fix(session): close the review gaps the boundary opened

- `SessionSummary.updatedAt`'s wire doc still said "Persisted file mtime",
  which stopped being true for attached sessions.
- The core invariant let `session/inherited` fall through the merge-extensible
  default. It is core-owned, so it gets an explicit case; an unbalanced seed
  legally places it inside an open turn, which the relation permits.
- The Agent Note claimed the boundary reaches disk via `live.pending`/
  `scheduleDrain`. Verified false: the constructor append precedes `enter()`,
  so it never publishes on `session/event` and rides the creation seed instead.
  Attaching is therefore a write where none happened before — recorded, since
  only `load()` stays a pure read.
- The deferred-index proposal asserted this change documented the cold-mtime
  skew on `dsh-host-apiproxy`. It did not; the README entry now exists.
- `firstLiveSeq`'s firehose gap runs through its own seq, not below it.
- The boundary is not always at `firstLiveSeq` (the idempotence guard), so
  consumers scan for the last one.
- `lastActivityTime` excludes by type, so a pickup time still leaks onto a
  synthetic closer when a boundary ends an open turn. Documented.
- Pin the fork claim end-to-end: a child inherits a still-running parent's
  open bracket below its own boundary, while the parent has none. Fails if the
  write moves back to the load path.
- Fix the telemetry title that contradicted its own assertions.

The `/status` call site cannot be pinned the way the other two are: the
command appends its own `command/run` before rendering, so the boundary is
never the log tail there. Its fixture now at least renders over a
boundary-bearing log.

packages/ui/tui/tests/snapshots/status-diagnostics-narrow.expected.txt

@@ -48,7 +48,7 @@ buffer
 17| "│                                                      │"
   style 0-0 dim
   style 55-55 dim
-18| "│  Agent:      idle · 7 events · 1 turn · 1 step · 1   │"
+18| "│  Agent:      idle · 8 events · 1 turn · 1 step · 1   │"
   style 0-0 dim
   style 3-12 dim
   style 55-55 dim

packages/ui/tui/tests/snapshots/status-diagnostics.expected.txt

@@ -45,7 +45,7 @@ buffer
 16| "│                                                                                │"
   style 0-0 dim
   style 81-81 dim
-17| "│  Agent:      idle · 7 events · 1 turn · 1 step · 1 tool call                   │"
+17| "│  Agent:      idle · 8 events · 1 turn · 1 step · 1 tool call                   │"
   style 0-0 dim
   style 3-12 dim
   style 81-81 dim

packages/ui/tui/tests/tui.snapshot.ts

@@ -909,6 +909,12 @@ describe('TUI terminal-state snapshots', () => {
           messageSeqs: [1],
           source: { kind: 'fallback' },
         })
+        // Renders over a boundary-bearing log. It cannot pin the exclusion:
+        // `/status` appends its own `command/run` first, so the boundary is
+        // never the tail here. The other two call sites pin it.
+        dateNow.mockReturnValue(Date.parse('2026-07-22T10:10:11.000Z'))
+        session.append('session/inherited', {})
+        dateNow.mockReturnValue(Date.parse('2026-07-22T09:10:11.000Z'))
       },
     }, { columns: 92, rows: 32 })
     await renderAfter(harness, () => {