diff --git a/AGENTS.md b/AGENTS.md
index 509972621..5c6c8c8da 100644
--- a/AGENTS.md
+++ b/AGENTS.md
@@ -273,7 +273,7 @@ stops startup instead of leaving a healthy-looking partial schema, and
 application code must not compensate for a missing table.
 
 Period leftover pairs (ADR 0017 / 0018 / 0048 / 0049 / 0119 / 0158 / 0162 /
-0163 / 0164 / 0182 / 0185 / 0201 / 0233 / 0266 / 0267 / 0268 / 0269 / 0270 / 0271 / 0272 / 0273 / 0274 / 0275 / 0276 / 0277 / 0278 / 0279 / 0280 / 0281 / 0282 / 0283 / 0284 / 0285 / 0286 / 0287 / 0288 / 0289 / 0290 / 0291) are computed in `lineageweave/leftover_pairs.py` from the
+0163 / 0164 / 0182 / 0185 / 0201 / 0233 / 0266 / 0267 / 0268 / 0269 / 0270 / 0271 / 0272 / 0273 / 0274 / 0275 / 0276 / 0277 / 0278 / 0279 / 0280 / 0281 / 0282 / 0283 / 0284 / 0285 / 0286 / 0287 / 0288 / 0289 / 0290 / 0291 / 0292) are computed in `lineageweave/leftover_pairs.py` from the
 residual after a real GRM/GPCM score, never invented. Distances are
 Euclidean on the two-dimensional Gabriel leftover map; missing cells stay
 out of the factorization. Closest and farthest post–criterion pairs
@@ -395,6 +395,7 @@ singular values `σ_k` so leftover-map axis share is not read as leftover-map
 structure without the Gabriel scale that produced it; a missing, non-finite,
 or negative singular value omits that `σ` badge independently of leftover-map
 axis share and does not invent `σ_k` from leftover-map axis share.
+ADR 0292 captions leftover-axis report badges with persisted leftover-map singular values `σ_k` when finite and non-negative, including rank-0 zero singular values; missing, non-finite, or negative singular values omit only the `σ` caption and never derive `σ_k` from leftover-map axis share.
 ADR 0290 captions the grouping comparison strip with persisted leftover-map post complete-case coverage only when the caller can see the full persisted grouping population. Partial-visibility groupings omit that aggregate; never reconstruct the psychometric denominator from visible members, leftover pairs, or plotted markers.
 ADR 0291 captions the grouping comparison strip with persisted leftover-map item complete-case coverage through leftoverMapItemCoverageCounts only when the caller can see the full persisted grouping population. Partial-visibility rows inherit ADR 0290's fail-closed coverage boundary and never recompute psychometric coverage from visible members, leftover pairs, or plotted criterion markers.
 When `R`, `R̂`, `U`, `x`,
diff --git a/ARCHITECTURE.md b/ARCHITECTURE.md
index deb6e6867..b9325e8e6 100644
--- a/ARCHITECTURE.md
+++ b/ARCHITECTURE.md
@@ -637,7 +637,7 @@ information at the group's mean θ (Lord, 1980 max-info CAT). Rankings
 persist to `report_item_information`. After those IRT main effects,
 residual SVD leftover pairs on two Gabriel axes (Jeon et al., 2021;
 ADR 0017 / 0048 / 0049 / 0119 / 0148 / 0158 / 0162 / 0163 / 0164 / 0168 /
-0182 / 0185 / 0201 / 0233 / 0266 / 0267 / 0268 / 0269 / 0270 / 0271 / 0272 / 0273 / 0274 / 0275 / 0276 / 0277 / 0278 / 0279 / 0280 / 0281 / 0282 / 0283 / 0284 / 0285 / 0286 / 0287 / 0288 / 0289 / 0290 / 0291) persist to `report_leftover_pair` with signed residual `R`,
+0182 / 0185 / 0201 / 0233 / 0266 / 0267 / 0268 / 0269 / 0270 / 0271 / 0272 / 0273 / 0274 / 0275 / 0276 / 0277 / 0278 / 0279 / 0280 / 0281 / 0282 / 0283 / 0284 / 0285 / 0286 / 0287 / 0288 / 0289 / 0290 / 0291 / 0292) persist to `report_leftover_pair` with signed residual `R`,
 observed `Y`, expected `E[Y|θ, item]`, full leftover-map rank, unexplained
 leftover, ADR 0201 reconstruction evidence, ADR 0185 cross-share evidence,
 ADR 0233 unexplained leftover share `s`, ADR 0266 explained leftover
@@ -671,6 +671,7 @@ incomplete item coverage. ADR 0288 fail-closes leftover-map post complete-case c
 on the leftover pair list through leftoverMapCoverageCounts. ADR 0289 captions leftover-map
 graphic axes with persisted leftover-map singular values `σ_k`. ADR 0290 captions the grouping comparison strip with persisted leftover-map post complete-case coverage under a full-visible-grouping rule; partial-visibility groupings fail closed that aggregate to avoid disclosing a hidden population denominator.
