diff --git a/AGENTS.md b/AGENTS.md
index 0bb84cfac..b4c64b52b 100644
--- a/AGENTS.md
+++ b/AGENTS.md
@@ -193,17 +193,20 @@ against a live local stack (`make up`) and self-skip without one -- see
 [README.md](README.md#local-product-stack-docker-compose).
 
 Period leftover pairs (ADR 0017 / 0018 / 0048 / 0049 / 0119 / 0162 / 0163 /
-0164) 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 persist to
-`report_leftover_pair` with signed residual `R`, observed `Y`, and
-expected `E[Y|θ, item]` so `R = Y − E` remains auditable, plus
-leftover-map rank so rank 0 is not read as structure. They sit above the
-member list so a click opens that post. Leftover-map axis share
-(ADR 0148) is Gabriel inertia of residual SVD axes 1 and 2 and persists
-to `report_leftover_map_axis`. Rank-0 residuals emit two zero-share
-axes; the shares are report-level and are not a leftover score.
+0164 / 0182) 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
+persist to `report_leftover_pair` with signed residual `R`, observed
+`Y`, and expected `E[Y|θ, item]` so `R = Y − E` remains auditable,
+plus leftover-map rank so rank 0 is not read as structure, and
+unexplained leftover `U = R − R̂` next to leftover-map distance `d`
+after two-axis Gabriel reconstruction. They sit above the member
+list so a click opens that post. Two-axis reconstruction `R̂` is not
+persisted. Leftover-map axis share (ADR 0148) is Gabriel inertia of
+residual SVD axes 1 and 2 and persists to `report_leftover_map_axis`.
+Rank-0 residuals emit two zero-share axes; the shares are report-level
+and are not a leftover score.
 
 `frontend/` has its own toolchain (Node pinned via `frontend/mise.toml`,
 pnpm via Corepack -- do not add a second Node package manager or a
diff --git a/ARCHITECTURE.md b/ARCHITECTURE.md
index 9dbf19f31..b682cadee 100644
--- a/ARCHITECTURE.md
+++ b/ARCHITECTURE.md
@@ -594,10 +594,11 @@ on those same fixed parameters (Kim, 2006 FIPC). After scoring,
 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 / 0119 / 0162 / 0163 / 0164) persist to
+ADR 0017 / 0048 / 0119 / 0162 / 0163 / 0164 / 0182) persist to
 `report_leftover_pair` with signed residual `R`, observed `Y`, expected
-`E[Y|θ, item]`, and full leftover-map rank. Leftover-map axis share
-(Gabriel inertia of residual SVD axes 1 and 2; ADR 0148) persists to
+`E[Y|θ, item]`, full leftover-map rank, and unexplained leftover
+`U = R − R̂` named on the pair row. Leftover-map axis share (Gabriel
+inertia of residual SVD axes 1 and 2; ADR 0148) persists to
 `report_leftover_map_axis`. Results persist to
 `report_period_score` / `report_member_score`.
 `GET /api/reports/{grouping}` lists the trend;
diff --git a/CHANGELOG.d/2.12.26-leftover-map-unexplained.md b/CHANGELOG.d/2.12.26-leftover-map-unexplained.md
new file mode 100644
index 000000000..a64e67997
--- /dev/null
+++ b/CHANGELOG.d/2.12.26-leftover-map-unexplained.md
@@ -0,0 +1,8 @@
+## 2.12.26 — Leftover-map unexplained leftover
+
+- Persist unexplained leftover `U = R − R̂` on leftover post–criterion
+  pairs (ADR 0182). After `make seed`, closest and farthest leftover
+  pairs sit above the member list with `U` next to leftover-map
+  distance `d`; click opens that post. Omit the badge when unexplained
+  leftover is missing. Never invent a leftover score. Do not persist
+  leftover-map reconstruction `R̂`.
diff --git a/CHANGELOG.md b/CHANGELOG.md
index 5d67eb0e5..393125f3c 100644
--- a/CHANGELOG.md
+++ b/CHANGELOG.md
@@ -40,6 +40,16 @@ All notable changes to this project are documented here. Format follows
   environment, so local OIDC and synthetic-data workflows resolve the same
   pinned dependencies as CI.
 
