fix: make circular and spiral presets truthful

tests/integration/array-builder-ui.test.js

@@ -572,6 +572,53 @@ test("Array Builder가 preset·직접 편집·검증·표준 JSON round trip을
       () => documentObject.documentElement.dataset.arrayValid === "true",
       "ellipse ring validation",
     );
+    assert(
+      documentObject.querySelector("#preset-hint").textContent.includes("실제 내접원"),
+      "ring hint describes a true circle instead of the boundary perimeter",
+    );
+    for (let index = 0; index < microphoneCount(documentObject); index += 1) {
+      microphoneButtons(documentObject)[index].click();
+      const [x, y] = selectedCoordinates(documentObject);
+      assertNear(Math.hypot(x, y), 0.215, 1e-8, "visible ring radius " + index);
+    }
+
+    dispatchValue(
+      windowObject,
+      documentObject.querySelector("#preset-type"),
+      "archimedean",
+    );
+    assert(
+      !documentObject.querySelector("#preset-turns").closest(".field").hidden,
+      "Archimedean preset exposes its turn count",
+    );
+    assert(
+      documentObject.querySelector("#preset-hint").textContent.includes("r∝θ"),
+      "Archimedean hint identifies the curved spiral law",
+    );
+    dispatchValue(windowObject, documentObject.querySelector("#preset-count"), 9);
+    dispatchValue(windowObject, documentObject.querySelector("#preset-turns"), 1.5);
+    documentObject.querySelector("#generate-preset-button").click();
+    await waitFor(() => microphoneCount(documentObject) === 9, "Archimedean generation");
+    microphoneButtons(documentObject)[0].click();
+    assertEqual(
+      selectedCoordinates(documentObject).join(","),
+      "0,0,0",
+      "Archimedean spiral visibly starts at the center",
+    );
+    microphoneButtons(documentObject).at(-1).click();
+    const archimedeanEnd = selectedCoordinates(documentObject);
+    assertNear(archimedeanEnd[0], -0.215, 1e-8, "1.5-turn spiral ends at -x");
+    assertNear(archimedeanEnd[1], 0, 1e-8, "1.5-turn spiral endpoint y");
+
+    dispatchValue(windowObject, documentObject.querySelector("#preset-type"), "spiral");
+    assert(
+      documentObject.querySelector("#preset-turns").closest(".field").hidden,
+      "Fermat preset does not claim an Archimedean turn count",
+    );
+    assert(
+      documentObject.querySelector("#preset-hint").textContent.includes("해바라기형"),
+      "Fermat hint explains why its points do not look like one connected curve",
+    );
 
     dispatchValue(windowObject, documentObject.querySelector("#preset-type"), "random");
     dispatchValue(windowObject, documentObject.querySelector("#preset-count"), 12);
@@ -585,7 +632,7 @@ test("Array Builder가 preset·직접 편집·검증·표준 JSON round trip을
     assertEqual(repeatedRandom, firstRandom, "same seed repeats the visible layout");
 
     documentObject.querySelector("#undo-button").click();
-    assertEqual(microphoneCount(documentObject), 8, "preset replacement is undoable");
+    assertEqual(microphoneCount(documentObject), 9, "preset replacement is undoable");
     documentObject.querySelector("#redo-button").click();
     assertEqual(microphoneCount(documentObject), 12, "preset replacement is redoable");