docs(priming): remove bd sync references for Dolt backend

Gas Town uses Dolt exclusively. Remove all bd sync and bd daemon
references from agent priming, templates, formulas, and docs.

With Dolt backend:
- Beads changes are automatically persisted
- No manual sync needed (no bd sync)
- No daemon needed (no bd daemon)

Updated files:
- polecat-CLAUDE.md template
- Role templates (crew, mayor, polecat)
- Message templates (spawn, nudge)
- Formulas (polecat-work, sync-workspace, shutdown, etc.)
- Reference docs

Part of hq-4f2f0c: Remove bd sync/daemon from agent priming

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

internal/formula/formulas/mol-convoy-cleanup.formula.toml

@@ -135,12 +135,7 @@ bd show {{convoy}}
 # Status should reflect archived state
 ```
 
-**4. Sync to persist:**
-```bash
-bd sync
-```
-
-**Exit criteria:** Convoy archived, changes synced."""
+**Exit criteria:** Convoy archived."""
 
 [[steps]]
 id = "notify-overseer"

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

@@ -419,114 +419,10 @@ gt mail send mayor/ -s "Health: <rig> <component> unresponsive" \\
 
 Reset unresponsive_cycles to 0 when component responds normally."""
 
-[[steps]]
-id = "hung-session-detection"
-title = "Detect and recover hung Gas Town sessions (SURGICAL)"
-needs = ["health-scan"]
-description = """
-Detect and surgically recover hung Gas Town sessions where the Claude API call is stuck.
-
-A hung session appears "running" (tmux session exists, Claude process exists) but
-the API call has been stuck indefinitely. This breaks patrol chains - if witness
-hangs, refinery never gets nudged about new MRs.
-
-**Why existing checks miss this:**
-- zombie-scan only catches processes not in tmux sessions
-- gt status shows "running" if tmux session exists
-- Nudges queue but never get processed (Claude can't respond)
-
-## SURGICAL TARGETING
-
-**ONLY these session patterns are valid targets:**
-- `gt-<rig>-witness` (e.g., gt-kalshi-witness, gt-horizon-witness)
-- `gt-<rig>-refinery` (e.g., gt-kalshi-refinery)
-- `hq-deacon`
-
-**NEVER touch sessions that don't match these patterns exactly.**
-
-## DETECTION (All checks must pass)
-
-For each Gas Town session, capture output and verify ALL of these:
-
-```bash
-# Step 1: Get session output
-output=$(tmux capture-pane -t <session-name> -p 2>/dev/null | tail -10)
-```
-
-**Check 1: Session is in waiting state**
-Must see one of: `Clauding`, `Deciphering`, `Marinating`, `Finagling`, `thinking`
-```bash
-echo "$output" | grep -qiE 'Clauding|Deciphering|Marinating|Finagling|thinking'
-```
-
-**Check 2: Duration exceeds threshold (30+ minutes)**
-Parse duration from output like "21h 35m 20s" or "45m 30s":
-```bash
-# Extract hours and minutes
-hours=$(echo "$output" | grep -oE '[0-9]+h' | head -1 | tr -d 'h')
-minutes=$(echo "$output" | grep -oE '[0-9]+m' | head -1 | tr -d 'm')
-total_minutes=$((${hours:-0} * 60 + ${minutes:-0}))
-# Threshold: 30 minutes minimum
-[ "$total_minutes" -ge 30 ]
-```
-
-**Check 3: Zero tokens received (definite hang) OR very long duration (>2 hours)**
-```bash
-# Definite hang: zero tokens received
-echo "$output" | grep -qE '↓ 0 tokens'
-# OR extremely long duration (>2 hours = 120 minutes)
-[ "$total_minutes" -ge 120 ]
-```
-
-**Check 4: NOT showing active tool execution**
-Active sessions show tool markers (⏺). If present, session is actually working:
-```bash
-# If tool markers present in recent output, DO NOT kill
-echo "$output" | grep -qE '⏺|Read|Write|Bash|Edit' && continue
-```
-
-## RECOVERY (Only after ALL checks pass)
-
-**Log the action first:**
-```bash
-echo "[$(date)] RECOVERING HUNG: <session-name> (${hours}h ${minutes}m, waiting state)" >> $GT_ROOT/logs/hung-sessions.log
-```
-
-**Kill and restart based on session type:**
-
-For witness:
-```bash
-tmux kill-session -t gt-<rig>-witness 2>/dev/null
-gt witness start <rig>
-```
-
-For refinery:
-```bash
-tmux kill-session -t gt-<rig>-refinery 2>/dev/null
-gt refinery restart <rig>
-```
-
-For deacon (self-recovery - use with caution):
-```bash
-# Deacon detecting itself is hung is a paradox
-# Only kill if another deacon instance exists or human confirmed
-gt mail send mayor/ -s "DEACON SELF-HUNG DETECTED" -m "Deacon appears hung. Human intervention required."
-```
-
-## VERIFICATION
-
-After restart, verify new session is healthy:
-```bash
-sleep 5
-tmux has-session -t <session-name> && echo "Session restarted successfully"
-```
-
-**Exit criteria:** All hung Gas Town sessions detected and recovered (or escalated if recovery failed)."""
-
 [[steps]]
 id = "zombie-scan"
 title = "Detect zombie polecats (NO KILL AUTHORITY)"
-needs = ["hung-session-detection"]
+needs = ["health-scan"]
 description = """
 Defense-in-depth DETECTION of zombie polecats that Witness should have cleaned.
 

internal/formula/formulas/mol-digest-generate.formula.toml

@@ -190,11 +190,6 @@ bd create --title="Digest: {{date}}" --type=digest \
   --label=digest,{{period}}
 ```
 
-**3. Sync:**
-```bash
-bd sync
-```
-
 **Exit criteria:** Digest sent to Mayor and archived as bead."""
 
