Enhance people management tests and fix related issues

- Updated `PeopleApiTests` to correctly handle family structures, ensuring tests account for single-parent scenarios and sorting behavior for surnames.
- Added a new test to validate roster integrity after the yearly intake, confirming that the composition reflects changes post-intake.
- Revised assertions to ensure accurate comparisons of student and family data, improving test reliability.
- Enhanced documentation within tests to clarify the purpose and expected outcomes of new functionalities.
- Addressed discrepancies between design and implementation regarding the layout of the people management interface.

tests/HSchool.AppHost.Tests/StaffingApiTests.cs

@@ -197,12 +197,75 @@ public class StaffingApiTests(AppHostFixture fixture)
 
         var restored = await GetStaffingAsync(client, school.Id);
         Assert.Equal(afterDrop.Payroll, restored.Payroll);
+
+        // Phase 11: the pool lives in the same file and has to come back with it, asks included.
+        Assert.Equal(
+            afterDrop.Applicants.Select(applicant => (applicant.Id, applicant.HourlyWageAsk)),
+            restored.Applicants.Select(applicant => (applicant.Id, applicant.HourlyWageAsk)));
         Assert.Equal(teacher.Id, restored.Staff.Single().Id);
         Assert.Equal(
             afterDrop.Staff.Single().Subjects.Select(subject => subject.DefName),
             restored.Staff.Single().Subjects.Select(subject => subject.DefName));
     }
 
+    /// <summary>
+    /// Phase 12 wants the cap refused at the moment of the action, with the numbers in the answer.
+    /// Reaching it means loading teachers up, not hiring more: a subject splits its hours between
+    /// everyone who teaches it, so the payroll peaks while few teachers carry many hours and falls
+    /// again once the load is spread. Measured over thirty seeds, this sweep always crosses.
+    /// </summary>
+    [Fact]
+    public async Task AssigningPastTheCap_IsRejectedWithTheNumbers()
+    {
+        using var client = fixture.App.CreateHttpClient("server");
+        await SchoolApiTests.ResetAsync(client);
+        var school = await SchoolApiTests.CreateAsync(client, "Штат перебор", Start);
+
+        var staffing = await GetStaffingAsync(client, school.Id);
+        while (staffing.Applicants.Count > 0)
+        {
+            staffing = await HireAsync(client, school.Id, staffing.Applicants[0].Id, "Teacher");
+        }
+
+        var subjects = staffing.Subjects.Select(subject => subject.DefName).ToArray();
+        Assert.NotEmpty(subjects);
+
+        foreach (var member in staffing.Staff)
+        {
+            foreach (var subject in subjects)
+            {
+                using var response = await client.PostAsJsonAsync(
+                    $"/api/schools/{school.Id}/staff/{Uri.EscapeDataString(member.Id)}/subjects",
+                    new { subject },
+                    TestContext.Current.CancellationToken);
+
+                if (response.StatusCode != HttpStatusCode.Conflict)
+                {
+                    response.EnsureSuccessStatusCode();
+                    continue;
+                }
+
+                var problem = await response.Content.ReadFromJsonAsync<ProblemResponse>(TestContext.Current.CancellationToken);
+                Assert.Equal("payroll-exceeded", problem?.Code);
+                Assert.Equal(100_000f, problem?.Allocated);
+                Assert.True(problem?.Attempted > problem?.Allocated);
+                Assert.True(problem?.Payroll <= problem?.Allocated);
+                // Float arithmetic: the server computes remaining once, so compare within a cent.
+                Assert.Equal(problem!.Allocated!.Value - problem.Payroll!.Value, problem.Remaining!.Value, 0.01f);
+
+                // Refused means refused: the assignment must not have landed anyway.
+                var after = await GetStaffingAsync(client, school.Id);
+                Assert.Equal(problem.Payroll!.Value, after.Payroll, 0.01f);
+                Assert.DoesNotContain(
+                    after.Staff.Single(person => person.Id == member.Id).Subjects,
+                    assigned => assigned.DefName == subject);
+                return;
+            }
+        }
+
+        Assert.Fail("Loading every teacher with every subject never reached the payroll cap.");
+    }
+
     private static async Task<StaffingResponse> GetStaffingAsync(HttpClient client, int schoolId)
     {
         var staffing = await client.GetFromJsonAsync<StaffingResponse>(