diff --git a/CHANGELOG.d/2.79.0-leftover-map-axis-singular.md b/CHANGELOG.d/2.79.0-leftover-map-axis-singular.md
new file mode 100644
index 000000000..d83609121
--- /dev/null
+++ b/CHANGELOG.d/2.79.0-leftover-map-axis-singular.md
@@ -0,0 +1,17 @@
+## 2.79.0 — Leftover-map singular values on leftover-axis report badges
+
+- Caption leftover-axis report badges with persisted leftover-map singular
+  values `σ_k` (ADR 0322). After `make seed`, leftover-axis badges name
+  persisted leftover-map singular values when finite together with persisted
+  leftover-map axis share; click a leftover pair opens that post. A missing,
+  non-finite, or negative singular value omits that `σ` badge and keeps
+  `leftover axis {k} {share}%`. Rank-0 unused axes still name `σ 0.00`.
+  Pair-list leftover-axis badges stay `leftover axis {k} σ {value} {share}%`
+  so they stay distinct from hyphen `leftover-map axis {k} σ {value} ({share}%)`
+  and from comparison copy
+  `leftover map comparison graphic leftover-map axis {k} σ {value} ({share}%)`.
+  Do not invent `σ_k` from leftover-map axis share. This increment does not
+  caption leftover-map singular values on leftover-axis badges on the grouping
+  comparison strip with a distinct name. A finite negative leftover is shown,
+  never clamped. Never invent a leftover score. Never invent a theta. No new
+  columns.
diff --git a/docs/adr/0322-leftover-map-axis-singular.md b/docs/adr/0322-leftover-map-axis-singular.md
new file mode 100644
index 000000000..f1c259954
--- /dev/null
+++ b/docs/adr/0322-leftover-map-axis-singular.md
@@ -0,0 +1,98 @@
+# ADR 0322 — Name leftover-map singular values on leftover-axis report badges
+
+**Decision status:** Proposed
+**Date:** 2026-08-31
+
+Amends leftover-map axis share persistence
+([ADR 0148](0148-leftover-map-axis-share.md)). Independent of leftover-map
+singular values on the grouping comparison leftover-map graphic
+([ADR 0321](0321-leftover-map-compare-plot-singular.md)), leftover-map
+coordinate ticks on that comparison graphic
+([ADR 0320](0320-leftover-map-compare-plot-ticks.md)), leftover-map
+axis share on that comparison graphic
+([ADR 0305](0305-leftover-map-compare-plot-axis-share.md)), leftover-map
+graphic display on the grouping comparison strip
+([ADR 0304](0304-leftover-map-compare-graphic.md)), leftover-map
+coordinates ([ADR 0267](0267-leftover-map-coordinates.md)), leftover-map
+graphic display ([ADR 0268](0268-leftover-map-graphic-display.md)), and
+leftover-map axis share on the graphic display
+([ADR 0269](0269-leftover-map-axis-share-plot.md)).
+
+## Context
+
+ADR 0148 already persists leftover-map singular values `σ_k` and leftover-map
+axis share `σ_k² / Σ_j σ_j²` and captions leftover-axis report badges with
+share only. ADR 0321 already captions the grouping comparison leftover-map
+graphic with persisted `σ_k`. Those leftover-axis badges still hide `σ_k`, so
+Gabriel (1971) singular values appear on the comparison graphic and never on
+the report-level leftover-axis badges that name the same axes. Hiding `σ_k` on
+the badges lets leftover-map axis share be read as leftover-map structure even
+when the buyer cannot tell whether axis 1 is large because `σ_1` is large or
+because axis 2 collapsed. A missing, non-finite, or negative singular value is
+not a leftover score and must omit independently of leftover-map axis share.
+Rank-0 unused axes still persist `σ_k = 0`. A finite negative leftover is
+shown, never clamped. Do not invent `σ_k` from leftover-map axis share.
+
+This increment captions leftover-axis report badges with already persisted
+leftover-map singular values. Pair-list leftover-axis badges stay
+`leftover axis {k} σ {value} {share}%` so they stay distinct from hyphen
+`leftover-map axis {k} σ {value} ({share}%)` and from comparison copy
+`leftover map comparison graphic leftover-map axis {k} σ {value} ({share}%)`.
+It does not add columns. Do not invent a leftover score. Do not invent a
+theta.