 ADR 0291 captions grouping-comparison persisted item complete-case coverage through leftoverMapItemCoverageCounts only under ADR 0290's full-visible-grouping authorization boundary.
+ADR 0292 captions leftover-axis report badges with persisted finite, non-negative leftover-map singular values `σ_k`; missing or invalid singular values omit only that caption, and axis share remains a distinct persisted measure.
 Click a post marker or a pair button
 opens that post. Those ADRs are the normative mathematical, storage, and
 display contracts. Leftover-map axis share
diff --git a/CHANGELOG.d/2.49.0-leftover-map-axis-singular.md b/CHANGELOG.d/2.49.0-leftover-map-axis-singular.md
new file mode 100644
index 000000000..e5b522a4f
--- /dev/null
+++ b/CHANGELOG.d/2.49.0-leftover-map-axis-singular.md
@@ -0,0 +1,13 @@
+## 2.49.0 — Leftover-map singular values on leftover-axis badges
+
+- Caption leftover-axis report badges with persisted leftover-map singular
+  values `σ_k` (ADR 0292). After `make seed`, leftover-axis badges name
+  persisted leftover-map singular values when finite together with persisted
+  leftover-map axis share, leftover-map graphic axes name those same singular
+  values, leftover-map axis ticks match `ξ (x, y) ζ (x, y)` on the pair row,
+  and leftover pairs sit above the member list; 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 distinct from leftover-map
+  graphic axis captions. Do not invent `σ_k` from leftover-map axis share.
+  Never invent a leftover score. Never invent a theta. No new columns.
diff --git a/CHANGELOG.md b/CHANGELOG.md
index ddade3aa4..04342510a 100644
--- a/CHANGELOG.md
+++ b/CHANGELOG.md
@@ -8,6 +8,8 @@ All notable changes to this project are documented here. Format follows
 
 ### Added
 
+- Period leftover pairs now caption leftover-axis report badges with persisted leftover-map singular values `σ_k` (ADR 0292 / v2.49.0). Missing, non-finite, or negative singular values omit only that `σ` caption; rank-0 zero remains `σ 0.00`, and axis share remains distinct.
+
 - Grouping comparison rows now name persisted leftover-map item complete-case coverage (ADR 0291 / v2.48.0) only when the caller can see the full persisted grouping population; partial-visibility rows omit the aggregate rather than leaking a denominator or recomputing subset coverage.
 
 - Grouping comparison rows now expose persisted leftover-map post complete-case coverage through `leftoverMapCoverageCounts` (ADR 0290 / v2.47.0) only for fully visible grouping populations. Partial-visibility rows omit the aggregate instead of leaking the full-group denominator or recomputing psychometric coverage from the visible subset.
