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/beads-release.formula.toml

@@ -305,28 +305,25 @@ Should show {{version}}.
 """
 
 [[steps]]
-id = "restart-daemons"
-title = "Restart daemons"
+id = "verify-install"
+title = "Verify installation"
 needs = ["local-install"]
 description = """
-Restart bd daemons to pick up new version.
+Verify the new bd version is working.
 
 ```bash
-bd daemons killall
+bd --version         # Should show {{version}}
+bd doctor            # Verify database connectivity
 ```
 
-Daemons will auto-restart with new version on next bd command.
-
-Verify:
-```bash
-bd daemons list
-```
+**Note:** Gas Town uses Dolt backend - there are no bd daemons to restart.
+The Dolt sql-server runs independently.
 """
 
 [[steps]]
 id = "release-complete"
 title = "Release complete"
-needs = ["restart-daemons"]
+needs = ["verify-install"]
 description = """
 Release v{{version}} is complete!
 
@@ -335,8 +332,7 @@ Summary:
 - Git tag pushed
 - CI artifacts built
 - npm and PyPI packages published
-- Local installation updated
-- Daemons restarted
+- Local installation updated and verified
 
 Optional next steps:
 - Announce on social media

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

@@ -281,9 +281,8 @@ 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 (final sync)
-2. Nukes your sandbox
-3. Exits your session immediately
+1. Nukes your sandbox
+2. Exits your session immediately
 
 **Run gt done:**
 ```bash
@@ -297,7 +296,7 @@ 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.scope]

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.
 

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

@@ -136,17 +136,19 @@ Old logs are moved to `$GT_ROOT/logs/archive/` with timestamps.
 """
 
 [[steps]]
-id = "sync-state"
+id = "verify-state"
 title = "Verify beads state"
 needs = ["rotate-logs"]
 description = """
-Verify beads state is healthy.
+Verify beads database is accessible.
 
 ```bash
-bd status
+bd doctor
+bd list --limit 5    # Quick connectivity check
 ```
 
-With Dolt backend, beads changes are automatically persisted.
+Note: With Dolt backend, changes are persisted immediately to the sql-server.
+There is no separate sync step needed.
 
 Note: We do NOT force-commit polecat work here. Their sandboxes
 are preserved with whatever state they had. They'll commit their
@@ -156,7 +158,7 @@ own work when they resume.
 [[steps]]
 id = "handoff-mayor"
 title = "Send Mayor handoff"
-needs = ["sync-state"]
+needs = ["verify-state"]
 description = """
 Record shutdown context for the fresh Mayor session.