(state): Drive the single spinner from the conversation head turn #625

Draft: D5.

Problem and evidence. Audit §2 R1/R9 describes a continuously spinning OSC title across eight completed head turns; this is audit-only runtime evidence. The mechanism remains: main.js:2507 directly sets _cliBusy from the title and :2536 also sets it from progress; main.js:1852 exposes the same flag to reconciliation (public/session-activity.js:206). public/session-state.js:4 puts busy above agentsBusy, reproduced by an executed combined-state render. transcript-turn.js:31 already recognizes closure with stop reason plus turn_duration and queue checks (:98), but an empty tail returns closed:false (:114), also reproduced. That false is not positive evidence of an open turn.

Expected behavior. Implement the maintainer's one-glyph contract. A live head with positive main-turn evidence displays an animated spinner/Working, including a silent synchronous tool call. A confirmed head at its prompt with known background work displays the same spinner family statically, with prompt/background wording. At the prompt without background work retain the waiting glyph. Attention retains priority in the one slot; liveness and archive semantics remain distinct. Background agents, shells, Monitor, or cron must not animate the head on their own. OSC spinner/progress and job working are coarse in-flight hints, not sufficient turn evidence; OSC idle remains a fast prompt hint reconciled against ordering and newer head evidence.

Use or extend the existing turn reader without weakening trigger/compaction guarantees. Distinguish open, closed, and unknown; filter task notifications, tool results, meta records, sidechains, and command-only output correctly. A new real prompt/assistant turn after a closure reopens it. For unavailable/malformed/truncated evidence, show the established neutral/unknown state rather than manufacturing Working or confirmed prompt status. Verified growth requests classification; it alone cannot prove which thread is in a turn. Remote data unavailable through the existing mirror remains a documented degraded case; no new SSH polling per row. Remove competing 3s/20s agent clocks from authoritative classification.

Acceptance criteria a reviewer can run. Add head-turn fixtures for open prompt → silent synchronous tool → valid closure with one background task → later genuine prompt. Assert animated → animated → static → animated, including a background shell/Monitor/cron with zero agents. Test empty/truncated tail, task notification at rest, queued/dequeued prompts, local commands, manual/automatic compaction, dead parent, stale OSC/progress, and attention during a turn. With #564, old member busy/new head at prompt stays static; reversing the head states animates. DOM/CSS tests assert one .session-icon, no second activity badge, no animation on static mode, and matching accessible text. Run node --test --test-concurrency=1 test/trigger-transcript-fallback.test.js test/session-state.test.js test/local-pty-adapter.test.js test/local-transcript-adapter.test.js test/remote-session-adapter.test.js test/dom-sidebar-icon-slot.test.js after adding focused cases.

Dependencies/order. Wave 3 after D4's background facts and D1. #564 head selection is required for merged rows; #616 owns scan placement. D6 owns title color and unseen state, D7 owns the number.

Out of scope. Replacing the terminal CLI spinner, a second glyph/badge, changing trigger authorization, merging Idle/Stale vocabulary, header/status-bar redesign, or calling every unclosed tail Working.