+## [2.12.26] - 2026-08-24
+
+### Added
+
+- Period leftover pair rows now name unexplained leftover `U = R − R̂`
+  next to leftover-map distance `d`, then open that post (Gabriel, 1971;
+  Jeon et al., 2021, eq. 3; ADR 0182). A missing unexplained leftover
+  omits the badge rather than inventing a leftover score. Two-axis
+  reconstruction `R̂` stays internal and is not persisted.
+
 ## [2.12.19] - 2026-08-24
 
 ### Added
diff --git a/CLAUDE.md b/CLAUDE.md
index 6f1af23b5..992753de8 100644
--- a/CLAUDE.md
+++ b/CLAUDE.md
@@ -55,8 +55,9 @@ mean θ. The period-report panel says Demo Corp is the opened grouping
 and to read its mean θ and member posts, then open a post. Those
 members land immediately under that next action, ahead of Other Corp
 and the week strip. After `make seed`, leftover closest/farthest pairs
-sit above the member list with leftover-map rank; rank 0 names no
-leftover structure. Leftover-map axis share badges name Gabriel inertia
+sit above the member list with leftover-map rank (rank 0 names no
+leftover structure) and unexplained leftover `U` next to leftover-map
+distance `d`. Leftover-map axis share badges name Gabriel inertia
 of axes 1 and 2; open a leftover pair to read the post–criterion cell.
 The shares do not invent a leftover score. Opening Public post names the next action: read
 Event Lineage, Keyman, and evaluation on that post. The popup Event
diff --git a/backend/app/report_ingestion.py b/backend/app/report_ingestion.py
index 022daed3e..deca8d3d7 100644
--- a/backend/app/report_ingestion.py
+++ b/backend/app/report_ingestion.py
@@ -444,8 +444,9 @@ async def persist_period_report(
             insert into report_leftover_pair (
                 grouping_kind, grouping_key, period_code, rubric_version,
                 pair_kind, post_id, criterion_code, leftover_distance, leftover_residual,
-                observed_response, expected_response, leftover_map_rank
-            ) values ($1,$2,$3,$4,$5,$6,$7,$8,$9,$10,$11,$12)
+                observed_response, expected_response, leftover_map_rank,
+                leftover_map_unexplained
+            ) values ($1,$2,$3,$4,$5,$6,$7,$8,$9,$10,$11,$12,$13)
             """,
             grouping_kind,
             grouping_key,
@@ -459,6 +460,7 @@ async def persist_period_report(
             pair.observed_response,
             pair.expected_response,
             pair.leftover_map_rank,
+            pair.leftover_map_unexplained,
         )
     for axis in report.leftover_map_axes:
         await conn.execute(
@@ -622,7 +624,8 @@ async def fetch_period_reports(
         f"""
         select lp.grouping_key, lp.pair_kind, lp.post_id, lp.criterion_code,
                lp.leftover_distance, lp.leftover_residual,
-               lp.observed_response, lp.expected_response, lp.leftover_map_rank, p.post_title,
+               lp.observed_response, lp.expected_response, lp.leftover_map_rank,
+               lp.leftover_map_unexplained, p.post_title,
                p.visibility_code, p.corporate_entity_id,
                ({_SOURCE_CONTEXT_PRESENT_SQL}) as has_real_source_context
         from report_leftover_pair lp
@@ -748,6 +751,11 @@ async def fetch_period_reports(
                             if row["leftover_map_rank"] is None
                             else int(row["leftover_map_rank"])
                         ),
+                        "leftover_map_unexplained": (
+                            None
+                            if row["leftover_map_unexplained"] is None
+                            else float(row["leftover_map_unexplained"])
+                        ),
                         "visibility_code": row["visibility_code"],
                         "corporate_entity_id": str(row["corporate_entity_id"]),
                         "has_real_source_context": bool(row["has_real_source_context"]),
diff --git a/backend/tests/test_api.py b/backend/tests/test_api.py
index db59bdea1..2b1eda9ca 100644
--- a/backend/tests/test_api.py
+++ b/backend/tests/test_api.py
@@ -137,6 +137,11 @@
     / "migrations"
     / "0169_report_leftover_map_axis.sql"
 )
+_LEFTOVER_MAP_UNEXPLAINED_MIGRATION = (
+    Path(__file__).resolve().parents[2]
+    / "migrations"
+    / "0182_report_leftover_map_unexplained.sql"
+)
 
