feat(booth): overstay sessions, top-up pricing, and session/feed filters

Rework paid-but-grace-expired sessions and add booth filters.

Overstay (was "stuck"):
- Stop silently aging out a paid transient whose walk-back grace lapsed with no
  signed exit. Keep it listed with an OVERSTAY badge — a new parking period began
  (re-parked) or the car is faulty/abandoned; it is not a system fault.
- No free exit: reopenBarrier refuses server-side once a transient's payment grace
  has expired (allow only subscription OR paid-and-within-grace); the UI hides the
  Open-barrier button on overstay rows and routes to the pay/exit modal. Closes a
  hole where a stale payment authorized a free multi-day exit (operator-as-adversary).
- Price the overstay as a NEW period from grace-expiry -> now with its own daily-cap
  ladder, NOT "full stay minus paid" (which a daily cap collapsed to 0 — ticket
  1245791632490 owed ALL 0; now owes its real overstay). quote() gains periodStart +
  overstay; SessionLookup/ActiveSession gain `overstay`. handlePayAndExit charges
  whenever the session is payable (was: only if !alreadyPaid, skipping the overstay).

Filters (new ui/FilterBar): Active Sessions — search + status
(unpaid/paid/exiting/overstay) + transient-vs-subscriber. Live feed — search +
event (entry/exit/pay/void/anomaly) + direction + source (booth=manual vs reader).
All client-side over already-fetched data; matched/total count shown.

i18n parity (sq+en). Wiki: booth-exit-flow updated (overstay model, naming history,
no-free-exit security fix, new-period pricing; open question on grace-renewal noted).

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

apps/web/src/BoothPayModal.tsx

@@ -52,9 +52,16 @@ export function BoothPayModal({ identity, onClose }: { identity: string; onClose
 
   const alreadyPaid = s?.paidAt != null;
   const isSubscription = s?.subscription === true;
+  // OVERSTAY = paid but walk-back grace expired with no exit → a NEW period began; owes
+  // a fresh TOP-UP. Treat it as payable even though it's "already paid": the car must
+  // settle the new period's fee (s.amountMinor, priced from grace-expiry) before any
+  // exit. A normal within-grace paid session is NOT payable (it's settled). See
+  // booth-exit-flow.md / reopenBarrier server guard.
+  const isOverstay = s?.overstay === true;
   // A subscription is prepaid: never charged. The only booth action is an audited
   // barrier open to ASSIST (faulty exit reader / lost card). Transient pay path is off.
-  const canPay = shiftReady && s?.found && s.open && !alreadyPaid && !isSubscription;
+  // Allow pay for an unpaid session OR an overstay (new-period top-up) one.
+  const canPay = !!(shiftReady && s?.found && s.open && (!alreadyPaid || isOverstay) && !isSubscription);
 
   async function handleOpenBarrier() {
     if (!s) return;
@@ -103,8 +110,11 @@ export function BoothPayModal({ identity, onClose }: { identity: string; onClose
     if (!s) return;
     setError(null);
     try {
-      // 1. Take payment (unless already paid — e.g. paid earlier at a kiosk).
-      if (!alreadyPaid) {
+      // 1. Take payment. For a first stay this is the only charge; for an OVERSTAY the
+      //    session is "already paid" but a new period accrued — we still charge (canPay
+      //    is true). A settled within-grace session is not payable (canPay false) and is
+      //    skipped. The server re-quotes authoritatively (overstay → from grace-expiry).
+      if (canPay) {
         setPhase("paying");
         await paySession(identity, tender);
       }
@@ -221,15 +231,32 @@ export function BoothPayModal({ identity, onClose }: { identity: string; onClose
                   />
                   <Row
                     label={t("pay.statusLabel")}
-                    value={isSubscription ? t("pay.subscription") : alreadyPaid ? t("pay.paid") : t("pay.unpaid")}
-                    valueClass={isSubscription ? "text-term-cyan" : alreadyPaid ? "text-term-green" : "text-term-amber"}
+                    value={
+                      isSubscription
+                        ? t("pay.subscription")
+                        : isOverstay
+                          ? t("pay.overstay")
+                          : alreadyPaid
+                            ? t("pay.paid")
+                            : t("pay.unpaid")
+                    }
+                    valueClass={
+                      isSubscription
+                        ? "text-term-cyan"
+                        : isOverstay
+                          ? "text-term-red"
+                          : alreadyPaid
+                            ? "text-term-green"
+                            : "text-term-amber"
+                    }
                   />
                 </div>
 
-                {/* Total — a subscription is prepaid (no amount); show a badge. */}
+                {/* Total — a subscription is prepaid (no amount); show a badge. For an
+                    overstay the amount is the TOP-UP delta, not the whole stay. */}
                 <div className="flex items-end justify-between rounded-term bg-term-panel-2 px-3 py-2">
                   <span className="text-[11px] uppercase tracking-wider text-term-muted">
-                    {isSubscription ? t("pay.plan") : t("pay.total")}
+                    {isSubscription ? t("pay.plan") : isOverstay ? t("pay.topUp") : t("pay.total")}
                   </span>
                   <span className="text-3xl font-bold text-term-cyan">
                     {isSubscription
@@ -249,6 +276,13 @@ export function BoothPayModal({ identity, onClose }: { identity: string; onClose
                   </div>
                 )}
 
+                {/* For an overstay, explain why a top-up is required (no free exit). */}
+                {isOverstay && (
+                  <div className="rounded-term border border-term-red/40 bg-term-red/5 px-3 py-2 text-[12px] text-term-text">
+                    {t("pay.overstayHint")}
+                  </div>
+                )}
+
                 {/* Snapshots */}
                 <SnapshotStrip identity={identity} />