feat(shared): give a payment a tab to settle

A payment has only ever recorded from, to, amount and date. That is why
the per-trip owed figure did not exist — getTripAnalytics said so where
the number should have been: "split_payments carries no trip attribution,
so a payment cannot be assigned to a trip. Settlement is a property of the
whole relationship." Every trip therefore read 100% unsettled, including
trips paid in full.

It is also why the Shared page silently drops payments under a tag filter.
With one global pool there was nothing honest to subtract, so it showed
gross splits under the same label. A tag is a view; a scope is a ledger.

The scope is a trip, not a new settlement_contexts table. trips already
has owner_id, dates and archived, and transaction_overrides.trip_id
already decides membership. A second grouping beside it would be two
unsynchronised scopes over the same rows, with no invariant saying which
governs. NULL means the ongoing household tab, which never closes.

settled answers a different question and the two must not be collapsed:
trip_id is which tab, settled is whether the obligation is still live.
Critically, a live obligation is NOT settled by flipping the flag — it is
settled by recording the payment, and the balance nets to zero on its own.
Doing both would subtract the settlement twice. So settled is written only
by the historical import, for repayments made on a platform we no longer
run, and there is deliberately no "mark settled" action.

Both owed figures now exclude settled splits and the trip figure nets its
own payments. Spend analytics (myShare/mySplitOf) deliberately still count
settled rows: my half of a 2025 grocery shop is my spend whether or not the
other half was ever repaid, and filtering them would re-inflate exactly the
figures importing settled history exists to correct.

Also drops /api/participants/[id]/balance. It had no consumers, no owner
scoping, no debit/credit signs and no EXCLUDE_RECONCILED_SOURCE — a fourth
balance implementation that disagreed with the others and would have
imported three bugs if anything had aligned to it.

getTripAnalytics had no test at all. It has five now, including the one
that matters: a household payment must not make a trip look paid. Verified
by mutation — neutering the settled filter fails three, and dropping the
trip filter on payments fails that one.

src/lib/analytics-sql.ts

@@ -171,3 +171,42 @@ export const SPEND_SIGNED = `CASE
   WHEN t.transaction_type IN ('refund', 'credit') THEN -(${SPEND_BASE})
   ELSE (${SPEND_BASE})
 END`;
+
+/**
+ * A split that still counts towards what someone owes.
+ *
+ * This is the line between the two questions the same table answers, and
+ * conflating them is what made three different "owed" figures disagree:
+ *
+ *  - **Owed** — what is still outstanding between two people. Must apply this.
+ *  - **Spend** — what a purchase cost me. Must NOT apply this.
+ *
+ * A settled split is still a real expense: my half of a 2025 grocery shop is my
+ * spend whether or not the other half was ever repaid. Filtering settled rows
+ * out of `myShare`/`mySplitOf` would re-inflate exactly the figures that
+ * importing settled history exists to correct.
+ *
+ * `settled` marks obligations discharged OUTSIDE this app — imported
+ * SplitMyExpenses history, whose repayments happened on a platform we no longer
+ * run and which therefore has no `split_payments` row here.
+ *
+ * A live obligation is NOT settled by flipping this. It is settled by recording
+ * the payment, and the balance nets to zero on its own. Doing both would
+ * subtract the settlement twice — the splits leave the sum AND the payment is
+ * deducted — driving the balance negative by the amount repaid. So there is
+ * deliberately no "mark settled" action anywhere: settling up is recording a
+ * payment, and this column is only ever written by the historical import.
+ *
+ * Assumes the `transaction_splits` alias is `ts`.
+ */
+export const ACTIVE_OBLIGATION = `ts.settled = false`;
+
+/**
+ * The tab a split belongs to: its transaction's trip, else the household.
+ *
+ * Membership already lives on `transaction_overrides.trip_id`, so this is a
+ * read of existing data rather than a new grouping key. Assumes an
+ * `transaction_overrides` alias `o` is joined (LEFT — a transaction with no
+ * override row has no trip, which is the common case and means household).
+ */
+export const SPLIT_SCOPE = `o.trip_id`;