diff --git a/CHANGELOG.d/2.12.25-leftover-map-length.md b/CHANGELOG.d/2.12.25-leftover-map-length.md
new file mode 100644
index 000000000..62842aab4
--- /dev/null
+++ b/CHANGELOG.d/2.12.25-leftover-map-length.md
@@ -0,0 +1,8 @@
+## 2.12.25 — Leftover-map length
+
+- Persist leftover-map lengths `‖ξ‖` and `‖ζ‖` on leftover
+  post–criterion pairs (ADR 0181). After `make seed`, closest and
+  farthest leftover pairs sit above the member list with `‖ξ‖` and
+  `‖ζ‖` next to leftover-map distance `d`; click opens that post.
+  Omit the badges when lengths are missing. Never invent a leftover
+  score.
diff --git a/CHANGELOG.md b/CHANGELOG.md
index c20009878..76a792679 100644
--- a/CHANGELOG.md
+++ b/CHANGELOG.md
@@ -262,6 +262,17 @@ All notable changes to this project are documented here. Format follows
   Jeon et al., 2021, eq. 3; ADR 0180). A missing cosine omits the
   badge rather than inventing leftover-map alignment.
 
+## [2.12.25] - 2026-08-24
+
+### Added
+
+- Period leftover pairs now name leftover-map lengths `‖ξ‖` and `‖ζ‖`
+  next to leftover-map distance `d`. After `make seed`, closest and
+  farthest leftover pairs sit above the member list with named
+  leftover-map magnitude; click opens that post (Gabriel, 1971;
+  Jeon et al., 2021, eq. 3; ADR 0181). A missing length omits the
+  badge. Never invent a leftover score.
+
 ## [2.12.6] - 2026-08-20
 
 ### Added
diff --git a/docs/adr/0049-leftover-pair-report-ui.md b/docs/adr/0049-leftover-pair-report-ui.md
index f2b6f905b..642fea7aa 100644
--- a/docs/adr/0049-leftover-pair-report-ui.md
+++ b/docs/adr/0049-leftover-pair-report-ui.md
@@ -32,6 +32,9 @@ criterion.
 
 Missing leftover rows render nothing — never a placeholder pair.
 A hidden post never appears as a leftover pair.
+A missing leftover-map length omits the `‖ξ‖` / `‖ζ‖` badges and
+keeps the closest/farthest next action. See
+[ADR 0181](0181-leftover-map-length.md).
 
