diff --git a/CHANGELOG.d/2.12.24-leftover-map-cosine.md b/CHANGELOG.d/2.12.24-leftover-map-cosine.md
new file mode 100644
index 000000000..3856da7a3
--- /dev/null
+++ b/CHANGELOG.d/2.12.24-leftover-map-cosine.md
@@ -0,0 +1,7 @@
+## 2.12.24 — Leftover-map cosine
+
+- Persist leftover-map cosine on leftover post–criterion pairs
+  (ADR 0180). After `make seed`, closest and farthest leftover pairs
+  sit above the member list with cosine next to leftover-map distance
+  `d`; click opens that post. Omit the badge when cosine is missing.
+  Never invent a leftover score.
diff --git a/CHANGELOG.md b/CHANGELOG.md
index 6e8b45a1c..c20009878 100644
--- a/CHANGELOG.md
+++ b/CHANGELOG.md
@@ -253,6 +253,15 @@ All notable changes to this project are documented here. Format follows
   (Gabriel, 1971; Jeon et al., 2021, eq. 3; ADR 0179). A missing
   inner product omits the badge rather than inventing a leftover score.
 
+## [2.12.24] - 2026-08-24
+
+### Added
+
+- Period leftover pair rows now name leftover-map cosine next to
+  leftover-map distance `d`, then open that post (Gabriel, 1971;
+  Jeon et al., 2021, eq. 3; ADR 0180). A missing cosine omits the
+  badge rather than inventing leftover-map alignment.
+
 ## [2.12.6] - 2026-08-20
 
 ### Added
diff --git a/docs/adr/0048-persist-lsirm-leftover-pairs.md b/docs/adr/0048-persist-lsirm-leftover-pairs.md
index b1db26002..2b1ab699b 100644
--- a/docs/adr/0048-persist-lsirm-leftover-pairs.md
+++ b/docs/adr/0048-persist-lsirm-leftover-pairs.md
@@ -43,7 +43,8 @@ the IRT matrix is unusable. A rank-0 residual still emits a
 stable pair so `make seed` is not empty; the stored distance is
 then zero, not a fabricated interaction.
 
-The UI contract is ADR 0049.
+The UI contract is ADR 0049. Leftover-map cosine on those pair rows
+is [ADR 0180](0180-leftover-map-cosine.md).
 
