feat(booth): rework Active Sessions + pay/exit modal around barrier re-open

Move the audited barrier re-open out of the inline Active-Sessions row button
and into the modal, and turn the modal's dead-ends into useful views.

- Remove the inline per-row "Open barrier" button. Clicking a row opens the
  modal, which carries the action.
- Modal recognizes a closed-within-grace transient (found && !open &&
  withinGrace) and shows the session view + Open barrier instead of dead-ending
  on "already closed" — the exact case (paid, barrier unconfirmed) that needs a
  re-pulse. Server reopenBarrier guard unchanged.
- Active-Sessions rows show a live grace-remaining countdown badge
  (exited - M:SS, 1s tick off graceExpiresAt) via new formatCountdown helper.
- Settled sessions show the ACTUAL sum paid (new SessionLookup.paidMinor,
  summed across payment events) instead of a flat "PAID" badge.
- A fully-closed (grace-expired) session's modal is no longer a dead-end: it
  shows a read-only review view (figures + paid amount + entry/exit snapshot
  strip) for dispute/audit review, with no pay/exit/open controls.

i18n sq+en parity kept; web build/lint/test green.

Claude-Session: https://claude.ai/code/session_01Xcm6ikLgGoCxxHrxtjkk5V

apps/web/src/lib/format.ts

@@ -22,6 +22,22 @@ export function formatDuration(fromIso: string, toIso: string): string {
   return h > 0 ? `${h}h ${m}m` : `${m}m`;
 }
 
+/** Remaining time until `untilIso`, as a live countdown: "M:SS" (or "H:MM:SS" past an
+ *  hour). Returns null once expired (or for a bad/empty input) so callers can drop the
+ *  badge. Pass `nowMs` (a ticking clock) to make it update each second. */
+export function formatCountdown(untilIso: string | null, nowMs: number = Date.now()): string | null {
+  if (!untilIso) return null;
+  const ms = Date.parse(untilIso) - nowMs;
+  if (!Number.isFinite(ms) || ms <= 0) return null;
+  const total = Math.ceil(ms / 1000);
+  const h = Math.floor(total / 3600);
+  const m = Math.floor((total % 3600) / 60);
+  const s = total % 60;
+  const ss = String(s).padStart(2, "0");
+  if (h > 0) return `${h}:${String(m).padStart(2, "0")}:${ss}`;
+  return `${m}:${ss}`;
+}
+
 /** Human duration from whole minutes, e.g. 134 → "2h 14m", 47 → "47m", 0 → "0m". */
 export function formatMinutes(mins: number): string {
   if (!Number.isFinite(mins) || mins < 0) return "—";

apps/web/src/lib/i18n/en.ts

@@ -160,6 +160,10 @@ export const en: Catalog = {
     fEvtVoid: "Void",
     fEvtAnomaly: "Anomaly",
     openPayExit: "Open pay / exit",
+    openReopenBarrier: "Open — paid, awaiting barrier",
+    exitedGrace: "exited · grace",
+    exitedGraceLeft: "exited · {{time}}",
+    exitedGraceTitle: "Paid and exited — barrier not confirmed; waiting out the grace period.",
     openBarrier: "Open barrier",
     openBarrierTitle: "Human-intervention barrier open (audited)",
     barrierOpened: "barrier opened",
@@ -874,14 +878,18 @@ export const en: Catalog = {
     ticket: "Ticket",
     entry: "Entry",
     now: "Now",
+    exit: "Exit",
     duration: "Duration",
     statusLabel: "Status",
     paid: "PAID",
     unpaid: "UNPAID",
     overstay: "OVERSTAY",
     overstayHint: "Earlier session paid. The customer failed to exit during the grace period. Payment for the new period is required. The total below is the new period's fee.",
+    closedWithinGrace: "EXITED · GRACE",
+    closedWithinGraceHint: "Paid and exit recorded — the barrier didn't confirm yet. The car stays listed until the grace period ends. Open the barrier manually if it's still waiting.",
     topUp: "New period due",
     total: "Total",
+    paidAmount: "Paid",
     noTariff: "no tariff",
     tender: "Tender",
     cash: "Cash",