 ## Consequences
 
diff --git a/docs/adr/0181-leftover-map-length.md b/docs/adr/0181-leftover-map-length.md
new file mode 100644
index 000000000..b81f2f6a2
--- /dev/null
+++ b/docs/adr/0181-leftover-map-length.md
@@ -0,0 +1,85 @@
+# ADR 0181 — Name leftover-map length 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 vector lengths
+`‖ξ‖` and `‖ζ‖`. Hiding those lengths lets a buyer read a close
+leftover-map pair as leftover-map aligned magnitude, or a distant pair
+as large leftover-map displacement, without the polar magnitude.
+
+A close leftover-map pair can sit at the origin (`‖ξ‖ = ‖ζ‖ = 0`) or
+share a non-origin position (`d = 0` with positive length). Those are
+different leftover-map facts.
+
+This increment does not persist leftover-map coordinates, does not name
+observed `Y` / expected `E`, does not name leftover-map cosine, does
+not name leftover-map inner product, 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 0121–0180. This protected-main increment uses **0181** so it does
+not collide with leftover-map cosine (0180), 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_person_length` and
+`leftover_map_item_length` — the Euclidean lengths `‖ξ‖` and `‖ζ‖` of
+the leftover-map person and item coordinates that produced leftover-map
+distance `d`. Migration `0181` is the single source of the columns on
+every install path, fresh or existing -- shipped migrations (`0001` /
+`0012`) are never edited after the fact. The columns are nullable so
+older leftover rows keep distance and residual without fabricating
+lengths. Fallback pairs that have no complete-case leftover map omit
+the values rather than inventing them. Origin coordinates persist
+length `0` because that is the measured leftover-map magnitude.
+
+The pair button shows `‖ξ‖ {person}` and `‖ζ‖ {item}` next to
+leftover-map distance `d` when both values are finite. Next action:
+leftover-map length names leftover-map magnitude independently of
+leftover-map distance; open this post to read the named criterion. A
+missing or non-finite length omits the badges 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_person_length` and `leftover_map_item_length`. After
+`make seed`, closest and farthest leftover pairs sit above the member
+list with named `‖ξ‖` and `‖ζ‖` 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, leftover-map inner product,
+and leftover-map cosine.
+
+## 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/leftoverMapLength.test.ts b/frontend/src/leftoverMapLength.test.ts
new file mode 100644
index 000000000..0b7277629
--- /dev/null
+++ b/frontend/src/leftoverMapLength.test.ts
@@ -0,0 +1,24 @@
+import { describe, expect, it } from "vitest";
+import {
+  formatLeftoverMapItemLength,
+  formatLeftoverMapLength,
+  formatLeftoverMapPersonLength,
+} from "./leftoverMapLength";
+
+describe("formatLeftoverMapLength", () => {
+  it("names leftover-map length without inventing a leftover score", () => {
+    expect(formatLeftoverMapPersonLength(0)).toBe("‖ξ‖ 0.00");
+    expect(formatLeftoverMapItemLength(1.25)).toBe("‖ζ‖ 1.25");
+    expect(formatLeftoverMapLength(0.4)).toBe("0.40");
+    expect(formatLeftoverMapPersonLength(0.4)).toBe("‖ξ‖ 0.40");
+    expect(formatLeftoverMapItemLength(0.9)).toBe("‖ζ‖ 0.90");
+  });
+
+  it("omits the badge when length is missing, negative, or non-finite", () => {
+    expect(formatLeftoverMapPersonLength(null)).toBeNull();
+    expect(formatLeftoverMapItemLength(undefined)).toBeNull();
+    expect(formatLeftoverMapLength(Number.NaN)).toBeNull();
+    expect(formatLeftoverMapLength(Number.POSITIVE_INFINITY)).toBeNull();
+    expect(formatLeftoverMapLength(-0.01)).toBeNull();
+  });
+});
diff --git a/frontend/src/leftoverMapLength.ts b/frontend/src/leftoverMapLength.ts
new file mode 100644
index 000000000..bb1f5c426
--- /dev/null
+++ b/frontend/src/leftoverMapLength.ts
@@ -0,0 +1,21 @@
+/** Gabriel leftover-map lengths ``‖ξ‖`` and ``‖ζ‖`` after IRT main effects. */
+
+export const LEFTOVER_MAP_LENGTH_ACTION =
+  "Leftover-map length ‖ξ‖ {person} and ‖ζ‖ {item} names leftover-map magnitude independently of leftover-map distance. Open this post to read {criterion}.";
+
+export function formatLeftoverMapLength(value: number | null | undefined): string | null {
+  if (value == null || !Number.isFinite(value) || value < 0) {
+    return null;
+  }
+  return value.toFixed(2);
+}
+
+export function formatLeftoverMapPersonLength(value: number | null | undefined): string | null {
+  const formatted = formatLeftoverMapLength(value);
+  return formatted === null ? null : `‖ξ‖ ${formatted}`;
+}
+
+export function formatLeftoverMapItemLength(value: number | null | undefined): string | null {
+  const formatted = formatLeftoverMapLength(value);
+  return formatted === null ? null : `‖ζ‖ ${formatted}`;
+}
diff --git a/migrations/0181_report_leftover_map_length.sql b/migrations/0181_report_leftover_map_length.sql
new file mode 100644
index 000000000..973ee68ed
--- /dev/null
+++ b/migrations/0181_report_leftover_map_length.sql
@@ -0,0 +1,25 @@
+-- ADR 0181: persist leftover-map lengths ‖ξ‖ and ‖ζ‖ on leftover
+-- post–criterion pairs. Distance stays Euclidean leftover-map d.
+-- Upgrade columns are nullable so older leftover rows keep distance
+-- and residual without fabricating lengths. This migration is the
+-- single source of the columns on fresh and existing installations.
+
+alter table report_leftover_pair
+    add column if not exists leftover_map_person_length numeric,
+    add column if not exists leftover_map_item_length numeric;
+
+do $$
+begin
+    if not exists (
+        select 1
+        from pg_constraint
+        where conname = 'report_leftover_pair_map_length_nonnegative'
+    ) then
+        alter table report_leftover_pair
+            add constraint report_leftover_pair_map_length_nonnegative
+            check (
+                (leftover_map_person_length is null or leftover_map_person_length >= 0)
+                and (leftover_map_item_length is null or leftover_map_item_length >= 0)
+            );
+    end if;
+end $$;
diff --git a/migrations/rollback/0181_report_leftover_map_length.sql b/migrations/rollback/0181_report_leftover_map_length.sql
new file mode 100644
index 000000000..add379776
--- /dev/null
+++ b/migrations/rollback/0181_report_leftover_map_length.sql
@@ -0,0 +1,8 @@
+-- Reverse 0181. Leftover distance and residual stay on the pair row.
+
+alter table report_leftover_pair
+    drop constraint if exists report_leftover_pair_map_length_nonnegative;
+
+alter table report_leftover_pair
+    drop column if exists leftover_map_person_length,
+    drop column if exists leftover_map_item_length;
diff --git a/tests/test_period_report.py b/tests/test_period_report.py
index 0f7ad763a..4cdbd9071 100644
--- a/tests/test_period_report.py
+++ b/tests/test_period_report.py
@@ -225,6 +225,10 @@ def test_leftover_residual_biplot_separates_aligned_and_opposed_cells() -> None:
     }
     assert farthest.leftover_residual == pytest.approx(-2.0)
     assert farthest.leftover_distance == pytest.approx(2.0 * np.sqrt(2.0), rel=1e-6)
+    assert closest.leftover_map_person_length == pytest.approx(0.0, abs=1e-9)
+    assert closest.leftover_map_item_length == pytest.approx(0.0, abs=1e-9)
+    assert farthest.leftover_map_person_length == pytest.approx(np.sqrt(2.0), rel=1e-6)
+    assert farthest.leftover_map_item_length == pytest.approx(np.sqrt(2.0), rel=1e-6)
 
 
 def test_leftover_missing_cells_are_not_treated_as_zero() -> None: