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-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"