 ## Consequences
 
diff --git a/docs/adr/0180-leftover-map-cosine.md b/docs/adr/0180-leftover-map-cosine.md
new file mode 100644
index 000000000..9c7cdbb8d
--- /dev/null
+++ b/docs/adr/0180-leftover-map-cosine.md
@@ -0,0 +1,78 @@
+# ADR 0180 — Name leftover-map cosine on period-report pair rows
+
+**Decision status:** Accepted
+**Date:** 2026-08-24
+
+Amends [ADR 0048](0048-persist-lsirm-leftover-pairs.md) and
+[ADR 0049](0049-leftover-pair-report-ui.md).
+
+## Context
+
+ADR 0048 already persists leftover-map distance `d = ‖ξ_p − ζ_i‖` and
+leftover residual `R = Y − E[Y|θ, item]` on `report_leftover_pair`.
+ADR 0049 already renders closest and farthest pairs above the member
+list and opens the named post. Distance is the Jeon et al. (2021,
+eq. 3) map gap. Gabriel (1971) also names leftover-map alignment as
+the cosine of the angle between `ξ_p` and `ζ_i`. Hiding that cosine
+lets a buyer read a large reconstructed leftover cell, or a close
+map pair, as leftover-map alignment without a scale-free value.
+
+This increment does not persist leftover-map coordinates, does not
+name leftover-map inner product, does not name observed `Y` / expected
+`E`, does not name leftover-map rank, does not split leftover-map
+distance onto two axes, and does not land Post quality on the leftover
+criterion.
+
+The unprotected-stack reconstructions for neighbouring leftover facts
+use 0162–0179. This protected-main increment uses **0180** so it does
+not collide with leftover-map inner product (0179), leftover residual
+disclosure (0178), leftover observed `Y` / expected `E` (0177),
+leftover-map rank (0172), two-axis leftover-map distance (0166),
+leftover coverage (0168), leftover-map axis share (0148), or leftover
+interaction-map persistence (0121).
+
+## Decision
+
+Each leftover pair names `leftover_map_cosine` — the cosine of the
+angle between leftover-map person and item coordinates that produced
+leftover-map distance `d`. Migration `0180` is the single source of
+the column on every install path, fresh or existing -- shipped
+migrations (`0001` / `0012`) are never edited after the fact. The
+column is nullable so older leftover rows keep distance and residual
+without fabricating alignment. Fallback pairs that have no
+complete-case leftover map, and pairs whose person or item vector
+sits at the origin, omit the value rather than inventing one.
+
+The pair button shows `cos {signed}` next to leftover-map distance
+`d` when the value is finite. Next action: leftover-map cosine names
+leftover-map alignment independent of distance; open this post to
+read the named criterion. A missing or non-finite cosine omits the
+badge and keeps the existing closest/farthest next action. Do not
+invent a leftover score. Do not invent a theta.
+
+## Consequences
+
+`GET /api/reports/{grouping}/{period}` returns `leftover_map_cosine`.
+After `make seed`, closest and farthest leftover pairs sit above the
+member list with named cosine next to `d`; click opens that post.
+Hidden posts stay hidden.
+
+## Related
+
+Independent of leftover interaction-map persistence, leftover-criterion
+evaluation landing, leftover residual disclosure, leftover observed
+`Y` / expected `E`, leftover-map complete-case coverage, leftover-map
+axis share, leftover pairs on the grouping comparison strip, two-axis
+leftover-map distance, leftover-map rank, and leftover-map inner
+product.
+
+## References
+
+Gabriel, K. R. (1971). The biplot graphic display of matrices with
+application to principal component analysis. *Biometrika, 58*(3),
+453–467. https://doi.org/10.1093/biomet/58.3.453
+
+Jeon, M., Jin, I. H., Schweinberger, M., & Baugh, S. (2021). Mapping
+unobserved item–respondent interactions: A latent space item response
+model with interaction map. *Psychometrika, 86*(2), 378–403.
+https://doi.org/10.1007/s11336-021-09762-5
diff --git a/frontend/src/i18n.test.ts b/frontend/src/i18n.test.ts
index a92b976f5..e29a86d67 100644
--- a/frontend/src/i18n.test.ts
+++ b/frontend/src/i18n.test.ts
@@ -270,6 +270,33 @@ describe("i18n", () => {
     expect(document.documentElement.lang).toBe(locale);
   });
 
