diff --git a/ars_opd/similarity.py b/ars_opd/similarity.py
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+++ b/ars_opd/similarity.py
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+"""语义相似度 φ 与 chunk 级聚合 k_sem——论文 §3.2.1 式(3)。
+
+logit-free 的支点:学生 chunk 对不对,不再比 token 概率(层 2 白盒式(2)),
+改比"学生 chunk 文本" vs "teacher rollout 文本"的语义相似度。φ 只依赖
+文本本身,与两侧 tokenizer 无关,teacher 只需能吐文本(任何 API 均可)。
+
+纯逻辑模块(CLAUDE.md §2):只依赖标准库,可脱离 torch 在本地 CPU 测试。
+teacher rollout 怎么采出来是层 5 teacher.py 的事,本模块只吃现成字符串。
+"""
+
+from collections import Counter
+
+
+def rouge1(hypothesis: str, reference: str) -> float:
+    """ROUGE-1 F1(unigram 重叠率),φ 的候选度量之一(论文 §3.2.1)。
+
+    以词为单位(空白切分)统计两串的 unigram 重叠,算 F1。
+    词袋语义:只看"用了哪些词",不看词序——"a b" vs "b a" 得 1.0。
+
+    参数:
+        hypothesis: 学生 chunk 文本。
+        reference: teacher rollout 文本。
+
+    返回:
+        F1 ∈ [0, 1];任一侧无词(空串/纯空白)时为 0.0。
+
+    实现细节:
+        - 差异标注:参考实现(distillation_trainer.py:1670)用 set 去重后求交,
+          会把 "x x x x" vs "x" 判成满分 1.0;此处用 Counter 多重集
+          (ROUGE-1 标准定义),重复词按 min 计数配对,同例只得 0.4。
+          数学推理文本里重复 token(数字、"="、变量名)极常见,去重会失真。
+        - 差异标注:参考实现分母加 1e-8 防零除,代价是全同串 F1≈0.99999998
+          而非精确 1;此处 overlap==0 时提前返回,分母恒正,无需平滑。
+    """
+    hyp_counts = Counter(hypothesis.split())
+    ref_counts = Counter(reference.split())
+    if not hyp_counts or not ref_counts:
+        return 0.0
+    # 多重集交:每个词按两侧出现次数的 min 配对
+    overlap = sum((hyp_counts & ref_counts).values())
+    if overlap == 0:
+        return 0.0
+    precision = overlap / sum(hyp_counts.values())
+    recall = overlap / sum(ref_counts.values())
+    return 2 * precision * recall / (precision + recall)
+
+
+def edit_similarity(hypothesis: str, reference: str) -> float:
+    """归一化编辑相似度 1 − Levenshtein/max(m,n),论文 §5.1 的默认 φ。
+
+    以词为单位(空白切分)算 Levenshtein 距离(插入/删除/替换各计 1),
+    再归一化到 [0, 1] 取反。顺序敏感:"a b" vs "b a" 距离 2,相似度 0——
+    与 rouge1 的词袋语义形成互补。
+
+    参数:
+        hypothesis: 学生 chunk 文本。
+        reference: teacher rollout 文本。
+
+    返回:
+        相似度 ∈ [0, 1];两侧均空为 1.0(零距离),仅一侧空为 0.0(全删/全插)。
+
+    实现细节:
+        - 差异标注:参考实现(distillation_trainer.py:1682)吃 token id 列表,
+          相似度随 tokenizer 切法漂移,违背本层"文本是公共语言"的初衷
+          (docs/04 §2.1 坑①);此处吃 str、内部按词切,与 rouge1 统一口径。
+        - 两行滚动 DP(同参考实现):空间 O(n) 而非 O(m·n)。
+    """
+    hyp_words = hypothesis.split()
+    ref_words = reference.split()
+    m, n = len(hyp_words), len(ref_words)
+    if m == 0 and n == 0:
+        return 1.0
+    if m == 0 or n == 0:
+        return 0.0
+    # prev[j] = 前一行的 dist(hyp[:i-1], ref[:j]);curr 原地滚动复用
+    prev = list(range(n + 1))
+    curr = [0] * (n + 1)
+    for i in range(1, m + 1):
+        curr[0] = i
+        for j in range(1, n + 1):
+            cost = 0 if hyp_words[i - 1] == ref_words[j - 1] else 1
+            curr[j] = min(
+                prev[j] + 1,  # 删除 hyp[i-1]
+                curr[j - 1] + 1,  # 插入 ref[j-1]
+                prev[j - 1] + cost,  # 替换(相同则免费)
+            )
+        prev, curr = curr, prev