 
 def _postgres_available() -> bool:
@@ -255,6 +260,7 @@ def seeded_db(demo_analyst_token):
             cur.execute(_LEFTOVER_OBSERVED_EXPECTED_MIGRATION.read_text())
             cur.execute(_LEFTOVER_MAP_RANK_MIGRATION.read_text())
             cur.execute(_LEFTOVER_MAP_AXIS_MIGRATION.read_text())
+            cur.execute(_LEFTOVER_MAP_UNEXPLAINED_MIGRATION.read_text())
             cur.execute(
                 "insert into common_lookup_value (lookup_category, lookup_code, lookup_label) values "
                 "('corporate_entity_level', 'group', 'Group'), "
@@ -4740,6 +4746,9 @@ def test_seed_period_report_surfaces_on_get_reports(client, demo_analyst_token,
     assert all(pair["leftover_distance"] >= 0 for pair in high_report.get("leftover_pairs", []))
     for pair in high_report.get("leftover_pairs", []):
         assert pair["leftover_map_rank"] >= 0
+        unexplained = pair.get("leftover_map_unexplained")
+        assert unexplained is None or isinstance(unexplained, (int, float))
+        assert "leftover_map_reconstruction" not in pair
         observed = pair.get("observed_response")
         expected = pair.get("expected_response")
         if observed is None or expected is None:
diff --git a/docs/adr/0003-fast-mlsirm-report-integration.md b/docs/adr/0003-fast-mlsirm-report-integration.md
index 439380d00..ed4ca22c8 100644
--- a/docs/adr/0003-fast-mlsirm-report-integration.md
+++ b/docs/adr/0003-fast-mlsirm-report-integration.md
@@ -100,9 +100,13 @@ than one large PR:
    `information_polytomous` (Lord, 1980 max-info). Persist the ranking
    (`report_item_information`) and show the rank-1 item on the Period
    reports panel. Do not reimplement an information function here.
-7. **Leftover-pair slice** (shipped in 0.71.2; ADR 0017 / 0018 / 0048 / 0049): after
+7. **Leftover-pair slice** (shipped in 0.71.2; ADR 0017 / 0018 / 0048 / 0049 / 0182): after
    IRT main effects, persist closest and farthest post–criterion pairs
-   from the residual leftover map. Do not fork LSIRM; do not invent a
+   from the residual leftover map, and name unexplained leftover
+   `U = R − R̂` when Gabriel coordinates exist so the leftover cell the
+   two-axis map does not reconstruct is not read as leftover residual
+   or leftover-map distance. Do not persist two-axis reconstruction
+   `R̂`. Do not fork LSIRM; do not invent a
    leftover-pair API inside `fast-mlsirm` in this slice.
 8. **Leftover-map axis-share slice** (ADR 0148): persist Gabriel inertia
    `σ_k² / Σ_j σ_j²` of leftover-map axes 1 and 2 on the same residual
diff --git a/docs/adr/0048-persist-lsirm-leftover-pairs.md b/docs/adr/0048-persist-lsirm-leftover-pairs.md
index bc79cb17e..fddc028e2 100644
--- a/docs/adr/0048-persist-lsirm-leftover-pairs.md
+++ b/docs/adr/0048-persist-lsirm-leftover-pairs.md
@@ -4,7 +4,8 @@
 **Date:** 2026-08-17
 **Amended by:** [ADR 0119](0119-leftover-map-two-dimensional-distance.md) (two leftover-map axes);
 [ADR 0163](0163-leftover-observed-expected.md) (observed Y and expected E);
-[ADR 0164](0164-leftover-map-rank.md) (full map rank)
+[ADR 0164](0164-leftover-map-rank.md) (full map rank);
+[ADR 0182](0182-leftover-map-unexplained.md) (unexplained leftover U)
 
 ## Context
 
@@ -32,12 +33,16 @@ they are never filled with zero. Persist exactly one `closest`
 and one `farthest` observed cell per period report in
 `report_leftover_pair` (3NF, two-or-more-word `snake_case`).
 