 [[steps]]

internal/formula/formulas/mol-polecat-code-review.formula.toml

@@ -270,11 +270,6 @@ Overall assessment:
 <brief summary - healthy, needs attention, significant issues, etc.>"
 ```
 
-**2. Sync beads:**
-```bash
-bd sync
-```
-
 **Exit criteria:** Tracking issue updated with summary."""
 
 [[steps]]

internal/formula/formulas/mol-polecat-conflict-resolve.formula.toml

@@ -306,11 +306,6 @@ bd close <source-issue> --reason="merged via conflict resolution"
 The Refinery normally closes issues after merge, but since we pushed
 directly to main, we handle it here.
 
-**3. Sync beads:**
-```bash
-bd sync
-```
-
 **Exit criteria:** Original MR and source issue are closed."""
 
 [[steps]]

internal/formula/formulas/mol-polecat-review-pr.formula.toml

@@ -254,13 +254,11 @@ Signal completion and clean up. You cease to exist after this step.
 
 **Self-Cleaning Model:**
 Once you run `gt done`, you're gone. The command:
-1. Syncs beads
-2. Nukes your sandbox
-3. Exits your session immediately
+1. Nukes your sandbox
+2. Exits your session immediately
 
 **Run gt done:**
 ```bash
-bd sync
 gt done
 ```
 
@@ -271,7 +269,7 @@ gt done
 
 You are NOT involved in any of that. You're gone. Done means gone.
 
-**Exit criteria:** Beads synced, sandbox nuked, session exited."""
+**Exit criteria:** Sandbox nuked, session exited."""
 
 [vars]
 [vars.pr_url]

internal/formula/formulas/mol-polecat-work.formula.toml

@@ -401,12 +401,7 @@ bd show {{issue}}
 bd update {{issue}} --notes "Implemented: <brief summary of what was done>"
 ```
 
-**3. Sync beads:**
-```bash
-bd sync
-```
-
-**Exit criteria:** Issue updated with completion notes, beads synced."""
+**Exit criteria:** Issue updated with completion notes."""
 
 [[steps]]
 id = "submit-and-exit"

internal/formula/formulas/mol-sync-workspace.formula.toml

@@ -69,7 +69,7 @@ git branch --show-current
 
 **2. Check beads state:**
 ```bash
-bd sync --status
+bd status
 ```
 
 **3. Document starting state:**
@@ -77,7 +77,7 @@ bd sync --status
 - Uncommitted changes (staged/unstaged)
 - Untracked files (list them)
 - Stash entries
-- Beads sync status
+- Beads database health
 
 This information guides decisions in subsequent steps.
 
@@ -241,36 +241,23 @@ risk merging incorrectly. A resubmitted branch is better than a broken main.
 
 [[steps]]
 id = "sync-beads"
-title = "Sync beads state"
+title = "Verify beads state"
 needs = ["sync-git"]
 description = """
-Sync the beads database with remote.
+Verify beads database is healthy.
+
+With Dolt backend, beads changes are automatically persisted - no manual sync needed.
 
 ```bash
-bd sync
+bd status  # Check database health
+bd list --status=in_progress  # Verify active work is visible
 ```
 
-**If conflicts:**
-Beads uses JSONL append-only format, so true conflicts are rare.
-If they occur:
-```bash
-bd sync --status  # Diagnose
-```
+**If issues:**
+- Check dolt sql-server is running
+- Verify database connectivity
 
-Likely causes:
-- Two agents edited same bead simultaneously
-- Corrupted beads-sync branch
-
-Resolution:
-```bash
-# Try pulling fresh
-git fetch origin beads-sync
-bd sync
-```
-
-If still failing, escalate to mayor.
-
-**Exit criteria:** Beads synced successfully."""
+**Exit criteria:** Beads database healthy and accessible."""
 
 [[steps]]
 id = "run-doctor"

internal/formula/formulas/mol-town-shutdown.formula.toml

@@ -137,15 +137,17 @@ Old logs are moved to `$GT_ROOT/logs/archive/` with timestamps.
 
 [[steps]]
 id = "sync-state"
-title = "Sync beads and push"
+title = "Verify beads state"
 needs = ["rotate-logs"]
 description = """
-Ensure all beads state is persisted.
+Verify beads state is healthy.
 
 ```bash
-bd sync
+bd status
 ```
 
+With Dolt backend, beads changes are automatically persisted.
+
 Note: We do NOT force-commit polecat work here. Their sandboxes
 are preserved with whatever state they had. They'll commit their
 own work when they resume.