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.

client/tests/unit/mobile/trip/MCostsTab.test.tsx

@@ -14,7 +14,7 @@ import { resetAllStores, seedStore } from '../../../helpers/store'
 import { server } from '../../../helpers/msw/server'
 import { fireEvent, render, screen, waitFor, within } from '../../../helpers/render'
 
-// FE-MOB-COSTT-001 to FE-MOB-COSTT-037
+// FE-MOB-COSTT-001 to FE-MOB-COSTT-043
 
 // The add/edit expense sheet is the shared desktop-sized form; the panel only
 // owns when it opens and what happens on save, so it is stubbed here.
@@ -580,41 +580,70 @@ describe('MCostsTab', () => {
     expect(within(row).getByText('$15.00')).toBeInTheDocument()
   })
 
-  it('FE-MOB-COSTT-041: shows what the trip costs each traveler and opens the arithmetic on tap', async () => {
+  it('FE-MOB-COSTT-043: shows what the trip costs each traveler and opens the arithmetic on tap', async () => {
+    // The rows come from the server in display cents. Bob's museum is a GBP expense
+    // the server booked at its frozen rate as $58.00; the live rate this test seeds
+    // would make it $60.00, and that figure must never reach the card.
     serveSettlement({
       ...SETTLEMENT,
-      settlements: [{ id: 5, from_user_id: 3, to_user_id: 1, amount: 15, currency: 'USD', created_at: '2026-05-02 10:00:00' }],
+      settlements: [{ id: 5, from_user_id: 3, to_user_id: 1, amount: 10, currency: 'USD', created_at: '2026-05-02 10:00:00' }],
       finalBudgets: [
-        { user_id: 1, username: 'Me', avatar_url: null, expenses: 100, reimbursed: 15, pending: 12.5, final: 72.5 },
+        {
+          user_id: 1, username: 'Me', avatar_url: null, expenses: 100, reimbursed: 10, pending: -25, final: 115,
+          sources: {
+            fronted: [{ item_id: 11, cents: 8000 }, { item_id: 14, cents: 2000 }],
+            moved: [{ settlement_id: 5, from_user_id: 3, to_user_id: 1, cents: 1000 }],
+            outstanding: [{ from_user_id: 1, to_user_id: 2, cents: -4000 }, { from_user_id: 3, to_user_id: 1, cents: 1500 }],
+          },
+        },
+        {
+          user_id: 3, username: 'Bob', avatar_url: null, expenses: 58, reimbursed: -10, pending: -10, final: 78,
+          sources: {
+            fronted: [{ item_id: 12, cents: 5800 }],
+            moved: [{ settlement_id: 5, from_user_id: 3, to_user_id: 1, cents: -1000 }],
+            outstanding: [{ from_user_id: 3, to_user_id: 1, cents: -1500 }, { from_user_id: 99, to_user_id: 3, cents: 500 }],
+          },
+        },
       ],
     })
     await renderTab()
     const card = up(screen.getByText('costs.finalBudget'), 1)
 
-    // One amount per traveler; the two the ledger left out cost nothing.
-    expect(within(card).getByText('$72.50')).toBeInTheDocument()
-    expect(within(card).getAllByText('$0.00')).toHaveLength(2)
+    // One amount per traveler; Ada, whom the ledger left out, costs nothing.
+    expect(within(card).getByText('$115.00')).toBeInTheDocument()
+    expect(within(card).getByText('$78.00')).toBeInTheDocument()
+    expect(within(card).getByText('$0.00')).toBeInTheDocument()
     expect(within(card).queryByText('costs.finalPending')).not.toBeInTheDocument()
 
-    const mine = within(card).getByRole('button', { name: /\$72\.50/ })
+    const mine = within(card).getByRole('button', { name: /\$115\.00/ })
     fireEvent.click(mine)
     expect(mine).toHaveAttribute('aria-expanded', 'true')
-    // Each line signed by what it does to the final.
+    // Each line signed by what it does to the final, and its rows signed the same
+    // way, so they add up to it on sight.
     expect(within(card).getByText('+$100.00')).toBeInTheDocument()
+    expect(within(card).getByText('+$80.00')).toBeInTheDocument()
+    expect(within(card).getByText('+$20.00')).toBeInTheDocument()
+    expect(within(card).getAllByText('−$10.00')).toHaveLength(2)
+    expect(within(card).getByText('+$25.00')).toBeInTheDocument()
+    expect(within(card).getByText('+$40.00')).toBeInTheDocument()
     expect(within(card).getByText('−$15.00')).toBeInTheDocument()
-    expect(within(card).getByText('−$12.50')).toBeInTheDocument()
-    // The rows behind it: only the expenses I fronted, the transfer I received,
-    // and the flows still open on my side.
     expect(within(card).getByText('Ramen')).toBeInTheDocument()
     expect(within(card).getByText('Taxi')).toBeInTheDocument()
     expect(within(card).queryByText('Museum')).not.toBeInTheDocument()
     expect(within(card).getAllByText('Bob → costs.you')).toHaveLength(2)
     expect(within(card).getByText('costs.you → Ada')).toBeInTheDocument()
-    expect(within(card).queryByText('? → Bob')).not.toBeInTheDocument()
 
-    fireEvent.click(mine)
+    // Opening Bob closes me. His museum row is the server's figure, not a live conversion.
+    const bob = within(card).getByRole('button', { name: /\$78\.00/ })
+    fireEvent.click(bob)
     expect(mine).toHaveAttribute('aria-expanded', 'false')
-    expect(within(card).queryByText('+$100.00')).not.toBeInTheDocument()
+    expect(within(card).getAllByText('+$58.00')).toHaveLength(2)
+    expect(within(card).queryByText(/60\.00/)).not.toBeInTheDocument()
+    expect(within(card).getByText('Museum')).toBeInTheDocument()
+
+    fireEvent.click(bob)
+    expect(bob).toHaveAttribute('aria-expanded', 'false')
+    expect(within(card).queryByText('+$58.00')).not.toBeInTheDocument()
   })
 
   it('FE-MOB-COSTT-040: the expense filters apply to payments the same way they do on desktop', async () => {