedIssue: linkedIssue
                 ? { number: linkedIssue.number, title: linkedIssue.title, url: linkedIssue.url, contextText: linkedIssue.contextText }
                 : null,