-The biplot lives in `lineageweave/leftover_pairs.py` so leftover
 tests do not import `period_report` or `fast_mlsirm`. Distances are
 Euclidean on the two leftover-map axes (ADR 0119). Each leftover row
 also names observed `Y` and expected `E[Y|θ, item]` so residual
 reconciles to `Y − E` (ADR 0163), and names the full singular-value
-rank while distance remains on the first two axes (ADR 0164).
+rank while distance remains on the first two axes (ADR 0164). Each
+leftover row also names unexplained leftover `U = R − R̂` when
+Gabriel coordinates exist so the leftover cell the two-axis map does
+not reconstruct is not read as leftover residual `R` or leftover-map
+distance `d` (ADR 0182). Two-axis reconstruction `R̂` is computed
+internally and is not persisted.
 
 Cascade the rows with `report_period_score`. A leftover post must
 also be a `report_member_score` row, and the leftover criterion
diff --git a/docs/adr/0049-leftover-pair-report-ui.md b/docs/adr/0049-leftover-pair-report-ui.md
index 3b6e4ce43..fdae8f2ee 100644
--- a/docs/adr/0049-leftover-pair-report-ui.md
+++ b/docs/adr/0049-leftover-pair-report-ui.md
@@ -4,7 +4,8 @@
 **Date:** 2026-08-17
 **Amended by:** [ADR 0162](0162-leftover-residual-disclosure.md) (signed residual R);
 [ADR 0163](0163-leftover-observed-expected.md) (observed Y and expected E);
-[ADR 0164](0164-leftover-map-rank.md) (full map rank)
+[ADR 0164](0164-leftover-map-rank.md) (full map rank);
+[ADR 0182](0182-leftover-map-unexplained.md) (unexplained leftover U)
 
 ## Context
 
@@ -21,14 +22,19 @@ second navigation surface.
 On each period-report group, render leftover pairs **above** the
 member list. Each pair is a button: closest or farthest label, post
 title, criterion short label, signed residual `R`, two-axis leftover-map
-distance, full map rank, observed `Y`, and expected `E` when finite.
+distance, full map rank, observed `Y`, expected `E` when finite, and
+unexplained leftover `U` when finite.
 The next action names every available measurement before opening the
-post; no amendment hides another, and rank 0 explicitly names no
-leftover structure.
+post; no amendment hides another, rank 0 explicitly names no
+leftover structure, and unexplained leftover names "leftover map leaves
+unexplained `U` after IRT main effects; open this post to read the
+named criterion" when present. A missing unexplained leftover keeps
+the existing next action.
 Clicking the button opens that post with the same handler as a member
 row. Residual naming is [ADR 0162](0162-leftover-residual-disclosure.md),
 observed/expected naming is [ADR 0163](0163-leftover-observed-expected.md),
-rank naming is [ADR 0164](0164-leftover-map-rank.md).
+rank naming is [ADR 0164](0164-leftover-map-rank.md), unexplained
+leftover naming is [ADR 0182](0182-leftover-map-unexplained.md).
 