diff --git a/CLAUDE.md b/CLAUDE.md
index ed11730d7..3a6fbb514 100644
--- a/CLAUDE.md
+++ b/CLAUDE.md
@@ -49,7 +49,7 @@ cutoff. Global Ask optional `knowledge_cutoff` uses the same cover
 
 Create/start endpoint rules (ADR 0017 / 0021), tie-vs-miss similarity
 (ADR 0026), R&R catalog ids (ADR 0019 / 0027), leftover pairs
-(ADR 0048–0164 / 0182 / 0185 / 0201 / 0233 / 0266 / 0267 / 0268 / 0269 / 0270 / 0271 / 0272 / 0273 / 0274 / 0275 / 0276 / 0277 / 0278 / 0279 / 0280 / 0281 / 0282 / 0283 / 0284 / 0285 / 0286 / 0287 / 0288 / 0289 / 0290 / 0291), occupational construct catalog search
+(ADR 0048–0164 / 0182 / 0185 / 0201 / 0233 / 0266 / 0267 / 0268 / 0269 / 0270 / 0271 / 0272 / 0273 / 0274 / 0275 / 0276 / 0277 / 0278 / 0279 / 0280 / 0281 / 0282 / 0283 / 0284 / 0285 / 0286 / 0287 / 0288 / 0289 / 0290 / 0291 / 0292), occupational construct catalog search
 (ADR 0257), the text-channel embedding swap and cosine
 clamp (ADR 0190), per-edge channel-score persistence (ADR 0195),
 token-backed status notices (ADR 0220),
diff --git a/docs/adr/0148-leftover-map-axis-share.md b/docs/adr/0148-leftover-map-axis-share.md
index 4b4e4d485..130a1c99e 100644
--- a/docs/adr/0148-leftover-map-axis-share.md
+++ b/docs/adr/0148-leftover-map-axis-share.md
@@ -5,7 +5,9 @@
 **Amended by:** [ADR 0269](0269-leftover-map-axis-share-plot.md)
 (leftover-map axis share on the graphic display);
 [ADR 0289](0289-leftover-map-plot-singular.md)
-(leftover-map singular values on the graphic display)
+(leftover-map singular values on the graphic display);
+[ADR 0292](0292-leftover-map-axis-singular.md)
+(leftover-map singular values on leftover-axis badges)
 
 ## Context
 
diff --git a/docs/adr/0292-leftover-map-axis-singular.md b/docs/adr/0292-leftover-map-axis-singular.md
new file mode 100644
index 000000000..d979f01b3
--- /dev/null
+++ b/docs/adr/0292-leftover-map-axis-singular.md
@@ -0,0 +1,163 @@
+# ADR 0292 — Name leftover-map singular values on leftover-axis badges
+
+**Decision status:** Proposed
+**Date:** 2026-08-30
+
+Amends leftover-map axis share persistence
+([ADR 0148](0148-leftover-map-axis-share.md)). Independent of leftover-map
+singular values on the graphic display
+([ADR 0289](0289-leftover-map-plot-singular.md)), fail-close leftover-map post
+complete-case coverage on the pair list
+([ADR 0288](0288-leftover-map-list-post-coverage-helper.md)), leftover-map
+incomplete item coverage on the pair list
+([ADR 0287](0287-leftover-map-list-incomplete-item.md)), leftover-map incomplete
+post coverage on the pair list
+([ADR 0286](0286-leftover-map-list-incomplete-post.md)), leftover-map item
+complete-case coverage on the pair list
+([ADR 0285](0285-leftover-map-list-item-coverage.md)), leftover-map incomplete
+item coverage on the graphic display
+([ADR 0284](0284-leftover-map-plot-incomplete-item.md)), leftover-map incomplete
+post coverage on the graphic display
+([ADR 0283](0283-leftover-map-plot-incomplete.md)), leftover-map item
+complete-case coverage on the graphic display
+([ADR 0282](0282-leftover-map-plot-item-coverage.md)), leftover-map complete-case
+coverage on the graphic display
+([ADR 0281](0281-leftover-map-plot-coverage.md)), leftover-map rank on pair
+segments ([ADR 0280](0280-leftover-map-segment-rank.md)), leftover expected on pair
+segments ([ADR 0279](0279-leftover-map-segment-expected.md)), leftover observed on
+pair segments ([ADR 0278](0278-leftover-map-segment-observed.md)), leftover
+residual on pair segments ([ADR 0277](0277-leftover-map-segment-residual.md)),
+leftover-map unexplained leftover on pair segments
+([ADR 0276](0276-leftover-map-segment-unexplained-leftover.md)), leftover-map
+cross share on pair segments ([ADR 0275](0275-leftover-map-segment-cross-share.md)),
+leftover-map unexplained leftover share on pair segments
+([ADR 0274](0274-leftover-map-segment-unexplained-share.md)), leftover-map
+explained leftover share on pair segments
+([ADR 0273](0273-leftover-map-segment-explained-share.md)), leftover-map
+reconstruction on pair segments
+([ADR 0272](0272-leftover-map-segment-reconstruction.md)), leftover-map
+distance on pair segments ([ADR 0271](0271-leftover-map-segment-distance.md)),