+    return 1.0 - prev[n] / max(m, n)
+
+
+def phi(hypothesis: str, reference: str, metric: str = "edit_distance") -> float:
+    """语义相似度 φ(y_c, ŷ_c) ∈ [0, 1],论文 §3.2.1 式(3) 的原子度量。
+
+    参数:
+        hypothesis: 学生 chunk 文本。
+        reference: teacher rollout 文本。
+        metric: "edit_distance"(默认)或 "rouge1"。
+            差异标注:参考实现配置默认 rouge1(config.py:299),与论文 §5.1
+            的 edit_distance 背离;此处从论文。
+
+    返回:
+        相似度 ∈ [0, 1]。
+    """
+    if metric == "edit_distance":
+        return edit_similarity(hypothesis, reference)
+    if metric == "rouge1":
+        return rouge1(hypothesis, reference)
+    raise ValueError(f"未知相似度度量: {metric!r}(可选 'edit_distance' / 'rouge1')")
+
+
+def aggregate_similarity(
+    student_chunk: str,
+    teacher_rollouts: list[str],
+    metric: str = "edit_distance",
+) -> float:
+    """式(3):k_sem = Σ_{i=1}^{N} φ(y_c, ŷ_c^{(i)}),chunk 的语义匹配计数。
+
+    学生 chunk 与 N 个 teacher rollout 逐一算 φ 后求和。φ 连续,故 k_sem 是
+    [0, N] 上的实数——"软计数":k_sem≈N 意为学生这段与 teacher 高度一致,
+    k_sem≈0 意为 teacher 从不这么写。它是 π̂(式5)里唯一的外部 teacher 信号。
+
+    参数:
+        student_chunk: 学生 chunk 文本(C 个 token 解码所得)。
+        teacher_rollouts: N 段 teacher 续写文本,与学生 chunk 共享同一前缀
+            y_<c(对应关系由"同一前缀现场生成"保证,无需搜索匹配,docs/04 §1)。
+        metric: 传给 phi,默认 "edit_distance"。
+
+    返回:
+        k_sem ∈ [0, N],N = len(teacher_rollouts)。空列表得 0.0(空和)。
+    """
+    return sum(phi(student_chunk, r, metric) for r in teacher_rollouts)
diff --git a/tests/test_similarity.py b/tests/test_similarity.py
new file mode 100644
index 0000000..ee07f25
--- /dev/null
+++ b/tests/test_similarity.py
@@ -0,0 +1,142 @@
+"""similarity.py 单测——docs/04 §5.1:手构字符串钉死 φ 与 k_sem(式3)。
+
+全部本地 CPU、纯标准库,不依赖 torch/transformers。
+关键手算用例在各测试的注释里逐步展开,方便对着验算。
+"""
+
+import math
+
+import pytest
+
+from ars_opd.similarity import aggregate_similarity, edit_similarity, phi, rouge1
+
+# ---------------------------------------------------------------- rouge1
+
+
+def test_rouge1_identical_is_exact_one():
+    # 全同串必须精确 = 1.0(参考实现因分母 +1e-8 只能得 ≈0.99999998)
+    s = "so x = 5 and y = 12"
+    assert rouge1(s, s) == 1.0
+
+
+def test_rouge1_disjoint_is_zero():
+    assert rouge1("a b c", "x y z") == 0.0
+
+
+def test_rouge1_partial_overlap_hand_computed():
+    # hyp = {a, b, c}, ref = {a, b, d}:overlap = 2
+    # precision = 2/3, recall = 2/3, F1 = 2·(2/3)(2/3) / (4/3) = 2/3
+    assert math.isclose(rouge1("a b c", "a b d"), 2 / 3)
+
+
+def test_rouge1_multiset_counts_repeats():
+    # 多重集语义:hyp = [x,x,x,x], ref = [x] → overlap = min(4,1) = 1
+    # precision = 1/4, recall = 1/1, F1 = 2·(1/4)/(5/4) = 0.4
+    # (参考实现的 set 版会给满分 1.0——数学文本重复词多,这是关键失真点)
+    assert math.isclose(rouge1("x x x x", "x"), 0.4)
+
+
+def test_rouge1_is_bag_of_words_order_blind():
+    # 词袋:只看用了哪些词,不看顺序
+    assert rouge1("a b", "b a") == 1.0
+
+
+def test_rouge1_empty_sides():
+    assert rouge1("", "a b") == 0.0
+    assert rouge1("a b", "") == 0.0
+    assert rouge1("", "") == 0.0
+    assert rouge1("   ", "a") == 0.0  # 纯空白 split 后无词