 After `make seed`, closest and farthest leftover pairs sit above the
 member list. Click a pair to open that post.
diff --git a/docs/adr/0182-leftover-map-unexplained.md b/docs/adr/0182-leftover-map-unexplained.md
new file mode 100644
index 000000000..e7cc46fca
--- /dev/null
+++ b/docs/adr/0182-leftover-map-unexplained.md
@@ -0,0 +1,92 @@
+# ADR 0182 — Name unexplained leftover 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) reconstructs a matrix from the biplot
+as the inner product of person and item coordinates. The leftover map
+buyers read is two-axis: unused axes pad with zero, and hidden SVD
+axes after the second are dropped. Two-axis reconstruction
+`R̂ = ξ_{1:2} · ζ_{1:2}` is therefore the leftover cell the map
+shows. Hiding unexplained leftover `U = R − R̂` lets a buyer read
+leftover residual `R` or leftover-map distance `d` as the leftover
+the two-axis map does not reconstruct.
+
+This increment does not persist leftover-map reconstruction `R̂`, does
+not persist leftover-map coordinates, does not name leftover-map inner
+product as a separate full-rank column, does not name leftover-map
+cosine, does not name leftover-map length, 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. Leftover-map distance stays full-rank Euclidean.
+Reconstruction `R̂` is computed internally so `U` is honest, then
+discarded.
+
+The unprotected-stack reconstructions for neighbouring leftover facts
+use 0162–0181. This protected-main increment uses **0182** so it does
+not collide with leftover-map reconstruction (0181), leftover-map
+length (0181 on the length stack), 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_unexplained` — unexplained
+leftover `U = R − R̂` after two-axis Gabriel reconstruction
+`R̂ = ξ_{1:2} · ζ_{1:2}`. Migration `0182` 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 unexplained leftover. Fallback pairs that have no
+complete-case leftover map omit the value rather than inventing one.
+A rank-0 origin map stores `0.0` (`R = 0` and `R̂ = 0`), not a
+missing value. A non-finite unexplained leftover stores null rather
+than inventing a leftover score. Do not persist
+`leftover_map_reconstruction`.
+
+The pair button shows `U {signed}` next to leftover-map distance `d`
+when the value is finite. Next action: leftover map leaves unexplained
+`U` after IRT main effects; open this post to read the named criterion.
+A missing or non-finite unexplained leftover 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_unexplained`. After `make seed`, closest and farthest
+leftover pairs sit above the member list with named `U` 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,
+leftover-map cosine, leftover-map length, and leftover-map
+reconstruction.
+
+## 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/package.json b/frontend/package.json
index 844794550..64cf88197 100644
--- a/frontend/package.json
+++ b/frontend/package.json
@@ -1,7 +1,7 @@
 {
   "name": "frontend",
   "private": true,
-  "version": "2.12.19",
+  "version": "2.12.26",
   "type": "module",
   "scripts": {
     "dev": "vite",
diff --git a/frontend/src/App.test.tsx b/frontend/src/App.test.tsx
index b738e142e..bc40dcdaf 100644
--- a/frontend/src/App.test.tsx
+++ b/frontend/src/App.test.tsx
@@ -950,6 +950,7 @@ describe("App, authenticated", () => {
                     criterion_code: "sales_lead_specificity",
                     leftover_distance: 0.12,
                     leftover_residual: 0.4,
+                    leftover_map_unexplained: 0.05,
                     observed_response: 2.4,
                     expected_response: 2.0,
                     leftover_map_rank: 1,
@@ -961,6 +962,7 @@ describe("App, authenticated", () => {
                     criterion_code: "general_sentiment_negative",
                     leftover_distance: 1.84,
                     leftover_residual: -1.1,
+                    leftover_map_unexplained: -0.25,
                     observed_response: 0.9,
                     expected_response: 2.0,
                     leftover_map_rank: 1,
@@ -3565,19 +3567,21 @@ describe("App, authenticated", () => {
     });
     expect(closestPair).toHaveTextContent("Closest leftover: Public post · sales-lead");
     expect(closestPair).toHaveTextContent(
-      "Read leftover map rank 1, observed Y 2.40, and expected E 2.00 after IRT main effects, then open this post.",
+      "Leftover map leaves unexplained U +0.05 after IRT main effects. Open this post to read sales-lead.",
     );
     expect(closestPair).toHaveTextContent("R +0.40");
     expect(closestPair).toHaveTextContent("Y 2.40 · E 2.00");
     expect(closestPair).toHaveTextContent("rank 1");
+    expect(closestPair).toHaveTextContent("U +0.05");
     expect(closestPair).toHaveTextContent("d 0.12");
     expect(farthestPair).toHaveTextContent("Farthest leftover: Specification revision requested · negative");
     expect(farthestPair).toHaveTextContent(
-      "Read leftover map rank 1, observed Y 0.90, and expected E 2.00 after IRT main effects, then open this post.",
+      "Leftover map leaves unexplained U −0.25 after IRT main effects. Open this post to read negative.",
     );
     expect(farthestPair).toHaveTextContent("R −1.10");
     expect(farthestPair).toHaveTextContent("Y 0.90 · E 2.00");
     expect(farthestPair).toHaveTextContent("rank 1");
+    expect(farthestPair).toHaveTextContent("U −0.25");
     expect(farthestPair).toHaveTextContent("d 1.84");
     const memberButton = screen.getByRole("button", { name: /open report post: public post/i });
     expect(closestPair.compareDocumentPosition(memberButton) & Node.DOCUMENT_POSITION_FOLLOWING).toBeTruthy();
diff --git a/frontend/src/api.ts b/frontend/src/api.ts
index 98c7bb91a..5532bb3fa 100644
--- a/frontend/src/api.ts
+++ b/frontend/src/api.ts
@@ -774,6 +774,7 @@ export interface LeftoverPair {
   observed_response?: number | null;
   expected_response?: number | null;
   leftover_map_rank?: number | null;
+  leftover_map_unexplained?: number | null;
 }
 