+  it.each([
+    [
+      "ko",
+      "잔여 지도 코사인 +0.95이(가) 거리와 무관한 잔여 지도 정렬을 이름합니다. sales-lead 기준을 읽으려면 이 글을 여세요.",
+    ],
+    [
+      "zh",
+      "残差图余弦 +0.95 命名与距离无关的残差图对齐。打开这篇帖子阅读 sales-lead。",
+    ],
+    [
+      "ja",
+      "残差マップ余弦 +0.95 は距離に依存しない残差マップの向きを示します。この投稿を開いて sales-lead を読んでください。",
+    ],
+    [
+      "vi",
+      "Cosin bản đồ phần dư +0.95 đặt tên sự thẳng hàng độc lập với khoảng cách. Mở bài viết này để đọc sales-lead.",
+    ],
+  ] as const)("formats leftover-map cosine next action in %s", (locale, expected) => {
+    setLocale(locale);
+    expect(
+      tf(
+        "Leftover-map cosine {value} names leftover-map alignment independent of distance. Open this post to read {criterion}.",
+        { value: "+0.95", criterion: "sales-lead" },
+      ),
+    ).toBe(expected);
+  });
+
   it.each([
     ["ko", "DEMO은(는) 이벤트 계보의 현재 항목입니다. 다음으로 Keyman과 평가를 읽으세요."],
     ["zh", "DEMO 是事件谱系中的当前记录。接下来查看关键联系人和评估。"],
diff --git a/frontend/src/i18n.ts b/frontend/src/i18n.ts
index a320f8e9a..8767adac1 100644
--- a/frontend/src/i18n.ts
+++ b/frontend/src/i18n.ts
@@ -644,6 +644,8 @@ const TRANSLATIONS: Partial<Record<Locale, Record<string, string>>> = {
       "잔여 맵 내적 ξ·ζ {value}가 IRT 주효과 이후 잔여를 재구성합니다. {criterion}을(를) 읽으려면 이 글을 여세요.",
     "Open leftover {kind} pair: {title} · {criterion}":
       "잔여 {kind} 쌍 열기: {title} · {criterion}",
+    "Leftover-map cosine {value} names leftover-map alignment independent of distance. Open this post to read {criterion}.":
+      "잔여 지도 코사인 {value}이(가) 거리와 무관한 잔여 지도 정렬을 이름합니다. {criterion} 기준을 읽으려면 이 글을 여세요.",
   },
   zh: {
     "Unknown": "未知",
@@ -1198,6 +1200,8 @@ const TRANSLATIONS: Partial<Record<Locale, Record<string, string>>> = {
       "残余图内积 ξ·ζ {value} 重建了 IRT 主效应后的残余。打开这篇帖子以阅读 {criterion}。",
     "Open leftover {kind} pair: {title} · {criterion}":
       "打开残余{kind}对：{title} · {criterion}",
+    "Leftover-map cosine {value} names leftover-map alignment independent of distance. Open this post to read {criterion}.":
+      "残差图余弦 {value} 命名与距离无关的残差图对齐。打开这篇帖子阅读 {criterion}。",
   },
   ja: {
     "Unknown": "不明",
@@ -1752,6 +1756,8 @@ const TRANSLATIONS: Partial<Record<Locale, Record<string, string>>> = {
       "残差マップ内積 ξ·ζ {value} が IRT 主効果後の残差を再構成します。{criterion} を読むにはこの投稿を開いてください。",
     "Open leftover {kind} pair: {title} · {criterion}":
       "残差{kind}ペアを開く: {title} · {criterion}",
+    "Leftover-map cosine {value} names leftover-map alignment independent of distance. Open this post to read {criterion}.":
+      "残差マップ余弦 {value} は距離に依存しない残差マップの向きを示します。この投稿を開いて {criterion} を読んでください。",
   },
   vi: {
     "Unknown": "Không rõ",
@@ -2306,6 +2312,8 @@ const TRANSLATIONS: Partial<Record<Locale, Record<string, string>>> = {
       "Tích trong bản đồ phần dư ξ·ζ {value} tái tạo phần dư sau hiệu ứng chính IRT. Mở bài viết này để đọc {criterion}.",
     "Open leftover {kind} pair: {title} · {criterion}":
       "Mở cặp phần dư {kind}: {title} · {criterion}",
+    "Leftover-map cosine {value} names leftover-map alignment independent of distance. Open this post to read {criterion}.":
+      "Cosin bản đồ phần dư {value} đặt tên sự thẳng hàng độc lập với khoảng cách. Mở bài viết này để đọc {criterion}.",
   },
 };
 
