feat(tasks): SessionStart reads the claim — a compaction gets its work back (milestone 381 step 3)

The claim now pays rent to the session that set it. The SessionStart hook
sends the host's `source` and the session id; the server renders a claim
section by source (task_claims.render_claims):

- compact / clear: the work this session had claimed, each with its two
  latest log entries, so a compacted session resumes from the record
  rather than from a count of open tasks.
- startup / clear: other sessions' live claims (may still be running) and
  in-progress tasks whose claim went quiet (abandoned mid-task).
- fork: the same, framed as "two sessions may now hold this".
- resume, or no source sent: nothing.

The task sidebar shows a live claim as "Being worked" and a dead one on
open work as "Went quiet", so the operator sees a session die mid-task.

Co-Authored-By: Claude Opus 5.5 <noreply@anthropic.com>

frontend/src/types/note.ts

@@ -29,6 +29,10 @@ export interface Note {
   due_date: string | null;
   started_at: string | null;
   completed_at: string | null;
+  // Which session is working this task now (milestone 381). Null when no
+  // session ever claimed it; `live` false when the claim's lease ran out —
+  // a session that went quiet mid-task, which is worth seeing, not hiding.
+  claim?: TaskClaim | null;
   recurrence_rule: Record<string, unknown> | null;
   recurrence_next_spawn_at: string | null;
   is_task: boolean;
@@ -53,3 +57,11 @@ export interface NoteListResponse {
   notes: Note[];
   total: number;
 }
+
+export interface TaskClaim {
+  held_by: number | null;
+  session: string | null;
+  since: string | null;
+  touched: string | null;
+  live: boolean;
+}

frontend/src/views/TaskEditorView.vue

@@ -15,7 +15,8 @@ import type { TaskStatus, TaskPriority } from "@/types/task";
 import type { TaskKind } from "@/types/note";
 import { useSystemsStore } from "@/stores/systems";
 import type { System } from "@/api/systems";
-import type { Note } from "@/types/note";
+import type { Note, TaskClaim } from "@/types/note";
+import { relativeTime } from "@/composables/useRelativeTime";
 import type { Editor } from "@tiptap/vue-3";
 import MarkdownToolbar from "@/components/MarkdownToolbar.vue";
 import TiptapEditor from "@/components/TiptapEditor.vue";
@@ -54,6 +55,7 @@ const parentId = ref<number | null>(null);
 const parentTitle = ref("");
 const startedAt = ref<string | null>(null);
 const completedAt = ref<string | null>(null);
+const claim = ref<TaskClaim | null>(null);
 const recurrenceRule = ref<Record<string, unknown> | null>(null);
 const parentSearchQuery = ref("");
 const parentSearchResults = ref<{ id: number; title: string }[]>([]);
@@ -318,6 +320,7 @@ onMounted(async () => {
       const noteTask = store.currentTask as unknown as Note;
       startedAt.value = noteTask.started_at ?? null;
       completedAt.value = noteTask.completed_at ?? null;
+      claim.value = noteTask.claim ?? null;
       recurrenceRule.value = noteTask.recurrence_rule ?? null;
       savedTitle = title.value;
       savedBody = body.value;
@@ -592,7 +595,21 @@ useEditorGuards(dirty, save);
               <option v-if="kind === 'plan'" value="plan">Plan (legacy)</option>
             </select>
           </div>
-          <div v-if="startedAt || completedAt" class="sb-timestamps">
+          <div v-if="startedAt || completedAt || claim" class="sb-timestamps">
+            <div v-if="claim && claim.live" class="sb-timestamp">
+              <span class="sb-ts-label">Being worked</span>
+              <span class="sb-ts-value">
+                by a session{{ claim.session ? ` (${claim.session.slice(0, 8)})` : "" }},
+                last active {{ claim.touched ? relativeTime(claim.touched) : "recently" }}
+              </span>
+            </div>
+            <div v-else-if="claim" class="sb-timestamp">
+              <span class="sb-ts-label">Went quiet</span>
+              <span class="sb-ts-value">
+                the session working this stopped
+                {{ claim.touched ? relativeTime(claim.touched) : "" }} without finishing it
+              </span>
+            </div>
             <div v-if="startedAt" class="sb-timestamp">
               <span class="sb-ts-label">Started</span>
               <span class="sb-ts-value">{{ new Date(startedAt).toLocaleString() }}</span>