+
+This protected increment uses **0322** so it does not collide with leftover-map
+singular values on the grouping comparison leftover-map graphic (0321),
+leftover-map coordinate ticks on that graphic (0320), leftover-map axis share
+on that graphic (0305), leftover-map graphic display on the grouping
+comparison strip (0304), leftover-map axis share persistence (0148), or the
+dashboard stacks.
+
+## Decision
+
+On leftover-axis report badges, caption leftover-map axis `k` with persisted
+leftover-map singular value `σ_k` when that singular value is finite and
+non-negative, including rank-0 zero singular values. Pair-list labels stay
+distinct from graphic labels: leftover-axis badges stay
+`leftover axis {k} σ {value} {share}%`; leftover-map graphic axes stay
+`leftover-map axis {k} ({share}%)` until a later increment captions `σ_k`
+there; grouping comparison leftover-map graphic axes stay
+`leftover map comparison graphic leftover-map axis {k} σ {value} ({share}%)`.
+A missing, non-finite, or negative singular value omits that `σ` badge and
+keeps the existing `leftover axis {k} {share}%` text. Do not invent `σ_k`
+from leftover-map axis share. Axis 1 and axis 2 stay independently named.
+Click a leftover pair to open that post. The grouping comparison strip
+(ADR 0149) stays on its reduced leftover payload and does not gain leftover-axis
+report badges with a distinct comparison-strip name.
+
+Do not add SQL. Do not edit shipped migrations. Do not persist inner
+product, cosine, or length as separate columns. Do not invent a leftover
+score. Do not invent a theta.
+
+## Consequences
+
+After `make seed`, leftover-axis report badges name persisted leftover-map
+singular values when finite together with persisted leftover-map axis share;
+click a leftover pair opens that post. Hidden posts stay hidden. Rank-0 unused
+axes still name `σ 0.00`. Grouping comparison leftover-map graphic leftover-map
+singular values (ADR 0321) remain.
+
+## Related
+
+Independent of leftover-map singular values on the grouping comparison
+leftover-map graphic. This increment does not caption leftover-map singular
+values on leftover-axis badges on the grouping comparison strip with a
+distinct name.
+
+## 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
+(LSIRM interaction `−γ‖ξ_j − ζ_i‖` after main effects
+`α_j − β_i`; typically `p = 2` for the interaction map.)
diff --git a/frontend/src/App.tsx b/frontend/src/App.tsx
index dfa30ed1b..ecd927785 100644
--- a/frontend/src/App.tsx
+++ b/frontend/src/App.tsx
@@ -137,6 +137,12 @@ import {
   LEFTOVER_MAP_PLOT_INCOMPLETE_POST,
   LEFTOVER_MAP_PLOT_ITEM_COVERAGE,
 } from "./leftoverMapCoverage";
+import {
+  leftoverMapAxisBadgeShare,
+  leftoverMapAxisBadgeSingular,
+  LEFTOVER_MAP_AXIS_BADGE_SHARE,
+  LEFTOVER_MAP_AXIS_BADGE_SINGULAR,
+} from "./leftoverMapAxisBadge";
 import {
   formatLeftoverMapReconstruction,
   LEFTOVER_MAP_COMPARE_RECONSTRUCTION_LABEL,
@@ -3923,18 +3929,25 @@ function ReportsPanel({
                 {tf(LEFTOVER_MAP_PLOT_INCOMPLETE_ITEM, incompleteItemCount)}
               </p>
             ) : null}
-            {report.leftover_map_axes?.map((axis) => (
-              <span key={axis.axis_index} className="post-badge">
-                {tf("leftover axis {axis} {share}%", {
-                  axis: axis.axis_index,
-                  share: (axis.leftover_share * 100).toFixed(0),
-                })}
-              </span>
-            ))}
+            {report.leftover_map_axes?.map((axis) => {
+              const singular = leftoverMapAxisBadgeSingular(axis);
+              const share = leftoverMapAxisBadgeShare(axis.leftover_share);
+              return (
