fix(costs): derive the final budget breakdown on the server, in the same cents

The three figures under a traveler's final budget came from the server, netted
in integer trip cents at the rate frozen on each expense. The rows listed under
them did not: both shells converted the expense list again with whatever live
rate they had, and with none loaded yet (first render, offline) the raw foreign
amount went out under the display currency's symbol. On any trip with a
foreign-currency expense the rows could not add up to the line above them.

The settlement response now carries the rows with the figure: `sources` on each
finalBudgets entry lists what the person fronted per expense, the recorded
transfers on their side and the open flows, in display cents. They are filled
by the same two loops that fill the totals, so an expense with no split members
stays out of them as it stays out of the ledger, and each list is spread over its
figure's already allocated cents with the same largest-remainder split, so the
sum is exact in every display currency. The shells print those cents and only
resolve names; the rows carry the sign of the line they sit under.

server/tests/unit/nest/budget.service.calc.test.ts

@@ -668,6 +668,82 @@ describe('calculateSettlement — finalBudgets', () => {
     expect(result.finalBudgets.find(f => f.user_id === 2)!).toMatchObject({ expenses: 0, reimbursed: -40, pending: 40, final: 0 });
     checkIdentity(result.finalBudgets);
   });
+
+  it('lists the rows each figure is made of, signed the way the figure is', () => {
+    // The first case read row by row, with 20 of Bob's 50 already sent: Alice
+    // fronted the one expense, received the 20, and the open flow is the 30 coming
+    // to her; on Bob's side the same transfer and flow are going out.
+    setupDb(
+      [makeItem(1, 100)],
+      [makeMember(1, 1, 'alice'), makeMember(1, 2, 'bob')],
+      [makePayer(1, 1, 100, 'alice')],
+      [makeSettlementRow(1, 2, 1, 20)],
+    );
+    const result = budget.calculateSettlement(1);
+
+    expect(result.finalBudgets.find(f => f.user_id === 1)!.sources).toEqual({
+      fronted: [{ item_id: 1, cents: 10000 }],
+      moved: [{ settlement_id: 1, from_user_id: 2, to_user_id: 1, cents: 2000 }],
+      outstanding: [{ from_user_id: 2, to_user_id: 1, cents: 3000 }],
+    });
+    expect(result.finalBudgets.find(f => f.user_id === 2)!.sources).toEqual({
+      fronted: [],
+      moved: [{ settlement_id: 1, from_user_id: 2, to_user_id: 1, cents: -2000 }],
+      outstanding: [{ from_user_id: 2, to_user_id: 1, cents: -3000 }],
+    });
+  });
+
+  it('keeps an expense with no split members out of the rows, as it is out of the ledger', () => {
+    // Item 1 has a payer but nobody to split it with: a planning-only entry that
+    // charges nobody, so it cannot be listed as something Alice fronted either.
+    setupDb(
+      [makeItem(1, 100), makeItem(2, 40)],
+      [makeMember(2, 1, 'alice'), makeMember(2, 2, 'bob')],
+      [makePayer(1, 1, 100, 'alice'), makePayer(2, 1, 40, 'alice')],
+    );
+    const result = budget.calculateSettlement(1);
+
+    const alice = result.finalBudgets.find(f => f.user_id === 1)!;
+    expect(alice.expenses).toBe(40);
+    expect(alice.sources.fronted).toEqual([{ item_id: 2, cents: 4000 }]);
+  });
+
+  it('spreads a foreign-currency figure over its rows so they still add up in the display currency', () => {
+    // Two USD expenses booked at their own frozen rates and one in the trip's euros,
+    // viewed in pounds: every row goes through the conversion its figure went
+    // through, and the cents lost to rounding land on rows instead of between them.
+    const sum = (rows: { cents: number }[]) => rows.reduce((a, r) => a + r.cents, 0);
+    setupDb(
+      [
+        { ...makeItem(1, 100), currency: 'USD', exchange_rate: 1.08 },
+        { ...makeItem(2, 33.33), currency: 'USD', exchange_rate: 1.1 },
+        makeItem(3, 50),
+      ],
+      [
+        makeMember(1, 1, 'alice'), makeMember(1, 2, 'bob'), makeMember(1, 3, 'carol'),
+        makeMember(2, 1, 'alice'), makeMember(2, 2, 'bob'), makeMember(2, 3, 'carol'),
+        makeMember(3, 1, 'alice'), makeMember(3, 2, 'bob'), makeMember(3, 3, 'carol'),
+      ],
+      [makePayer(1, 1, 100, 'alice'), makePayer(2, 1, 33.33, 'alice'), makePayer(3, 2, 50, 'bob')],
+      [makeSettlementRow(1, 3, 1, 20, 'GBP', 0.8547), makeSettlementRow(2, 3, 2, 7.77, 'GBP', 0.8547)],
+    );
+    for (const eurPerGbp of [1.17, 1.1523, 1.2]) {
+      const result = budget.calculateSettlement(1, { base: 'GBP', tripCurrency: 'EUR', rates: { GBP: 1, EUR: eurPerGbp } });
+
+      checkIdentity(result.finalBudgets);
+      for (const f of result.finalBudgets) {
+        expect(sum(f.sources.fronted)).toBe(Math.round(f.expenses * 100));
+        expect(sum(f.sources.moved)).toBe(Math.round(f.reimbursed * 100));
+        expect(sum(f.sources.outstanding)).toBe(Math.round(f.pending * 100));
+      }
+      // Alice's two rows each stay within a cent of their own conversion: the
+      // remainder is handed out, not rounded away one row at a time.
+      const alice = result.finalBudgets.find(f => f.user_id === 1)!;
+      expect(alice.sources.fronted.map(r => r.item_id)).toEqual([1, 2]);
+      expect(Math.abs(alice.sources.fronted[0].cents - Math.round(100 / 1.08 * 100) / eurPerGbp)).toBeLessThan(1);
+      expect(Math.abs(alice.sources.fronted[1].cents - Math.round(33.33 / 1.1 * 100) / eurPerGbp)).toBeLessThan(1);
+    }
+  });
 });
 
 describe('splitEqualShares — client parity (#2176)', () => {