diff --git a/admin-frontend-vue/tests/task-34.test.ts b/admin-frontend-vue/tests/task-34.test.ts
new file mode 100644
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+++ b/admin-frontend-vue/tests/task-34.test.ts
@@ -0,0 +1,55 @@
+import test from 'node:test'
+import assert from 'node:assert/strict'
+import { readSource } from './helpers.ts'
+import { firstVisiblePath } from '../src/router/helpers.ts'
+
+test('test_task_034_unauthorized_fallback_normal_primary_path', () => {
+  // 正常主路径：越权页面回退到菜单树首个可见页面。
+  const tree = [{ key: 'g', name: '组', children: [{ key: 'users', name: '用户管理', route: '/account/users' }] }]
+  assert.equal(firstVisiblePath(tree), '/account/users')
+})
+
+test('test_task_034_unauthorized_fallback_normal_variant_input', () => {
+  // 正常变体：多个可路由页面时取深度优先的首个。
+  const tree = [
+    { key: 'a', name: 'A', route: '/a', sort: 9 },
+    { key: 'b', name: 'B', route: '/b', sort: 1 },
+  ]
+  assert.equal(firstVisiblePath(tree), '/a', '前端保序，不按 sort 重排')
+})
+
+test('test_task_034_unauthorized_fallback_normal_repeated_operation_is_idempotent', () => {
+  // 正常重复：查找不修改树、结果稳定。
+  const tree = [{ key: 'g', name: '组', children: [{ key: 'p', name: '页', route: '/p' }] }]
+  assert.equal(firstVisiblePath(tree), firstVisiblePath(tree))
+})
+
+test('test_task_034_unauthorized_fallback_boundary_empty_input', () => {
+  // 边界空值：无任何可见页面时回退空串，由守卫回根路由空态。
+  assert.equal(firstVisiblePath([]), '')
+  assert.equal(firstVisiblePath(null), '')
+})
+
+test('test_task_034_unauthorized_fallback_boundary_single_item', () => {
+  // 边界单元素：单个页面即回退目标。
+  assert.equal(firstVisiblePath([{ key: 's', name: '单', route: '/solo' }]), '/solo')
+})
+
+test('test_task_034_unauthorized_fallback_boundary_limit_or_missing_field', () => {
+  // 边界上限/缺字段：深层叶页面也被当作回退目标。
+  const tree = [{ key: 'g', name: '组', children: [{ key: 'g2', name: '子组', children: [{ key: 'deep', name: '深', route: '/deep' }] }] }]
+  assert.equal(firstVisiblePath(tree), '/deep')
+})
+
+test('test_task_034_unauthorized_fallback_invalid_input_rejected', () => {
+  // 异常输入：畸形节点（无 route 无 children）被跳过，不影响回退目标。
+  const tree = [{ key: 'bad', name: '坏' }, { key: 'ok', name: '好', route: '/ok' }]
+  assert.equal(firstVisiblePath(tree), '/ok')
+})
+
+test('test_task_034_unauthorized_fallback_dependency_failure_returns_actionable_message', () => {
+  // 依赖失败：路由守卫对越权页以首个可见页/根路由作为回退目标。
+  const router = readSource('src/router/index.ts')
+  assert.match(router, /session\.firstVisible\s*\|\|\s*'\/'/, '越权页回退首个可见页，全无则根路由')
+  assert.match(router, /isRouteAllowed/)
+})