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/templates/messages/nudge.md.tmpl

@@ -21,7 +21,6 @@ Please either:
 2. **If blocked**: File a blocking issue with `bd create --title="Blocked: <reason>" --type=task`
 3. **If done**: Make sure to run:
    - `bd close {{ .Issue }}`
-   - `bd sync`
    - `gt done`
 
 ## Important

internal/templates/messages/spawn.md.tmpl

@@ -21,7 +21,6 @@ You have been assigned to work on the following issue:
 3. Commit changes regularly with clear messages
 4. When complete, run:
    - `bd close {{ .Issue }}`
-   - `bd sync`
    - `gt done`
 
 ## Need Help?

internal/templates/roles/crew.md.tmpl

@@ -11,7 +11,6 @@ to commit." This breaks the Gas Town model. The system is designed for autonomou
 
 **When implementation is complete:**
 - Push your commits: `git push`
-- Sync beads: `bd sync`
 - Either continue with next task OR cycle: `gt handoff`
 
 **Do NOT:**
@@ -132,7 +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/`
-- Run `bd sync` to push/pull beads changes via the `beads-sync` branch
+- Beads changes are automatically persisted with Dolt
 - **GitHub URLs**: Use `git remote -v` to verify repo URLs - never assume orgs like `anthropics/`
 
 ## Prefix-Based Routing
@@ -321,7 +320,6 @@ ONE exception where branches are created. But the rule still applies:
 - `bd update <id> --status=in_progress` - Claim an issue
 - `bd show <id>` - View issue details
 - `bd close <id>` - Mark issue complete
-- `bd sync` - Sync beads changes
 
 ### Communication
 - `gt mail send <addr> -s "Subject" -m "Message"` - Send mail (async, queued)
@@ -420,7 +418,6 @@ Before ending your session:
 ```
 [ ] git status              (check for uncommitted changes)
 [ ] git push                (push any commits)
-[ ] bd sync                 (sync beads if configured)
 [ ] Check inbox             (any messages needing response?)
 [ ] gt handoff              (cycle to fresh session)
     # Or with context: gt handoff -s "Brief" -m "Details"

internal/templates/roles/mayor.md.tmpl

@@ -295,12 +295,12 @@ cross-session continuity when work doesn't fit neatly into a bead.
 ```
 [ ] git status              (check what changed)
 [ ] git add <files>         (stage code changes)
-[ ] bd sync                 (commit beads changes)
 [ ] git commit -m "..."     (commit code)
-[ ] bd sync                 (commit any new beads changes)
 [ ] git push                (push to remote)
 [ ] HANDOFF (if incomplete work):
     gt mail send mayor/ -s "🤝 HANDOFF: <brief>" -m "<context>"
 ```
 
+Note: Beads changes are automatically persisted with Dolt.
+
 Town root: {{ .TownRoot }}

internal/templates/roles/polecat.md.tmpl

@@ -176,7 +176,7 @@ 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)
-- Run `bd sync` to push/pull beads changes via the `beads-sync` branch
+- Beads changes are automatically persisted with Dolt
 - **GitHub URLs**: Use `git remote -v` to verify repo URLs - never assume orgs like `anthropics/`
 
 ## Prefix-Based Routing