 export interface LeftoverMapAxis {
diff --git a/frontend/src/components/LeftoverPairList.tsx b/frontend/src/components/LeftoverPairList.tsx
index f718b6369..d31fbc05b 100644
--- a/frontend/src/components/LeftoverPairList.tsx
+++ b/frontend/src/components/LeftoverPairList.tsx
@@ -7,6 +7,11 @@ import {
 } from "../leftoverMapRank";
 import { formatLeftoverObservedExpected } from "../leftoverObservedExpected";
 import { formatLeftoverResidual } from "../leftoverResidual";
+import {
+  formatLeftoverMapUnexplained,
+  formatSignedLeftoverValue,
+  LEFTOVER_MAP_UNEXPLAINED_ACTION,
+} from "../leftoverMapUnexplained";
 
 export type LeftoverPairListProps = {
   pairs: LeftoverPair[];
@@ -18,9 +23,11 @@ export type LeftoverPairListProps = {
  * Closest and farthest leftover post–criterion pairs after IRT main effects.
  *
  * Distance is the two-axis leftover-map Euclidean gap. Residual is
- * ``R = Y − E[Y|θ, item]`` (Jeon et al., 2021, eq. 3 input).
- * The action preserves residual, observed/expected, and full-rank
- * amendments together before opening the named post.
+ * ``R = Y − E[Y|θ, item]`` (Jeon et al., 2021, eq. 3 input). Unexplained
+ * leftover ``U = R − R̂`` after two-axis Gabriel reconstruction (ADR 0182)
+ * takes priority over the residual/observed-expected/rank next action
+ * when finite; every badge still renders together before opening the
+ * named post.
  */
 export function LeftoverPairList({
   pairs,
@@ -42,8 +49,16 @@ export function LeftoverPairList({
           pair.expected_response,
         );
         const rankBadge = formatLeftoverMapRank(pair.leftover_map_rank);
+        const unexplained = formatLeftoverMapUnexplained(pair.leftover_map_unexplained);
         let nextAction: string;
-        if (rankBadge !== null && observedExpected !== null) {
+        if (unexplained !== null) {
+          const signedUnexplained =
+            formatSignedLeftoverValue(pair.leftover_map_unexplained ?? Number.NaN) ?? "—";
+          nextAction = tf(LEFTOVER_MAP_UNEXPLAINED_ACTION, {
+            value: signedUnexplained,
+            criterion,
+          });
+        } else if (rankBadge !== null && observedExpected !== null) {
           nextAction =
             pair.leftover_map_rank === 0
               ? tf(
@@ -104,6 +119,7 @@ export function LeftoverPairList({
               <span className="post-badge">R {residual}</span>
               {observedExpected ? <span className="post-badge">{observedExpected}</span> : null}
               {rankBadge ? <span className="post-badge">{rankBadge}</span> : null}
+              {unexplained ? <span className="post-badge">{unexplained}</span> : null}
               <span className="post-badge">d {pair.leftover_distance.toFixed(2)}</span>
             </button>
           </li>
diff --git a/frontend/src/i18n.test.ts b/frontend/src/i18n.test.ts
index 97f3c0c85..2ab5aa3b6 100644
--- a/frontend/src/i18n.test.ts
+++ b/frontend/src/i18n.test.ts
@@ -45,6 +45,7 @@ describe("i18n", () => {
     "Open leftover {kind} pair: {title} · {criterion}",
     "Open this post to read the criterion it sat closest to after main effects.",
     "Open this post to read the criterion it sat farthest from after main effects.",
+    "Leftover map leaves unexplained U {value} after IRT main effects. Open this post to read {criterion}.",
     "Read observed Y {observed} and expected E {expected} after IRT main effects, then open this post.",
     "Leftover map has no leftover structure after IRT main effects. Open this post.",
     "Leftover map rank {rank} after IRT main effects. Open this post.",
@@ -149,6 +150,33 @@ describe("i18n", () => {
       ),
     ).toBe(expected);
   });
+
+  it.each([
+    [
+      "ko",
+      "잔여 지도가 IRT 주효과 이후 설명되지 않은 U +0.05을(를) 남깁니다. sales-lead 기준을 읽으려면 이 글을 여세요.",
+    ],
+    [
+      "zh",
+      "残差图在 IRT 主效应后留下未解释的 U +0.05。打开这篇帖子阅读 sales-lead。",
+    ],
+    [
+      "ja",
+      "残差マップはIRT主効果後の未説明 U +0.05 を残します。この投稿を開いて sales-lead を読んでください。",
+    ],
+    [
+      "vi",
+      "Bản đồ phần dư để lại U +0.05 chưa giải thích sau hiệu ứng chính IRT. Mở bài viết này để đọc sales-lead.",
+    ],
+  ] as const)("formats leftover-map unexplained next action in %s", (locale, expected) => {
+    setLocale(locale);
+    expect(
+      tf(
+        "Leftover map leaves unexplained U {value} after IRT main effects. Open this post to read {criterion}.",
+        { value: "+0.05", criterion: "sales-lead" },
+      ),
+    ).toBe(expected);
+  });
 });
 