+leftover-map coordinate ticks ([ADR 0270](0270-leftover-map-coordinate-ticks.md)),
+leftover-map axis share on the graphic display
+([ADR 0269](0269-leftover-map-axis-share-plot.md)), leftover residual
+disclosure ([ADR 0162](0162-leftover-residual-disclosure.md)), leftover
+observed `Y` / expected `E` ([ADR 0163](0163-leftover-observed-expected.md)),
+leftover-map explained leftover share persistence
+([ADR 0266](0266-leftover-map-explained-share.md)), leftover-map
+unexplained leftover share persistence
+([ADR 0233](0233-leftover-map-unexplained-share.md)), leftover-map
+reconstruction persistence ([ADR 0201](0201-leftover-map-reconstruction.md)),
+leftover-map cross share persistence
+([ADR 0185](0185-leftover-map-cross-share.md)), leftover-map unexplained leftover
+persistence ([ADR 0182](0182-leftover-map-unexplained.md)), leftover-map rank
+persistence ([ADR 0164](0164-leftover-map-rank.md)), leftover-map complete-case
+coverage persistence ([ADR 0168](0168-leftover-map-complete-case-coverage.md)),
+and leftover-map graphic display ([ADR 0268](0268-leftover-map-graphic-display.md)).
+
+
+This successor identity is orthogonal to ADR 0290 (grouping comparison post coverage) and ADR 0291 (grouping comparison item coverage). Those authorization decisions remain unchanged; this UI decision does not reinterpret their persisted denominators.
+
+## 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 0289 already captions leftover-map graphic axes with persisted
+`σ_k`. Those leftover-axis badges still hide `σ_k`, so Gabriel (1971)
+singular values appear on the plot 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.
+
+This increment captions leftover-axis report badges with already persisted
+leftover-map singular values. It does not add columns. It does not invent
+`σ_k` from leftover-map axis share. It does not persist leftover-map inner
+product, cosine, or length. It does not land Post quality on the leftover
+criterion. Leftover-map distance stays two-axis Euclidean. Do not invent a
+leftover score. Do not invent a theta.
+
+The dashboard stack already used neighbouring leftover facts under other
+numbers. This protected increment uses **0290** so it does not collide with
+leftover-map singular values on the graphic display (0289), fail-close
+leftover-map post complete-case coverage on the pair list (0288), leftover-map
+incomplete item coverage on the pair list (0287), leftover-map incomplete post
+coverage on the pair list (0286), leftover-map item complete-case coverage on
+the pair list (0285), leftover-map incomplete item coverage on the graphic
+display (0284), leftover-map incomplete post coverage on the graphic display
+(0283), leftover-map item complete-case coverage on the graphic display (0282),
+leftover-map complete-case coverage on the graphic display (0281), leftover-map
+rank on pair segments (0280), leftover expected on pair segments (0279),
+leftover observed on pair segments (0278), leftover residual on pair segments
+(0277), leftover-map unexplained leftover on pair segments (0276), leftover-map
+cross share on pair segments (0275), leftover-map unexplained leftover share on
+pair segments (0274), leftover-map explained leftover share on pair segments
+(0273), leftover-map reconstruction on pair segments (0272), leftover-map
+distance on pair segments (0271), leftover-map coordinate ticks (0270),
+leftover-map axis share on the graphic display (0269), leftover-map graphic
+display (0268), leftover-map coordinates (0267 / migration 0245), leftover-map
+explained leftover share persistence (0266 / migration 0244), leftover-map
+unexplained leftover share persistence (0233 / migration 0233), leftover-map
+reconstruction persistence (0201 / migration 0206), leftover-map cross share
+(0185), leftover residual disclosure, leftover observed `Y` / expected `E`,
+leftover-map rank, two-axis leftover-map distance, leftover coverage,
+leftover-map axis share persistence (0148), leftover interaction-map
+persistence, occupational construct catalog search (0265), 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} 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 this caption.
