A proof is a run with a state, and a proof that cannot start or is cut off says why on the pull #7706

Split from #7673, which this serves (P0 Trust).

Today proving in src/checks/proof.ts is an in-memory map: a proof writes no run row, so the runs list never shows one. startProof returns false without saying why (no slot spare, head does not merge, one already running), and a proof cut off at GATE_DEADLINE_MS (interrupted) keeps nothing, so it can repeat with no trace. Those are the reasons #7673 took hours to diagnose.

Done when:

• [ ] A proof writes a run (kind proof) with a state (running, green, red, cut off) that the runs list, the pull and MCP runs show.
• [ ] A proof that cannot start says why on the pull, once per head: no check slot spare, or the head does not merge onto the base tip.
• [ ] A proof cut off at the deadline or by a restart says so on the pull, and the pull API's next names the wait instead of queue, approved and green.
• [ ] Tests: a proof waits while every check slot is taken and starts when one frees; an interrupted proof leaves a cut-off run and a comment, and is proved again on the next sweep.

Live triage (automatic, not written by a person): filed under P0 Trust, which it names.