 describe("locale-aware source labels", () => {
diff --git a/frontend/src/i18n.ts b/frontend/src/i18n.ts
index ef451f253..f4d93d9ed 100644
--- a/frontend/src/i18n.ts
+++ b/frontend/src/i18n.ts
@@ -389,6 +389,8 @@ const TRANSLATIONS: Partial<Record<Locale, Record<string, string>>> = {
       "주효과 이후 가장 가깝게 앉은 기준을 읽으려면 이 글을 여세요.",
     "Open this post to read the criterion it sat farthest from after main effects.":
       "주효과 이후 가장 멀리 앉은 기준을 읽으려면 이 글을 여세요.",
+    "Leftover map leaves unexplained U {value} after IRT main effects. Open this post to read {criterion}.":
+      "잔여 지도가 IRT 주효과 이후 설명되지 않은 U {value}을(를) 남깁니다. {criterion} 기준을 읽으려면 이 글을 여세요.",
     "Read observed Y {observed} and expected E {expected} after IRT main effects, then open this post.":
       "IRT 주효과 이후 관측 Y {observed}와 기대 E {expected}를 읽은 다음, 이 글을 여세요.",
     "Leftover map has no leftover structure after IRT main effects. Open this post.":
@@ -770,6 +772,8 @@ const TRANSLATIONS: Partial<Record<Locale, Record<string, string>>> = {
       "打开这篇帖子，阅读主效应后距离最近的准则。",
     "Open this post to read the criterion it sat farthest from after main effects.":
       "打开这篇帖子，阅读主效应后距离最远的准则。",
+    "Leftover map leaves unexplained U {value} after IRT main effects. Open this post to read {criterion}.":
+      "残差图在 IRT 主效应后留下未解释的 U {value}。打开这篇帖子阅读 {criterion}。",
     "Read observed Y {observed} and expected E {expected} after IRT main effects, then open this post.":
       "阅读 IRT 主效应后的观测 Y {observed} 与期望 E {expected}，然后打开这篇帖子。",
     "Leftover map has no leftover structure after IRT main effects. Open this post.":
@@ -1151,6 +1155,8 @@ const TRANSLATIONS: Partial<Record<Locale, Record<string, string>>> = {
       "主効果後に最も近くなった基準を読むには、この投稿を開いてください。",
     "Open this post to read the criterion it sat farthest from after main effects.":
       "主効果後に最も遠くなった基準を読むには、この投稿を開いてください。",
+    "Leftover map leaves unexplained U {value} after IRT main effects. Open this post to read {criterion}.":
+      "残差マップはIRT主効果後の未説明 U {value} を残します。この投稿を開いて {criterion} を読んでください。",
     "Read observed Y {observed} and expected E {expected} after IRT main effects, then open this post.":
       "IRT主効果後の観測 Y {observed} と期待 E {expected} を読んでから、この投稿を開いてください。",
     "Leftover map has no leftover structure after IRT main effects. Open this post.":
@@ -1532,6 +1538,8 @@ const TRANSLATIONS: Partial<Record<Locale, Record<string, string>>> = {
       "Mở bài viết này để đọc tiêu chí nằm gần nhất sau hiệu ứng chính.",
     "Open this post to read the criterion it sat farthest from after main effects.":
       "Mở bài viết này để đọc tiêu chí nằm xa nhất sau hiệu ứng chính.",
+    "Leftover map leaves unexplained U {value} after IRT main effects. Open this post to read {criterion}.":
+      "Bản đồ phần dư để lại U {value} chưa giải thích sau hiệu ứng chính IRT. Mở bài viết này để đọc {criterion}.",
     "Read observed Y {observed} and expected E {expected} after IRT main effects, then open this post.":