+
+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`. Leftover-map graphic axes (ADR 0289) remain.
+
+## 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 persistence, leftover pairs on the grouping comparison
+strip, two-axis leftover-map distance, leftover-map rank, leftover-map
+inner product, leftover-map cosine, leftover-map length, leftover-map
+reconstruction, leftover-map unexplained leftover, leftover-map cross
+share, leftover-map unexplained leftover share, leftover-map explained
+leftover share, leftover-map coordinate persistence, leftover-map
+graphic display, leftover-map coordinate ticks, leftover-map axis share
+on the graphic display, leftover-map complete-case coverage on the graphic
+display, leftover-map item complete-case coverage on the graphic display,
+leftover-map incomplete post coverage on the graphic display, leftover-map
+incomplete item coverage on the graphic display, leftover-map item
+complete-case coverage on the pair list, leftover-map incomplete post
+coverage on the pair list, leftover-map incomplete item coverage on the
+pair list, fail-close leftover-map post complete-case coverage on the
+pair list, and leftover-map singular values on the graphic display.
+
+## 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/package.json b/frontend/package.json
index 887c52179..8fa819fcf 100644
--- a/frontend/package.json
+++ b/frontend/package.json
@@ -1,7 +1,7 @@
 {
   "name": "frontend",
   "private": true,
-  "version": "2.48.0",
+  "version": "2.49.0",
   "type": "module",
   "scripts": {
     "dev": "vite",
diff --git a/frontend/src/App.test.tsx b/frontend/src/App.test.tsx
index 4c12ee10c..ec0b734f2 100644
--- a/frontend/src/App.test.tsx
+++ b/frontend/src/App.test.tsx
@@ -4143,8 +4143,8 @@ describe("App, authenticated", () => {
     expect(screen.getByText(/TEST-PU-REPORT/)).toBeInTheDocument();
     expect(screen.getAllByText("shared metric").length).toBeGreaterThan(0);
     expect(screen.getAllByText(/CAT: sales-lead I=0\.70/).length).toBeGreaterThan(0);
-    expect(screen.getAllByText(/leftover axis 1 82%/).length).toBeGreaterThan(0);
-    expect(screen.getAllByText(/leftover axis 2 18%/).length).toBeGreaterThan(0);
+    expect(screen.getAllByText(/leftover axis 1 σ 1\.84 82%/).length).toBeGreaterThan(0);
+    expect(screen.getAllByText(/leftover axis 2 σ 0\.86 18%/).length).toBeGreaterThan(0);
     expect(screen.getByLabelText("Leftover-map axis share")).toHaveTextContent(
       "Open a leftover pair to read the post–criterion cell",
     );
diff --git a/frontend/src/App.tsx b/frontend/src/App.tsx
index 3bd2f263f..e22eabc2d 100644
--- a/frontend/src/App.tsx
+++ b/frontend/src/App.tsx
@@ -135,6 +135,12 @@ 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 "./App.css";
 
 const AdminPanel = lazy(() => import("./components/AdminPanel").then((module) => ({ default: module.AdminPanel })));
@@ -3882,18 +3888,25 @@ 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/i18n.test.ts b/frontend/src/i18n.test.ts
index 97fdb11d1..f1e262bc7 100644
--- a/frontend/src/i18n.test.ts
+++ b/frontend/src/i18n.test.ts
@@ -43,8 +43,9 @@ describe("i18n", () => {
     "Page",
     "Answer",
     "leftover axis {axis} {share}%",
+    "leftover axis {axis} σ {value} {share}%",
     "Leftover-map axis share",