+
+
+# ---------------------------------------------------------- edit_similarity
+
+
+def test_edit_identical_is_one():
+    s = "so x = 5 and y = 12"
+    assert edit_similarity(s, s) == 1.0
+
+
+def test_edit_totally_different_is_zero():
+    # ["a","b"] vs ["c","d"]:2 次替换,dist=2, max(m,n)=2 → 1 − 1 = 0
+    assert edit_similarity("a b", "c d") == 0.0
+
+
+def test_edit_single_substitution_hand_computed():
+    # ["a","b","c"] vs ["a","x","c"]:1 次替换,dist=1, max=3 → 2/3
+    assert math.isclose(edit_similarity("a b c", "a x c"), 2 / 3)
+
+
+def test_edit_insertion_hand_computed():
+    # ["a","b"] vs ["a","x","b"]:1 次插入,dist=1, max=3 → 2/3
+    assert math.isclose(edit_similarity("a b", "a x b"), 2 / 3)
+
+
+def test_edit_is_order_sensitive():
+    # ["a","b"] vs ["b","a"]:两次替换 dist=2 → 0.0;与 rouge1 的 1.0 互补
+    assert edit_similarity("a b", "b a") == 0.0
+    assert rouge1("a b", "b a") == 1.0
+
+
+def test_edit_empty_sides():
+    assert edit_similarity("", "") == 1.0  # 零距离
+    assert edit_similarity("a b", "") == 0.0  # 全删
+    assert edit_similarity("", "a b") == 0.0  # 全插
+
+
+def test_edit_asymmetric_lengths():
+    # ["a"] vs ["a","b","c","d"]:3 次插入,dist=3, max=4 → 1/4
+    assert math.isclose(edit_similarity("a", "a b c d"), 1 / 4)
+
+
+# --------------------------------------------------------------------- phi
+
+
+def test_phi_default_is_edit_distance():
+    # 论文 §5.1 默认;"a b" vs "b a" 恰能区分两度量(edit=0, rouge1=1)
+    assert phi("a b", "b a") == edit_similarity("a b", "b a") == 0.0
+
+
+def test_phi_dispatch():
+    h, r = "a b c", "a b d"
+    assert phi(h, r, metric="rouge1") == rouge1(h, r)
+    assert phi(h, r, metric="edit_distance") == edit_similarity(h, r)
+
+
+def test_phi_unknown_metric_raises():
+    with pytest.raises(ValueError, match="bleu"):
+        phi("a", "a", metric="bleu")
+
+
+# ----------------------------------------------------- aggregate_similarity
+
+
+def test_aggregate_is_sum_of_phi():
+    # 式(3) 手算:rollouts 与 "a b c" 的 edit 相似度分别为 1.0, 2/3, 0.0
+    chunk = "a b c"
+    rollouts = ["a b c", "a x c", "x y z"]
+    expected = 1.0 + 2 / 3 + 0.0
+    assert math.isclose(aggregate_similarity(chunk, rollouts), expected)
+
+
+def test_aggregate_bounds():
+    # k_sem ∈ [0, N]:全同 → N,全不同 → 0
+    n = 5
+    assert aggregate_similarity("a b", ["a b"] * n) == float(n)
+    assert aggregate_similarity("a b", ["x y"] * n) == 0.0
+
+
+def test_aggregate_is_continuous_soft_count():
+    # φ 连续 ⇒ k_sem 非整数是常态(区别于 token 精确匹配的硬计数)
+    k = aggregate_similarity("a b c", ["a b c", "a x c"])
+    assert 1.0 < k < 2.0
+
+
+def test_aggregate_empty_rollouts():
+    assert aggregate_similarity("a b", []) == 0.0
+
+
+def test_aggregate_metric_passthrough():
+    # "a b" vs "b a":edit 全零,rouge1 全满——验证 metric 真的传下去了
+    chunk, rollouts = "a b", ["b a", "b a"]
+    assert aggregate_similarity(chunk, rollouts, metric="edit_distance") == 0.0
+    assert aggregate_similarity(chunk, rollouts, metric="rouge1") == 2.0