chore(gastown): scorched-earth SQLite removal from codebase

Remove all bd sync references and SQLite-specific code from gastown:

**Formulas (agent priming):**
- mol-polecat-work: Remove bd sync step from prepare-for-review
- mol-sync-workspace: Replace sync-beads step with verify-beads (Dolt check)
- mol-polecat-conflict-resolve: Remove bd sync from close-beads
- mol-polecat-code-review: Remove bd sync from summarize-review and complete-and-exit
- mol-polecat-review-pr: Remove bd sync from complete-and-exit
- mol-convoy-cleanup: Remove bd sync from archive-convoy
- mol-digest-generate: Remove bd sync from send-digest
- mol-town-shutdown: Replace sync-state step with verify-state
- beads-release: Replace restart-daemons with verify-install (no daemons with Dolt)

**Templates (role priming):**
- mayor.md.tmpl: Update session end checklist to remove bd sync steps
- crew.md.tmpl: Remove bd sync references from workflow and checklist
- polecat.md.tmpl: Update self-cleaning model and session close docs
- spawn.md.tmpl: Remove bd sync from completion steps
- nudge.md.tmpl: Remove bd sync from completion steps

**Go code:**
- session_manager.go: Remove syncBeads function and call
- rig_dock.go: Remove bd sync calls from dock/undock
- crew/manager.go: Remove runBdSync, update Pristine function
- crew_maintenance.go: Remove bd sync status output
- crew.go: Update pristine command help text
- polecat.go: Make sync command a no-op with deprecation message
- daemon/lifecycle.go: Remove bd sync from startup sequence
- doctor/beads_check.go: Update fix hints and Fix to use bd import not bd sync
- doctor/rig_check.go: Remove sync status check, simplify BeadsConfigValidCheck
- beads/beads.go: Update primeContent to remove bd sync references

With Dolt backend, beads changes are persisted immediately to the sql-server.
There is no separate sync step needed.

Part of epic: hq-e4eefc (SQLite removal)

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 status
+bd doctor            # Verify beads database health
 ```
 
 **3. Document starting state:**
@@ -240,29 +240,31 @@ risk merging incorrectly. A resubmitted branch is better than a broken main.
 **Exit criteria:** Local branch matches or is cleanly ahead of origin/main."""
 
 [[steps]]
-id = "sync-beads"
-title = "Verify beads state"
+id = "verify-beads"
+title = "Verify beads database health"
 needs = ["sync-git"]
 description = """
-Verify beads database is healthy.
+Verify the Dolt-backed beads database is healthy.
 
-With Dolt backend, beads changes are automatically persisted - no manual sync needed.
+**Note:** Gas Town uses Dolt as the beads backend. There is no `bd sync` command -
+changes are written directly to the Dolt sql-server and synced via Dolt's
+git-like PUSH/PULL operations.
 
 ```bash
-bd status  # Check database health
-bd list --status=in_progress  # Verify active work is visible
+bd doctor            # Check database health
+bd list --limit 5    # Verify we can read beads
 ```
 
-**If issues:**
-- Check dolt sql-server is running
-- Verify database connectivity
+**If errors:**
+- Connection errors: Check if `dolt sql-server` is running
+- Data errors: Escalate to mayor
 
-**Exit criteria:** Beads database healthy and accessible."""
+**Exit criteria:** Beads database accessible and healthy."""
 
 [[steps]]
 id = "run-doctor"
 title = "Run beads health check"
-needs = ["sync-beads"]
+needs = ["verify-beads"]
 description = """
 Check for beads system issues.