-    "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.",
     "Leftover pairs",
     "Closest leftover",
     "Farthest leftover",
@@ -610,6 +611,18 @@ describe("i18n", () => {
     ).toBe(expected);
   });
 
+  it.each([
+    ["ko", "잔차 축 1 σ 1.84 82%"],
+    ["zh", "残差轴 1 σ 1.84 82%"],
+    ["ja", "残差軸 1 σ 1.84 82%"],
+    ["vi", "trục phần dư 1 σ 1.84 82%"],
+  ] as const)("formats leftover-axis badge singular value with share in %s", (locale, expected) => {
+    setLocale(locale);
+    expect(
+      tf("leftover axis {axis} σ {value} {share}%", { axis: 1, value: "1.84", share: "82" }),
+    ).toBe(expected);
+  });
+
   it.each([
     ["ko", "IRT 주효과 이후 관측 Y 2.40와 기대 E 2.00를 읽은 다음, 이 글을 여세요."],
     ["zh", "阅读 IRT 主效应后的观测 Y 2.40 与期望 E 2.00，然后打开这篇帖子。"],
diff --git a/frontend/src/i18n.ts b/frontend/src/i18n.ts
index 0682adf21..692d66bb4 100644
--- a/frontend/src/i18n.ts
+++ b/frontend/src/i18n.ts
@@ -568,9 +568,10 @@ const TRANSLATIONS: Partial<Record<Locale, Record<string, string>>> = {
     "Valid from": "유효 시작",
     "Valid to": "유효 종료",
     "leftover axis {axis} {share}%": "잔차 축 {axis} {share}%",
+    "leftover axis {axis} σ {value} {share}%": "잔차 축 {axis} σ {value} {share}%",
     "Leftover-map axis share": "잔차 지도 축 비율",
-    "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.":
-      "잔차 지도 축 비율은 잔차 SVD 축 1과 2의 Gabriel 관성입니다. 글–기준 셀을 읽으려면 잔차 쌍을 여세요. 이 비율은 잔차 점수를 만들어내지 않습니다.",
+    "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.":
+      "잔차 지도 축 비율은 잔차 SVD 축 1과 2의 Gabriel 관성입니다. 잔차 지도 특이값은 그 축의 Gabriel 척도입니다. 글–기준 셀을 읽으려면 잔차 쌍을 여세요. 이 비율과 특이값은 잔차 점수를 만들어내지 않습니다.",
     "Leftover pairs": "잔여 쌍",
     "Closest leftover": "가장 가까운 잔여",
     "Farthest leftover": "가장 먼 잔여",
@@ -1174,9 +1175,10 @@ const TRANSLATIONS: Partial<Record<Locale, Record<string, string>>> = {
     "Valid from": "有效起始",
     "Valid to": "有效结束",
     "leftover axis {axis} {share}%": "残差轴 {axis} {share}%",
+    "leftover axis {axis} σ {value} {share}%": "残差轴 {axis} σ {value} {share}%",
     "Leftover-map axis share": "残差图轴占比",
-    "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.":
-      "残差图轴占比是残差 SVD 第 1、2 轴的 Gabriel 惯量。打开一个残差配对可查看文章–准则单元格。这些占比不会虚构残差分数。",
+    "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.":
+      "残差图轴占比是残差 SVD 第 1、2 轴的 Gabriel 惯量。残差图奇异值是这些轴的 Gabriel 尺度。打开一个残差配对可查看文章–准则单元格。这些占比和奇异值不会虚构残差分数。",
     "Leftover pairs": "残余配对",
     "Closest leftover": "最近残余",
     "Farthest leftover": "最远残余",
@@ -1784,9 +1786,10 @@ const TRANSLATIONS: Partial<Record<Locale, Record<string, string>>> = {
     "Valid from": "有効開始",
     "Valid to": "有効終了",
     "leftover axis {axis} {share}%": "残差軸 {axis} {share}%",
+    "leftover axis {axis} σ {value} {share}%": "残差軸 {axis} σ {value} {share}%",
     "Leftover-map axis share": "残差マップ軸の比率",
-    "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.":
-      "残差マップ軸の比率は、残差 SVD の第1軸と第2軸の Gabriel 慣性です。投稿–基準セルを読むには残差ペアを開いてください。この比率から残差スコアを作りません。",
+    "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.":
+      "残差マップ軸の比率は、残差 SVD の第1軸と第2軸の Gabriel 慣性です。残差マップ特異値はそれらの軸の Gabriel 尺度です。投稿–基準セルを読むには残差ペアを開いてください。この比率と特異値から残差スコアを作りません。",
     "Leftover pairs": "残差ペア",
     "Closest leftover": "最も近い残差",
     "Farthest leftover": "最も遠い残差",
