feat(solve): 审查 markov_simulate：工程侧收口 + 9 条测试

用户实现的第一版 markov_simulate 算法本体经探针实测正确（终点频率 [0.657, 0.343]
对照平稳分布 [2/3, 1/3]，同种子可复现），本次只动工程侧，未改任何算法行：

- ruff 8 条全清（I001 导入排序 + 5 处 W293 / 2 处 W291）
- docstring 从一行补全为 Parameters / Returns / Raises / Notes / Examples
- 文案：马尔科夫→马尔可夫（3 处）、initail→initial、半角逗号→全角、paths[:,0]→paths[:, 0]
- arr, kind = as_array(...) 的 kind 全程未用 → arr, _
- tests/test_solve.py 新增 9 条 markov_simulate 测试（此前 0 条）
- STATUS.md 新增决策 56（initial 语义待用户拍板，A/B/C 三选项）与 5 条待办
- CHANGELOG 与 docs/dev/personal-pitfalls.md 易错点 27（入口分流后两条分支口径矛盾）

校验：pytest 188 passed / 31 skipped（219 收集）；ruff 0；pyright 0

tests/test_solve.py

@@ -407,6 +407,108 @@ def test_random_walk_docstring_example_runs():
     assert r.data.shape == (5, 101)
     assert np.all(np.abs(np.diff(r.data, axis=1)) == 1)
 
+def test_markov_simulate_shape_start_and_states():
+    """`data` 形状 (n_paths, steps + 1)，第 0 列是初始状态，取值只能是合法状态编号。"""
+    from mcm_toolkit.solve import markov_simulate
+
+    P = np.array([[0.9, 0.1], [0.2, 0.8]])
+    r = markov_simulate(P, initial=0, steps=100, n_paths=5, random_state=0)
+    assert r.success
+    assert r.data.shape == (5, 101)
+    assert np.all(r.data[:, 0] == 0)
+    assert set(np.unique(r.data)) <= {0, 1}
+    assert np.issubdtype(r.data.dtype, np.integer)
+
+
+def test_markov_simulate_absorbing_states_never_move():
+    """单位转移矩阵：状态自锁，整条路径恒等于初始状态（确定性自检）。"""
+    from mcm_toolkit.solve import markov_simulate
+
+    r = markov_simulate(np.eye(2), initial=1, steps=30, n_paths=4, random_state=0)
+    assert r.success
+    assert np.all(r.data == 1)
+    assert r.metrics["mean_terminal"] == 1.0
+    assert r.metrics["std_terminal"] == 0.0
+
+
+def test_markov_simulate_is_reproducible():
+    from mcm_toolkit.solve import markov_simulate
+
+    P = np.array([[0.7, 0.3], [0.4, 0.6]])
+    a = markov_simulate(P, initial=0, steps=40, n_paths=6, random_state=11)
+    b = markov_simulate(P, initial=0, steps=40, n_paths=6, random_state=11)
+    c = markov_simulate(P, initial=0, steps=40, n_paths=6, random_state=12)
+    assert np.array_equal(a.data, b.data)
+    assert not np.array_equal(a.data, c.data)
+
+
+def test_markov_simulate_terminal_frequency_matches_stationary():
+    """终点频率收敛到平稳分布 [2/3, 1/3] —— 马尔可夫链模拟的核心性质。"""
+    from mcm_toolkit.solve import markov_simulate
+
+    P = np.array([[0.9, 0.1], [0.2, 0.8]])
+    r = markov_simulate(P, initial=0, steps=200, n_paths=20000, random_state=0)
+    freq = np.bincount(r.data[:, -1], minlength=2) / r.data.shape[0]
+    assert np.allclose(freq, [2 / 3, 1 / 3], atol=0.02)
+
+
+def test_markov_simulate_metrics_are_terminal_states():
+    from mcm_toolkit.solve import markov_simulate
+
+    P = np.array([[0.9, 0.1], [0.2, 0.8]])
+    r = markov_simulate(P, initial=0, steps=50, n_paths=500, random_state=3)
+    assert r.metrics["mean_terminal"] == float(np.mean(r.data[:, -1]))
+    assert r.metrics["std_terminal"] == float(np.std(r.data[:, -1]))
+
+
+def test_markov_simulate_rejects_bad_transition():
+    """非方阵、行和不为 1 都要在入口直接报错，不能悄悄模拟。"""
+    from mcm_toolkit.solve import markov_simulate
+
+    with pytest.raises(ValueError):
+        markov_simulate(np.array([[0.5, 0.5, 0.0], [0.2, 0.8, 0.0]]), initial=0)
+    with pytest.raises(ValueError):
+        markov_simulate(np.array([[0.5, 0.4], [0.2, 0.8]]), initial=0)
+
+
+def test_markov_simulate_rejects_bad_arguments():
+    """§4.7：步数 / 路径数的类型与正负都要卡住（bool 先判，易错点 1）。"""
+    from mcm_toolkit.solve import markov_simulate
+
+    P = np.array([[0.9, 0.1], [0.2, 0.8]])
+    with pytest.raises(ValueError):
+        markov_simulate(P, initial=0, steps=0)
+    with pytest.raises(ValueError):
+        markov_simulate(P, initial=0, n_paths=0)
+    with pytest.raises(TypeError):
+        markov_simulate(P, initial=0, steps=1.5)  # type: ignore[arg-type]  # 就是要传错
+    with pytest.raises(TypeError):
+        markov_simulate(P, initial=0, n_paths=1.5)  # type: ignore[arg-type]  # 就是要传错
+    with pytest.raises(TypeError):
+        markov_simulate(P, initial=0, steps=True)  # type: ignore[arg-type]  # 就是要传错
+    with pytest.raises(TypeError):
+        markov_simulate(P, initial=0, n_paths=True)  # type: ignore[arg-type]  # 就是要传错
+
+
+def test_markov_simulate_accepts_numpy_integer():
+    from mcm_toolkit.solve import markov_simulate
+
+    P = np.array([[0.9, 0.1], [0.2, 0.8]])
+    r = markov_simulate(P, initial=0, steps=np.int64(10), n_paths=np.int64(2), random_state=0)
+    assert r.success
+    assert r.data.shape == (2, 11)
+
+
+def test_markov_simulate_docstring_example_runs():
+    from mcm_toolkit.solve import markov_simulate
+
+    P = np.array([[0.9, 0.1], [0.2, 0.8]])
+    r = markov_simulate(P, initial=0, steps=50, n_paths=4, random_state=0)
+    assert r.data.shape == (4, 51)
+    assert np.all(r.data[:, 0] == 0)
+
+
+
 
 @pytest.mark.skip(reason=_NOT_IMPLEMENTED)
 def test_markov_steady_state_two_state_chain():