+                <span key={axis.axis_index} className="post-badge">
+                  {singular === null
+                    ? tf(LEFTOVER_MAP_AXIS_BADGE_SHARE, { axis: axis.axis_index, share })
+                    : tf(LEFTOVER_MAP_AXIS_BADGE_SINGULAR, {
+                        axis: axis.axis_index,
+                        value: singular,
+                        share,
+                      })}
+                </span>
+              );
+            })}
             {report.leftover_map_axes && report.leftover_map_axes.length > 0 && (
               <p aria-label={t("Leftover-map axis share")}>
                 {t(
-                  "Leftover-map axis share is Gabriel inertia of residual SVD axes 1 and 2. Open a leftover pair to read the post–criterion cell. The shares do not invent a leftover score.",
+                  "Leftover-map axis share is Gabriel inertia of residual SVD axes 1 and 2. Leftover-map singular values are the Gabriel scale of those axes. Open a leftover pair to read the post–criterion cell. The shares and singular values do not invent a leftover score.",
                 )}
               </p>
             )}
diff --git a/frontend/src/leftoverMapAxisBadge.test.ts b/frontend/src/leftoverMapAxisBadge.test.ts
new file mode 100644
index 000000000..84ec57a39
--- /dev/null
+++ b/frontend/src/leftoverMapAxisBadge.test.ts
@@ -0,0 +1,65 @@
+import { describe, expect, it } from "vitest";
+import {
+  leftoverMapAxisBadgeShare,
+  leftoverMapAxisBadgeSingular,
+  LEFTOVER_MAP_AXIS_BADGE_SHARE,
+  LEFTOVER_MAP_AXIS_BADGE_SINGULAR,
+} from "./leftoverMapAxisBadge";
+import {
+  LEFTOVER_MAP_COMPARE_PLOT_AXIS_SINGULAR,
+  LEFTOVER_MAP_COMPARE_PLOT_AXIS_SINGULAR_SHARE,
+  LEFTOVER_MAP_PLOT_AXIS_SINGULAR,
+  LEFTOVER_MAP_PLOT_AXIS_SINGULAR_SHARE,
+} from "./leftoverMapPlotAxisSingular";
+
+describe("leftoverMapAxisBadgeShare", () => {
+  it("formats leftover-axis share percent without inventing a leftover score", () => {
+    expect(leftoverMapAxisBadgeShare(0.82)).toBe("82");
+    expect(leftoverMapAxisBadgeShare(0.18)).toBe("18");
+    expect(leftoverMapAxisBadgeShare(0)).toBe("0");
+  });
+});
+
+describe("leftoverMapAxisBadgeSingular", () => {
+  it("reads persisted leftover-map singular values without inventing a leftover score", () => {
+    expect(
+      leftoverMapAxisBadgeSingular({ axis_index: 1, leftover_singular_value: 1.84 }),
+    ).toBe("1.84");
+    expect(
+      leftoverMapAxisBadgeSingular({ axis_index: 2, leftover_singular_value: 0.86 }),
+    ).toBe("0.86");
+  });
+
+  it("names a rank-0 zero leftover-map singular value", () => {
+    expect(leftoverMapAxisBadgeSingular({ axis_index: 1, leftover_singular_value: 0 })).toBe("0.00");
+  });
+
+  it("omits leftover-map singular values that are missing, non-finite, or negative", () => {
+    expect(leftoverMapAxisBadgeSingular({ axis_index: 1 })).toBeNull();
+    expect(
+      leftoverMapAxisBadgeSingular({ axis_index: 1, leftover_singular_value: Number.NaN }),
+    ).toBeNull();
+    expect(
+      leftoverMapAxisBadgeSingular({ axis_index: 1, leftover_singular_value: Number.POSITIVE_INFINITY }),
+    ).toBeNull();
+    expect(
+      leftoverMapAxisBadgeSingular({ axis_index: 1, leftover_singular_value: -0.01 }),
+    ).toBeNull();
+  });
+
+  it("does not invent leftover-map singular values from leftover-map axis share", () => {
+    expect(leftoverMapAxisBadgeSingular({ axis_index: 1 })).toBeNull();
+  });
+
+  it("stays distinct from leftover-map graphic and comparison graphic leftover-map axis σ copy", () => {
+    expect(LEFTOVER_MAP_AXIS_BADGE_SINGULAR).toBe("leftover axis {axis} σ {value} {share}%");
+    expect(LEFTOVER_MAP_AXIS_BADGE_SHARE).toBe("leftover axis {axis} {share}%");
