feat(deacon): use await-signal with exponential backoff in patrol loop

The Deacon patrol formula now uses `gt mol step await-signal` with
exponential backoff instead of vague "sleep 60s" instructions.

How it works:
- Subscribes to `bd activity --follow` (beads activity feed)
- Returns IMMEDIATELY when any gt/bd command triggers activity
- On timeout, waits exponentially longer: 60s → 120s → 240s → max 10m
- Tracks idle:N label on hq-deacon bead across invocations

This connects the designed-but-unintegrated backoff mechanism to the
actual patrol loop. Idle towns let Deacon sleep; active work wakes it.

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

.beads/formulas/mol-deacon-patrol.formula.toml

@@ -23,7 +23,7 @@ Witnesses detect it and escalate to the Mayor.
 The Deacon's agent bead last_activity timestamp is updated during each patrol
 cycle. Witnesses check this timestamp to verify health."""
 formula = "mol-deacon-patrol"
-version = 6
+version = 8
 
 [[steps]]
 id = "inbox-check"
@@ -707,15 +707,39 @@ Burn and let daemon respawn, or exit if context high.
 Decision point at end of patrol cycle:
 
 If context is LOW:
-- **Sleep 60 seconds minimum** before next patrol cycle
+Use await-signal with exponential backoff to wait for activity:
-- If town is idle (no in_progress work), sleep longer (2-5 minutes)
-- Return to inbox-check step
 
-**Why longer sleep?**
+```bash
-- Idle agents should not be disturbed
+gt mol step await-signal --agent-bead hq-deacon \
-- Health checks every few seconds flood inboxes and waste context
+  --backoff-base 60s --backoff-mult 2 --backoff-max 10m
-- The daemon (10-minute heartbeat) is the safety net for dead sessions
+```
-- Active work triggers feed events, which wake agents naturally
+
+This command:
+1. Subscribes to `bd activity --follow` (beads activity feed)
+2. Returns IMMEDIATELY when any beads activity occurs
+3. If no activity, times out with exponential backoff:
+   - First timeout: 60s
+   - Second timeout: 120s
+   - Third timeout: 240s
+   - ...capped at 10 minutes max
+4. Tracks `idle:N` label on hq-deacon bead for backoff state
+
+**On signal received** (activity detected):
+Reset the idle counter and start next patrol cycle:
+```bash
+gt agent state hq-deacon --set idle=0
+```
+Then return to inbox-check step.
+
+**On timeout** (no activity):
+The idle counter was auto-incremented. Continue to next patrol cycle
+(the longer backoff will apply next time). Return to inbox-check step.
+
+**Why this approach?**
+- Any `gt` or `bd` command triggers beads activity, waking the Deacon
+- Idle towns let the Deacon sleep longer (up to 10 min between patrols)
+- Active work wakes the Deacon immediately via the feed
+- No polling or fixed sleep intervals
 
 If context is HIGH:
 - Write state to persistent storage

internal/formula/formulas/mol-deacon-patrol.formula.toml

@@ -23,7 +23,7 @@ Witnesses detect it and escalate to the Mayor.
 The Deacon's agent bead last_activity timestamp is updated during each patrol
 cycle. Witnesses check this timestamp to verify health."""
 formula = "mol-deacon-patrol"
-version = 7
+version = 8
 
 [[steps]]
 id = "inbox-check"
@@ -749,15 +749,39 @@ Burn and let daemon respawn, or exit if context high.
 Decision point at end of patrol cycle:
 
 If context is LOW:
-- **Sleep 60 seconds minimum** before next patrol cycle
+Use await-signal with exponential backoff to wait for activity:
-- If town is idle (no in_progress work), sleep longer (2-5 minutes)
-- Return to inbox-check step
 
-**Why longer sleep?**
+```bash
-- Idle agents should not be disturbed
+gt mol step await-signal --agent-bead hq-deacon \
-- Health checks every few seconds flood inboxes and waste context
+  --backoff-base 60s --backoff-mult 2 --backoff-max 10m
-- The daemon (10-minute heartbeat) is the safety net for dead sessions
+```
-- Active work triggers feed events, which wake agents naturally
+
+This command:
+1. Subscribes to `bd activity --follow` (beads activity feed)
+2. Returns IMMEDIATELY when any beads activity occurs
+3. If no activity, times out with exponential backoff:
+   - First timeout: 60s
+   - Second timeout: 120s
+   - Third timeout: 240s
+   - ...capped at 10 minutes max
+4. Tracks `idle:N` label on hq-deacon bead for backoff state
+
+**On signal received** (activity detected):
+Reset the idle counter and start next patrol cycle:
+```bash
+gt agent state hq-deacon --set idle=0
+```
+Then return to inbox-check step.
+
+**On timeout** (no activity):
+The idle counter was auto-incremented. Continue to next patrol cycle
+(the longer backoff will apply next time). Return to inbox-check step.
+
+**Why this approach?**
+- Any `gt` or `bd` command triggers beads activity, waking the Deacon
+- Idle towns let the Deacon sleep longer (up to 10 min between patrols)
+- Active work wakes the Deacon immediately via the feed
+- No polling or fixed sleep intervals
 
 If context is HIGH:
 - Write state to persistent storage