CB-307: lock push-loop injection to dedicated status-gated loop (mechanism b)

Chose a small dedicated scheduled loop over AgentControl.send guarded by an
injectable status check, instead of reusing the worker Injector (which couples
to WorkerPresence/StatusPoller). Isolated + unit-testable via injected clock.
Records live ground truth: this primary resolves to term_656c8cc03e1f0b1 (w2:pY)
— confirms the primary runs in a herdr pane so the push path is exercisable.

bridged/docs/CB-307-Push-Loop.md

@@ -78,9 +78,15 @@ A `ReplyPushLoop` component, notified at the single no-waiter call site
 - **Ack = drain.** The primary draining (`drainReplies` = peek + ack) is the acknowledgement.
   The loop's **stop condition is `inbox.peek(target).isEmpty()`** — the reply is gone from the
   inbox because it was acked. No new `bridge_ack` tool needed for v1 (see Increment 3).
-- **Status-gated injection.** Never inject mid-turn. Reuse the `Injector`/`StatusPoller`
-  discipline: deliver only when the primary's terminal samples **injectable** (IDLE/BLOCKED),
-  one message per turn. **Readiness caveat (below).**
+- **Status-gated injection (mechanism (b), chosen).** A dedicated lightweight scheduled loop,
+  **not** the worker `Injector`. It injects via `AgentControl.send(primaryTerminal, nudge)`
+  (the same herdr `agent.send` = `pane send-text` + submit that delivers to workers) only when
+  `AgentControl.status(primaryTerminal).injectable()` (IDLE/BLOCKED) — never mid-turn. This keeps
+  the primary path fully isolated from `WorkerPresence`/`StatusPoller` (which are worker-scoped),
+  and makes it unit-testable with a fake `AgentControl` + an injected clock (per the CB-306
+  `LongSupplier` clock + `Runnable` sleeper seam). Rejected (a) reuse-the-Injector: it would force
+  the primary terminal into the worker poller set and couple to worker-presence semantics — more
+  integration surface, harder to test, no real gain for a bounded reminder.
 - **Bounded reminder / backoff.** While `peek(target)` stays non-empty, re-inject on a
   backoff schedule up to a cap (N reminders or a max duration; config
   `primary.push_reminders` / `primary.push_backoff_ms`). After the cap, **stop reminding** —
@@ -102,16 +108,15 @@ in v1; the stop-on-empty loop is sufficient.
    and needs no new env var or argument (identity stays connection-derived, per the existing
    `BridgeMcp` invariant).
 
-2. **Readiness-gate mismatch.** The existing `Injector` gates delivery on
+2. **Readiness-gate mismatch → dedicated loop.** The existing `Injector` gates delivery on
    `ready.test(target)` = `WorkerPresence` (the *worker's* MCP connected). The primary is not
-   in `WorkerPresence`. So the primary push path uses a **different liveness signal**: the
-   primary is provably MCP-connected at the moment it calls us (that's how we learned its
-   terminal), and we still status-gate on its terminal sampling **injectable** via the poller.
-   Concretely, the primary push path either (a) uses a dedicated `Injector` instance whose
-   `ready` predicate is "primary terminal is known" (always true once registered), or
-   (b) a lighter direct `AgentControl.send` guarded by a `StatusPoller` injectable sample.
-   Decision: **(a)** — reuse the `Injector` queue/one-per-turn/backoff machinery with a
-   primary-appropriate `ready` predicate, rather than reimplement gating.
+   in `WorkerPresence`, so reusing `Injector` would mean forcing the primary terminal into the
+   worker `StatusPoller` set and swapping the `ready` predicate — extra integration surface with
+   worker-scoped machinery. Decision: **mechanism (b)** — a small dedicated scheduled loop that
+   calls `AgentControl.status(primaryTerminal).injectable()` then `AgentControl.send(...)`, with
+   an injected clock. Isolated from worker presence, trivially unit-testable, sufficient for a
+   bounded reminder. (Verified live: this primary resolves to `term_656c8cc03e1f0b1`, pane
+   `w2:pY` — the primary genuinely runs in a herdr pane on this host, so the path is exercisable.)
 
 ## Boundary note