feat(v5): add Inertia app shell with Flux health

Adds v5 routing, middleware, home page rendering, team context sharing,
project model/table support, and Flux health reporting. Installs Flux in
container builds with role-aware s6 services and documents runtime roles.

app/Services/Flux/FluxHealth.php

@@ -0,0 +1,66 @@
+<?php
+
+namespace App\Services\Flux;
+
+class FluxHealth
+{
+    /**
+     * @return array{available: bool, label: string, message: string, socket: string|null}
+     */
+    public function check(): array
+    {
+        $socketPath = config('flux.unix_socket_path');
+
+        if (! is_string($socketPath) || $socketPath === '') {
+            return $this->unavailable(null, 'Flux socket is not configured.');
+        }
+
+        if (! file_exists($socketPath)) {
+            return $this->unavailable($socketPath, 'Flux socket was not found.');
+        }
+
+        $timeout = (float) config('flux.health_timeout_seconds', 1.0);
+        $stream = @stream_socket_client("unix://{$socketPath}", $errorCode, $errorMessage, $timeout);
+
+        if ($stream === false) {
+            return $this->unavailable($socketPath, $errorMessage ?: "Could not connect to Flux socket ({$errorCode}).");
+        }
+
+        stream_set_timeout($stream, (int) ceil($timeout));
+
+        fwrite($stream, "GET /v1/health HTTP/1.1\r\nHost: flux\r\nAccept: application/json\r\nConnection: close\r\n\r\n");
+        $response = stream_get_contents($stream) ?: '';
+        fclose($stream);
+
+        if (! str_starts_with($response, 'HTTP/1.1 200') && ! str_starts_with($response, 'HTTP/1.0 200')) {
+            return $this->unavailable($socketPath, 'Flux health endpoint did not return HTTP 200.');
+        }
+
+        $body = str_contains($response, "\r\n\r\n") ? substr($response, strpos($response, "\r\n\r\n") + 4) : '';
+        $payload = json_decode($body, true);
+
+        if (! is_array($payload) || ($payload['ok'] ?? false) !== true) {
+            return $this->unavailable($socketPath, 'Flux health endpoint returned an invalid response.');
+        }
+
+        return [
+            'available' => true,
+            'label' => 'Running',
+            'message' => 'Flux is running.',
+            'socket' => $socketPath,
+        ];
+    }
+
+    /**
+     * @return array{available: false, label: string, message: string, socket: string|null}
+     */
+    private function unavailable(?string $socketPath, string $message): array
+    {
+        return [
+            'available' => false,
+            'label' => 'Unavailable',
+            'message' => $message,
+            'socket' => $socketPath,
+        ];
+    }
+}