@@ -2394,9 +2397,10 @@ const TRANSLATIONS: Partial<Record<Locale, Record<string, string>>> = {
     "Valid from": "Hiệu lực từ",
     "Valid to": "Hiệu lực đến",
     "leftover axis {axis} {share}%": "trục phần dư {axis} {share}%",
+    "leftover axis {axis} σ {value} {share}%": "trục phần dư {axis} σ {value} {share}%",
     "Leftover-map axis share": "Tỷ trọng trục bản đồ phần dư",
-    "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.":
-      "Tỷ trọng trục bản đồ phần dư là quán tính Gabriel của các trục SVD phần dư 1 và 2. Mở một cặp phần dư để đọc ô bài viết–tiêu chí. Các tỷ trọng này không tạo ra điểm phần dư.",
+    "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.":
+      "Tỷ trọng trục bản đồ phần dư là quán tính Gabriel của các trục SVD phần dư 1 và 2. Giá trị kỳ dị bản đồ phần dư là thang Gabriel của các trục đó. Mở một cặp phần dư để đọc ô bài viết–tiêu chí. Các tỷ trọng và giá trị kỳ dị này không tạo ra điểm phần dư.",
     "Leftover pairs": "Cặp phần dư",
     "Closest leftover": "Phần dư gần nhất",
     "Farthest leftover": "Phần dư xa nhất",
diff --git a/frontend/src/leftoverMapAxisBadge.test.ts b/frontend/src/leftoverMapAxisBadge.test.ts
new file mode 100644
index 000000000..821f8246f
--- /dev/null
+++ b/frontend/src/leftoverMapAxisBadge.test.ts
@@ -0,0 +1,38 @@
+import { describe, expect, it } from "vitest";
+import {
+  leftoverMapAxisBadgeShare,
+  leftoverMapAxisBadgeSingular,
+} from "./leftoverMapAxisBadge";
+
+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: -0.01 }),
+    ).toBeNull();
+  });
+});
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/lineageweave/__init__.py b/lineageweave/__init__.py
index 1a6851aca..6c50368ed 100644
--- a/lineageweave/__init__.py
+++ b/lineageweave/__init__.py
@@ -130,4 +130,4 @@
     "serialize_lineage_analysis_result",
 ]
 
-__version__ = "2.48.0"
+__version__ = "2.49.0"
diff --git a/pyproject.toml b/pyproject.toml
index 75a3f8f91..3da74d89c 100644
--- a/pyproject.toml
+++ b/pyproject.toml
@@ -1,6 +1,6 @@
 [project]
 name = "lineageweave"
-version = "2.48.0"
+version = "2.49.0"
 description = "Reconstructs git-branch-style lineage DAGs from scattered short records using multi-channel score fusion and LLM adjudication."
 readme = "README.md"
 license = { text = "MIT" }
diff --git a/tests/test_adr_0292_governance.py b/tests/test_adr_0292_governance.py
new file mode 100644
index 000000000..984b5adea
--- /dev/null
+++ b/tests/test_adr_0292_governance.py
@@ -0,0 +1,16 @@
+"""Governance regression for leftover-axis singular-value badges."""
+from pathlib import Path
+
+ROOT = Path(__file__).resolve().parents[1]
+
+def test_axis_singular_badges_use_distinct_proposed_adr_0292() -> None:
+    """Keep #823 distinct from #821/#822 decisions and Proposed while Draft."""
+    assert (ROOT / "docs/adr/0290-leftover-map-compare-coverage.md").exists()
+    assert (ROOT / "docs/adr/0291-leftover-map-compare-item-coverage.md").exists()
+    assert not (ROOT / "docs/adr/0290-leftover-map-axis-singular.md").exists()
+    successor = ROOT / "docs/adr/0292-leftover-map-axis-singular.md"
+    assert successor.exists()
+    text = successor.read_text(encoding="utf-8")
+    assert text.startswith("# ADR 0292 —")
+    assert "**Decision status:** Proposed" in text
+    assert "**Decision status:** Accepted" not in text