feat(patrol): add backoff test formula and fix await-signal

Add mol-backoff-test formula for integration testing exponential backoff
with short intervals (2s base, 10s max) to observe multiple cycles quickly.

Fix await-signal to use --since 1s when subscribing to activity feed.
Without this, historical events would immediately wake the signal,
preventing proper timeout and backoff behavior.

Co-Authored-By: Claude Opus 4.5 <noreply@anthropic.com>

internal/cmd/molecule_await_signal.go

@@ -237,8 +237,10 @@ func calculateEffectiveTimeout(idleCycles int) (time.Duration, error) {
 // waitForActivitySignal starts bd activity --follow and waits for any output.
 // Returns immediately when a line is received, or when context is canceled.
 func waitForActivitySignal(ctx context.Context, workDir string) (*AwaitSignalResult, error) {
-	// Start bd activity --follow
-	cmd := exec.CommandContext(ctx, "bd", "activity", "--follow")
+	// Start bd activity --follow --since 1s
+	// The --since flag ensures we only wait for NEW events, not historical ones.
+	// Without this, the feed would immediately return old activity and never timeout.
+	cmd := exec.CommandContext(ctx, "bd", "activity", "--follow", "--since", "1s")
 	cmd.Dir = workDir
 
 	stdout, err := cmd.StdoutPipe()

internal/formula/formulas/mol-backoff-test.formula.toml

@@ -0,0 +1,95 @@
+description = """
+Test molecule for verifying exponential backoff behavior.
+
+Uses SHORT intervals (2s base, 10s max) so you can observe multiple
+backoff cycles in under a minute.
+
+Expected backoff progression:
+- Cycle 0: 2s
+- Cycle 1: 4s
+- Cycle 2: 8s
+- Cycle 3+: 10s (capped)
+
+Full backoff cycle observable in ~24 seconds.
+
+## Running This Test
+
+1. Create a test agent bead (if not exists):
+   ```bash
+   bd create --type=agent --title="Backoff Test Agent" --id=hq-backoff-test
+   ```
+
+2. Run the deacon with this formula, or execute steps manually:
+   ```bash
+   gt mol step await-signal --agent-bead hq-backoff-test \
+     --backoff-base 2s --backoff-mult 2 --backoff-max 10s
+   ```
+
+3. Watch the idle counter increment on each timeout:
+   ```bash
+   bd show hq-backoff-test --json | jq '.[] | .labels'
+   ```
+
+4. Trigger activity to test signal wake:
+   ```bash
+   bd create --title="Wake signal" --silent  # Any bd command triggers activity
+   ```
+
+5. Reset idle counter after signal (caller responsibility):
+   ```bash
+   bd update hq-backoff-test --set-labels=idle:0
+   ```
+"""
+formula = "mol-backoff-test"
+version = 1
+
+[[steps]]
+id = "heartbeat"
+title = "Touch heartbeat"
+description = """
+Touch heartbeat to show we're alive.
+
+```bash
+gt deacon heartbeat "backoff test cycle"
+```
+
+This updates the deacon's heartbeat file to signal liveness.
+"""
+
+[[steps]]
+id = "await-signal"
+title = "Wait with exponential backoff (SHORT intervals)"
+needs = ["heartbeat"]
+description = """
+Wait for activity with exponential backoff using TEST intervals.
+
+**IMPORTANT**: This uses SHORT intervals for testing:
+- Base: 2s (production: 60s)
+- Multiplier: 2
+- Max: 10s (production: 10m)
+
+```bash
+gt mol step await-signal --agent-bead hq-backoff-test \
+  --backoff-base 2s --backoff-mult 2 --backoff-max 10s
+```
+
+**Expected behavior:**
+
+| Idle Cycles | Timeout |
+|-------------|---------|
+| 0           | 2s      |
+| 1           | 4s      |
+| 2           | 8s      |
+| 3+          | 10s     |
+
+**On timeout**: The command auto-increments the `idle:N` label on the agent bead.
+Continue to the next patrol cycle (loop back to heartbeat).
+
+**On signal**: Activity was detected. Reset the idle counter:
+```bash
+bd update hq-backoff-test --set-labels=idle:0
+```
+Then loop back to heartbeat for the next cycle.
+
+**To exit**: When context is high or testing is complete, exit cleanly.
+"""