feat(sync): record last_synced_version in state.json — whether sync ran for the current version is unverifiable #1273

Pre-flight Checks

• I have searched existing issues and this is not a duplicate
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Problem Description

There is no reliable way to answer the question "did gentle-ai sync run after the last binary upgrade?" — the instrument does not record its own state.

What a user (or an AI agent following orchestrator instructions that say "confirm sync ran after the upgrade, otherwise stop") has available today:

• ~/.gentle-ai/state.json — stores installed_agents, claude_phase_assignments, last_update_check… but not the version of the last sync.
• mtimes of synced assets (~/.claude/skills/...) — a false-negative trap. Sync is idempotent: it only rewrites files whose content differs. An indicator that only moves when there is a change cannot distinguish "sync never ran" from "sync ran and there was nothing to do". We hit exactly this: skills with pre-upgrade mtimes led to the wrong conclusion that sync was pending, when it had in fact run twice.
• gentle-ai sync --dry-run (2.1.4) — prints only step counts (Prepare steps: 1 / Apply steps: 9) without listing pending files or stating "0 changes / up to date", so it cannot settle the question either.
• ~/.gentle-ai/backups/<timestamp>/ — the only unconditional footprint (snapshot is created before writing), but correlating backup timestamps against the binary's install time is forensics, not a check.

This is the same family as #1261: the tool does not register its own state, so verifying it requires reconstructing evidence from side effects. It is also the verifiability counterpart of #227 (auto-sync after self-update): even if sync becomes automatic, orchestrators still need a one-line way to prove configs match the running binary.

Proposed Solution

1. Write last_synced_version (and ideally last_synced_at) into ~/.gentle-ai/state.json at the end of every successful sync. The check then becomes one line: compare against gentle-ai --version.
2. Make sync --dry-run decisive: list the files it would touch, or print an explicit "up to date — 0 changes" when there is nothing to do, instead of only step counts.

Affected Area

Scripts (setup, installation)

Alternatives Considered

• Checking mtimes of synced assets — false negatives due to idempotency (described above).
• Correlating ~/.gentle-ai/backups/ snapshot timestamps with the binary install time — works, but it is forensic reconstruction, not a check, and requires knowing that sync always snapshots.
• feat(ux): auto-sync configs after self-update — zero-maintenance experience #227's auto-sync would reduce how often the question matters, but does not make the state observable.

Additional Context

Observed on gentle-ai 2.1.4, macOS, Homebrew install, Claude Code agent.
Related: #227 (auto-sync after self-update), #1261 (instrument not recording its own state).