+    expect(LEFTOVER_MAP_AXIS_BADGE_SINGULAR).not.toBe(LEFTOVER_MAP_PLOT_AXIS_SINGULAR);
+    expect(LEFTOVER_MAP_AXIS_BADGE_SINGULAR).not.toBe(LEFTOVER_MAP_PLOT_AXIS_SINGULAR_SHARE);
+    expect(LEFTOVER_MAP_AXIS_BADGE_SINGULAR).not.toBe(LEFTOVER_MAP_COMPARE_PLOT_AXIS_SINGULAR);
+    expect(LEFTOVER_MAP_AXIS_BADGE_SINGULAR).not.toBe(
+      LEFTOVER_MAP_COMPARE_PLOT_AXIS_SINGULAR_SHARE,
+    );
+    expect(LEFTOVER_MAP_AXIS_BADGE_SINGULAR).not.toBe("leftover axis {axis} σ {value}");
+  });
+});
diff --git a/frontend/src/leftoverMapAxisBadge.ts b/frontend/src/leftoverMapAxisBadge.ts
new file mode 100644
index 000000000..7d2b377df
--- /dev/null
+++ b/frontend/src/leftoverMapAxisBadge.ts
@@ -0,0 +1,27 @@
+/** Caption leftover-axis report badges with persisted Gabriel singular values. */
+
+import type { LeftoverMapAxis } from "./api";
+import {
+  formatLeftoverMapPlotAxisSingular,
+  leftoverSingularForAxis,
+} from "./leftoverMapPlotAxisSingular";
+
+export const LEFTOVER_MAP_AXIS_BADGE_SHARE = "leftover axis {axis} {share}%";
+
+export const LEFTOVER_MAP_AXIS_BADGE_SINGULAR = "leftover axis {axis} σ {value} {share}%";
+
+export function leftoverMapAxisBadgeShare(
+  leftoverShare: LeftoverMapAxis["leftover_share"] | null | undefined,
+): string {
+  return ((leftoverShare ?? Number.NaN) * 100).toFixed(0);
+}
+
+export function leftoverMapAxisBadgeSingular(
+  axis: Pick<LeftoverMapAxis, "axis_index"> & {
+    leftover_singular_value?: LeftoverMapAxis["leftover_singular_value"] | null;
+  },
+): string | null {
+  return formatLeftoverMapPlotAxisSingular(
+    leftoverSingularForAxis([axis], axis.axis_index),
+  );
+}
diff --git a/tests/test_leftover_axis_report_singular_contract.py b/tests/test_leftover_axis_report_singular_contract.py
new file mode 100644
index 000000000..5319a7113
--- /dev/null
+++ b/tests/test_leftover_axis_report_singular_contract.py
@@ -0,0 +1,38 @@
+"""Executable contract for persisted singular values on report-axis badges."""
+
+from pathlib import Path
+
+
+ROOT = Path(__file__).resolve().parents[1]
+BADGE_SOURCE = ROOT / "frontend" / "src" / "leftoverMapAxisBadge.ts"
+SINGULAR_SOURCE = ROOT / "frontend" / "src" / "leftoverMapPlotAxisSingular.ts"
+
+
+def test_report_axis_badge_keeps_singular_value_and_share_semantically_distinct() -> None:
+    """Report badges must expose persisted σ without borrowing comparison-graphic copy."""
+    assert BADGE_SOURCE.exists(), "report-axis singular-value badge helper is missing"
+    badge_source = BADGE_SOURCE.read_text(encoding="utf-8")
+
+    assert 'LEFTOVER_MAP_AXIS_BADGE_SHARE = "leftover axis {axis} {share}%"' in badge_source
+    assert (
+        'LEFTOVER_MAP_AXIS_BADGE_SINGULAR = "leftover axis {axis} σ {value} {share}%"'
+        in badge_source
+    )
+    assert "leftoverSingularForAxis" in badge_source
+    assert "formatLeftoverMapPlotAxisSingular" in badge_source
+    assert "leftover map comparison graphic" not in badge_source
+
+
+def test_report_axis_badge_never_infers_singular_value_from_share() -> None:
+    """Missing or invalid σ omits independently while rank-0 σ=0 stays representable."""
+    assert BADGE_SOURCE.exists(), "report-axis singular-value badge helper is missing"
+    assert SINGULAR_SOURCE.exists(), "persisted singular-value helper is missing"
+    badge_source = BADGE_SOURCE.read_text(encoding="utf-8")
+    singular_source = SINGULAR_SOURCE.read_text(encoding="utf-8")
+
+    assert "leftover_singular_value" in singular_source
+    assert "Number.isFinite" in singular_source
+    assert "< 0" in singular_source
+    assert ".toFixed(2)" in singular_source
+    assert "Math.sqrt" not in badge_source
+    assert "Math.sqrt" not in singular_source