apps/web/src/lib/i18n/sq.ts

@@ -162,10 +162,14 @@ export const sq = {
     fEvtVoid: "Anulim",
     fEvtAnomaly: "Anomali",
     openPayExit: "Hap pagesën / daljen",
+    openReopenBarrier: "Hap — paguar, pret barrierën",
+    exitedGrace: "doli · në afat",
+    exitedGraceLeft: "doli · {{time}}",
+    exitedGraceTitle: "Paguar dhe dalur — barriera nuk u konfirmua; po pret afatin kohor.",
     openBarrier: "Hap barrierën",
     openBarrierTitle: "Hap barrierën manualisht",
     barrierOpened: "barriera u hap",
-    openManually: "hape me dorë",
+    openManually: "hape manualisht",
     // session row badges
     badgeExiting: "duke dalë",
     badgePaid: "paguar",
@@ -242,9 +246,9 @@ export const sq = {
     "exit.refused.noSession": "Dalja u refuzua — biletë e panjohur",
     "exit.refused.unpaid": "Dalja u refuzua — e papaguar (bëj pagesën në fillim)",
     "exit.refused.graceExpired": "Dalja u refuzua — afati i daljes skadoi (kërkohet pagesë shtesë)",
-    "exit.open.noBarrier": "Dalja u regjistrua, por nuk ka barrierë daljeje të konfiguruar — hape me dorë",
-    "exit.open.unavailable": "Dalja u regjistrua, por barriera është e padisponueshme — hape me dorë",
-    "exit.open.failed": "Dalja u regjistrua, por barriera nuk u hap — hape me dorë",
+    "exit.open.noBarrier": "Dalja u regjistrua, por nuk ka barrierë daljeje të konfiguruar — hape manualisht",
+    "exit.open.unavailable": "Dalja u regjistrua, por barriera është e padisponueshme — hape manualisht",
+    "exit.open.failed": "Dalja u regjistrua, por barriera nuk u hap — hape manualisht",
     "exit.freeGrace": "Periudhë pa pagesë në hyrje (pa tarifë)",
     "exit.manualOpen": "Hapje manuale e barrierës (ndërhyrje njerëzore)",
     "sub.refused.notFound": "Abonimi u refuzua — nuk u gjet",
@@ -890,20 +894,24 @@ export const sq = {
     ticket: "Bileta",
     entry: "Hyrja",
     now: "Tani",
+    exit: "Dalja",
     duration: "Kohëzgjatja",
     statusLabel: "Statusi",
     paid: "PAGUAR",
     unpaid: "PAPAGUAR",
     overstay: "TEJ AFATIT",
     overstayHint: "Sesion i mëparshëm i paguar. Klienti nuk doli brënda afatit kohor. Kërkohet pagesë për periudhën e re. Totali më poshtë është tarifa e periudhës së re.",
+    closedWithinGrace: "Paguar",
+    closedWithinGraceHint: "Pagesa dhe dalja u regjistruan — barriera nuk u konfirmua ende. Makina mbetet në listë derisa të mbarojë afati. Hapni barrierën manualisht nëse pret ende.",
     topUp: "Periudha e re për pagesë",
     total: "Totali",
+    paidAmount: "Paguar",
     noTariff: "pa tarifë",
     tender: "Mënyra",
     cash: "Para",
     card: "Kartë",
     printExitVoucher: "Printo biletë dalje",
-    selfExitHint: "(klienti del vetë te dalja)",
+    selfExitHint: "(klienti del duke skanuar biletën)",
     payAndOpen: "Paguaj + hap barrierën",
     payAndVoucher: "Paguaj + printo biletën",
     openBarrier: "Hap barrierën",
@@ -926,7 +934,7 @@ export const sq = {
     windowCharge: "JASHTË ORARIT",
     windowChargeHint: "Ky abonent parkoi jashtë orarit të lejuar të planit. Detyrohet të paguajë tarifën kalimtare për kohën jashtë orarit — merr pagesën, pastaj hap barrierën.",
     subBarrierOpened: "Barriera u hap për abonentin (ndërhyrje e regjistruar).",
-    voucherPrinted: "Bileta e daljes u printua në {{printer}}. Klienti del vetë te dalja.",
+    voucherPrinted: "Bileta e daljes u printua në {{printer}}. Klienti del duke skanuar biletën.",
     // payment receipt (transparency slip)
     receiptPrintFailed: "(fatura nuk u printua — provoni \"Riprinto faturën\".)",
     receiptReprinted: "Fatura u riprintua në {{printer}}.",