diff --git a/frontend/src/leftoverMapCosine.test.ts b/frontend/src/leftoverMapCosine.test.ts
new file mode 100644
index 000000000..b38c88bfc
--- /dev/null
+++ b/frontend/src/leftoverMapCosine.test.ts
@@ -0,0 +1,19 @@
+import { describe, expect, it } from "vitest";
+import { formatLeftoverMapCosine, formatSignedLeftoverValue } from "./leftoverMapCosine";
+
+describe("formatLeftoverMapCosine", () => {
+  it("names leftover-map cosine without inventing a leftover score", () => {
+    expect(formatLeftoverMapCosine(0.95)).toBe("cos +0.95");
+    expect(formatLeftoverMapCosine(-1)).toBe("cos −1.00");
+    expect(formatLeftoverMapCosine(0)).toBe("cos 0.00");
+    expect(formatSignedLeftoverValue(0.95)).toBe("+0.95");
+    expect(formatSignedLeftoverValue(-1)).toBe("−1.00");
+  });
+
+  it("omits the badge when cosine is missing or non-finite", () => {
+    expect(formatLeftoverMapCosine(null)).toBeNull();
+    expect(formatLeftoverMapCosine(undefined)).toBeNull();
+    expect(formatLeftoverMapCosine(Number.NaN)).toBeNull();
+    expect(formatLeftoverMapCosine(Number.POSITIVE_INFINITY)).toBeNull();
+  });
+});
diff --git a/frontend/src/leftoverMapCosine.ts b/frontend/src/leftoverMapCosine.ts
new file mode 100644
index 000000000..4683f8e92
--- /dev/null
+++ b/frontend/src/leftoverMapCosine.ts
@@ -0,0 +1,26 @@
+/** Gabriel leftover-map cosine after IRT main effects. */
+
+export const LEFTOVER_MAP_COSINE_ACTION =
+  "Leftover-map cosine {value} names leftover-map alignment independent of distance. Open this post to read {criterion}.";
+
+export function formatSignedLeftoverValue(value: number): string | null {
+  if (!Number.isFinite(value)) {
+    return null;
+  }
+  const magnitude = Math.abs(value).toFixed(2);
+  if (value > 0) {
+    return `+${magnitude}`;
+  }
+  if (value < 0) {
+    return `−${magnitude}`;
+  }
+  return magnitude;
+}
+
+export function formatLeftoverMapCosine(value: number | null | undefined): string | null {
+  if (value == null) {
+    return null;
+  }
+  const signed = formatSignedLeftoverValue(value);
+  return signed === null ? null : `cos ${signed}`;
+}
diff --git a/lineageweave/period_report.py b/lineageweave/period_report.py
index de79478b9..17b82e15e 100644
--- a/lineageweave/period_report.py
+++ b/lineageweave/period_report.py
@@ -252,7 +252,7 @@ def leftover_pairs_for_fit(
     theta: np.ndarray,
     fit: PolytomousFit,
 ) -> tuple[LeftoverPair, ...]:
-    """Leftover pairs from the already-fitted GRM/GPCM main effects."""
+    """Leftover pairs from the already-fitted GRM/GPCM main effects, with cosine."""
     probs = _category_probabilities(model, theta, fit)
     expected = expected_category_matrix(matrix, probs)
     return leftover_pairs_from_residual(post_ids, item_codes, matrix, expected)
diff --git a/migrations/0180_report_leftover_map_cosine.sql b/migrations/0180_report_leftover_map_cosine.sql
new file mode 100644
index 000000000..74bad32ea
--- /dev/null
+++ b/migrations/0180_report_leftover_map_cosine.sql
@@ -0,0 +1,8 @@
+-- ADR 0180: persist leftover-map cosine on leftover post–criterion
+-- pairs. Distance stays Euclidean leftover-map d. Upgrade column is
+-- nullable so older leftover rows keep distance and residual without
+-- fabricating alignment. This migration is the single source of the
+-- column on fresh and existing installations.
+
+alter table report_leftover_pair
+    add column if not exists leftover_map_cosine numeric;
diff --git a/migrations/rollback/0180_report_leftover_map_cosine.sql b/migrations/rollback/0180_report_leftover_map_cosine.sql
new file mode 100644
index 000000000..fb0a86fb6
--- /dev/null
+++ b/migrations/rollback/0180_report_leftover_map_cosine.sql
@@ -0,0 +1,4 @@
+-- Reverse 0180. Leftover distance and residual stay on the pair row.
+
+alter table report_leftover_pair
+    drop column if exists leftover_map_cosine;