feat(api): settle a cash_owed / partial_pending settlement off-machine

POST /api/v1/dca/settlements/{id}/settle-cash-owed: the operator paid
the customer by hand. Appends the provenance note, distributes at the
full amount (the customer is whole), and publishes a settle_transaction
op so the machine flips its own dispense_error/partial row to
remediated. Until now the only way to close an owed row was to dispense
more cash from the machine that had just jammed.

distribution.py

@@ -359,6 +359,31 @@ async def apply_partial_dispense_and_redistribute(
     return after if after is not None else updated
 
 
+async def settle_cash_owed_settlement(
+    settlement: DcaSettlement, note: str, author_user_id: str
+) -> DcaSettlement:
+    """ADR-005 §6 — operator paid the customer by hand. The sale is whole, so
+    the settlement distributes at the full amount it landed with. The note is
+    the provenance; `remediated` on the machine side comes from the
+    settle_transaction op the endpoint publishes."""
+    from .crud import append_settlement_note
+
+    if settlement.status not in ("cash_owed", "partial_pending"):
+        raise ValueError(f"settlement {settlement.id} is {settlement.status!r}")
+    await append_settlement_note(
+        settlement.id, f"Settled off-machine (full amount): {note}", author_user_id
+    )
+    await mark_settlement_status(settlement.id, "pending", None)
+    logger.info(
+        f"distribution: settlement {settlement.id} settled off-machine by "
+        f"{author_user_id[:8]}… — distributing in full"
+    )
+    await process_settlement(settlement.id)
+    after = await get_settlement(settlement.id)
+    assert after is not None
+    return after
+
+
 async def process_settlement(settlement_id: str) -> None:
     """Process a pending settlement end-to-end.