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/templates/roles/crew.md.tmpl

@@ -131,7 +131,7 @@ Town ({{ .TownRoot }})
 **Key points:**
 - Mail ALWAYS uses town beads - `gt mail` routes there automatically
 - Project issues use your clone's beads - `bd` commands use local `.beads/`
-- Beads changes are automatically persisted with Dolt
+- Beads changes are persisted immediately to Dolt - no sync step needed
 - **GitHub URLs**: Use `git remote -v` to verify repo URLs - never assume orgs like `anthropics/`
 
 ## Prefix-Based Routing
@@ -379,7 +379,6 @@ Raw `tmux send-keys` is unreliable. Always use `gt nudge` for agent-to-agent com
 - Managing your own progress
 - Asking for help when stuck
 - Keeping your git state clean
-- Syncing beads before long breaks
 
 ## Context Cycling (Handoff)
 

internal/templates/roles/mayor.md.tmpl

@@ -301,6 +301,6 @@ cross-session continuity when work doesn't fit neatly into a bead.
     gt mail send mayor/ -s "🤝 HANDOFF: <brief>" -m "<context>"
 ```
 
-Note: Beads changes are automatically persisted with Dolt.
+Note: Beads changes are persisted immediately to Dolt - no sync step needed.
 
 Town root: {{ .TownRoot }}

internal/templates/roles/polecat.md.tmpl

@@ -23,10 +23,9 @@ just `gt done`.
 ### The Self-Cleaning Model
 
 Polecats are **self-cleaning**. When you run `gt done`:
-1. Syncs beads
-2. Nukes your sandbox
-3. Exits your session
-4. **You cease to exist**
+1. Nukes your sandbox
+2. Exits your session
+3. **You cease to exist**
 
 There is no "idle" state. There is no "waiting for more work". Done means GONE.
 
@@ -176,7 +175,7 @@ Town ({{ .TownRoot }})
 **Key points:**
 - You're in a project git worktree - your `.beads/` is tracked in the project repo
 - The rig-level `{{ .RigName }}/.beads/` is **gitignored** (local runtime state)
-- Beads changes are automatically persisted with Dolt
+- Beads changes are persisted immediately to Dolt - no sync step needed
 - **GitHub URLs**: Use `git remote -v` to verify repo URLs - never assume orgs like `anthropics/`
 
 ## Prefix-Based Routing
@@ -364,8 +363,8 @@ git log --oneline -3    # Verify your commits are present
 
 Then submit: **`gt done`** ← MANDATORY FINAL STEP
 
-This single command verifies git is clean, syncs beads, and submits your branch
-to the merge queue. The Witness handles the rest.
+This single command verifies git is clean and submits your branch to the merge
+queue. The Witness handles the rest.
 
 **Note:** Do NOT manually close the root issue with `bd close`. The Refinery
 closes it after successful merge. This enables conflict-resolution retries.