feat(landing): hero Ara → /demo for empty input or example VIN match

Real visitors were repeatedly clicking the hero "Ara" button while only the
placeholder VIN (WVWZZZ1JZ3W597935) was showing — they mistook the example
for typed text. The click hit handleVinSearch's `if (!trimmed) return`
early-out and did nothing, leaving them stuck.

Now: empty input OR a value that exactly matches the example VIN routes to
/demo (the pre-warmed VW Golf 2003 catalog) instead of /register. The button
label also updates to "Örnek aracı dene" in that state so the outcome is
predictable before the click. Any other typed VIN still goes through the
existing /register?vin= flow unchanged.

PostHog: emits hero_ara_demo_redirect with {had_value, was_example_match}
so we can split the funnel by what triggered the demo entry.

apps/web/src/routes/index.tsx

@@ -31,6 +31,13 @@ import {
 } from "lucide-react";
 import { Suspense, lazy, useEffect, useRef, useState } from "react";
 
+// The example VIN shown as the hero input placeholder. Clicking "Ara" with
+// the input empty (or already containing this exact VIN — common when users
+// mistake the placeholder for typed text) opens /demo with the pre-warmed
+// VW Golf 2003 catalog instead of stalling at the signup wall. Any other
+// VIN takes the normal /register?vin= path.
+const EXAMPLE_VIN = "WVWZZZ1JZ3W597935";
+
 // Remotion — lazy loaded to keep initial bundle lean
 const RemotionSchemaPlayer = lazy(() =>
   Promise.all([import("@remotion/player"), import("@/remotion/SchemaDemo")]).then(
@@ -366,8 +373,25 @@ export function HomePage() {
   }, [vin]);
 
   const handleVinSearch = async () => {
-    const trimmed = vin.trim();
-    if (!trimmed) return;
+    const trimmed = vin.trim().toUpperCase();
+
+    // Empty input OR the example VIN itself → open the public demo instead
+    // of the signup wall. Real visitors were repeatedly clicking "Ara" with
+    // the placeholder showing because they mistook it for typed text; this
+    // turns that dead-end click into an "aha" catalog browse.
+    if (!trimmed || trimmed === EXAMPLE_VIN) {
+      capture("hero_ara_demo_redirect", {
+        had_value: !!trimmed,
+        was_example_match: trimmed === EXAMPLE_VIN,
+      });
+      navigate({
+        to: "/demo",
+        search: {
+          source: trimmed === EXAMPLE_VIN ? "hero_example_match" : "hero_empty",
+        },
+      });
+      return;
+    }
 
     // Read latest auth state directly from store (avoids stale closure)
     const { user, isLoading } = useAuthStore.getState();
@@ -430,7 +454,7 @@ export function HomePage() {
   };
 
   const fillExampleVin = () => {
-    setVin("WVWZZZ1JZ3W597935");
+    setVin(EXAMPLE_VIN);
     setIsExampleVin(true);
   };
 
@@ -933,7 +957,13 @@ export function HomePage() {
                   </>
                 ) : (
                   <>
-                    {t("landing.hero.searchCta")}
+                    {/* When the input is empty or already holds the example
+                        VIN, the button takes the visitor to /demo instead of
+                        the signup wall — surface that intent in the label so
+                        the click outcome is predictable. */}
+                    {vin.trim() === "" || vin.trim().toUpperCase() === EXAMPLE_VIN
+                      ? "Örnek aracı dene"
+                      : t("landing.hero.searchCta")}
                     <ArrowRight className="ml-2 size-4" />
                   </>
                 )}