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.

docs/persistence-catalog.md

@@ -78,7 +78,7 @@ export type SessionEvent<T extends SessionEventType = SessionEventType> = {
 }[T]
 ```
 
-Sources: [`packages/core/session/src/types.ts:270`](../packages/core/session/src/types.ts) · [`packages/core/session/src/types.ts:277`](../packages/core/session/src/types.ts) · [`packages/core/session/src/types.ts:306`](../packages/core/session/src/types.ts) · [`packages/core/session/src/types.ts:338`](../packages/core/session/src/types.ts)
+Sources: [`packages/core/session/src/types.ts:274`](../packages/core/session/src/types.ts) · [`packages/core/session/src/types.ts:281`](../packages/core/session/src/types.ts) · [`packages/core/session/src/types.ts:310`](../packages/core/session/src/types.ts) · [`packages/core/session/src/types.ts:342`](../packages/core/session/src/types.ts)
 
 ## Events
 
@@ -413,6 +413,10 @@ Source: [`packages/sandbox/sandbox-policy/src/session-mode.ts:33`](../packages/s
  * and no writer in this lifecycle produced it. Payload is empty — position
  * and `time` carry the meaning.
  *
+ * Locate the LAST one rather than reading `firstLiveSeq`: a seed already
+ * ending in a boundary is not re-marked, so reopening an untouched session
+ * does not grow its log per pickup.
+ *
  * An owner of a standalone open/close bracket (`compact/start` …
  * `compact/end`) reads it because inherited history and live work are
  * otherwise byte-identical: an unmatched opening marker below the boundary
@@ -423,7 +427,7 @@ Source: [`packages/sandbox/sandbox-policy/src/session-mode.ts:33`](../packages/s
 'session/inherited': Record<string, never>
 ```
 
-Source: [`packages/core/session/src/types.ts:266`](../packages/core/session/src/types.ts)
+Source: [`packages/core/session/src/types.ts:270`](../packages/core/session/src/types.ts)
 
 #### `session/title` — log-only