       "Đọc Y quan sát {observed} và E kỳ vọng {expected} sau hiệu ứng chính IRT, rồi mở bài viết này.",
     "Leftover map has no leftover structure after IRT main effects. Open this post.":
diff --git a/frontend/src/leftoverMapUnexplained.test.ts b/frontend/src/leftoverMapUnexplained.test.ts
new file mode 100644
index 000000000..aa23a125b
--- /dev/null
+++ b/frontend/src/leftoverMapUnexplained.test.ts
@@ -0,0 +1,22 @@
+import { describe, expect, it } from "vitest";
+import {
+  formatLeftoverMapUnexplained,
+  formatSignedLeftoverValue,
+} from "./leftoverMapUnexplained";
+
+describe("formatLeftoverMapUnexplained", () => {
+  it("names unexplained leftover without inventing a leftover score", () => {
+    expect(formatLeftoverMapUnexplained(0.05)).toBe("U +0.05");
+    expect(formatLeftoverMapUnexplained(-0.25)).toBe("U \u22120.25");
+    expect(formatLeftoverMapUnexplained(0)).toBe("U 0.00");
+    expect(formatSignedLeftoverValue(0.05)).toBe("+0.05");
+    expect(formatSignedLeftoverValue(-0.25)).toBe("\u22120.25");
+  });
+
+  it("omits the badge when unexplained leftover is missing or non-finite", () => {
+    expect(formatLeftoverMapUnexplained(null)).toBeNull();
+    expect(formatLeftoverMapUnexplained(undefined)).toBeNull();
+    expect(formatLeftoverMapUnexplained(Number.NaN)).toBeNull();
+    expect(formatLeftoverMapUnexplained(Number.POSITIVE_INFINITY)).toBeNull();
+  });
+});
diff --git a/frontend/src/leftoverMapUnexplained.ts b/frontend/src/leftoverMapUnexplained.ts
new file mode 100644
index 000000000..4fa23c935
--- /dev/null
+++ b/frontend/src/leftoverMapUnexplained.ts
@@ -0,0 +1,26 @@
+/** Unexplained leftover ``U = R − R̂`` after two-axis reconstruction. */
+
+export const LEFTOVER_MAP_UNEXPLAINED_ACTION =
+  "Leftover map leaves unexplained U {value} after IRT main effects. 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 `+${magnitude}`;
+  }
+  if (value < 0) {
+    return `\u2212${magnitude}`;
+  }
+  return magnitude;
+}
+
+export function formatLeftoverMapUnexplained(value: number | null | undefined): string | null {
+  if (value == null) {
+    return null;
+  }
+  const signed = formatSignedLeftoverValue(value);
+  return signed === null ? null : `U ${signed}`;
+}
diff --git a/lineageweave/leftover_pairs.py b/lineageweave/leftover_pairs.py
index 06a85d715..7531607e9 100644
--- a/lineageweave/leftover_pairs.py
+++ b/lineageweave/leftover_pairs.py
@@ -1,6 +1,6 @@
 """Jeon leftover post–criterion pairs after a main-effect IRT.
 
-Implements ADR 0048 as amended by ADR 0119, ADR 0163, and ADR 0164.
+Implements ADR 0048 as amended by ADR 0119, ADR 0163, ADR 0164, and ADR 0182.
 
 Does not import ``fast_mlsirm`` or ``period_report``. A Gabriel biplot
 of the residual ``R = Y − E[Y|θ, item]`` supplies person and item
@@ -13,6 +13,11 @@
 after the second are dropped. Each pair also names the full leftover-map
 rank so a rank-0 collapse is not read as leftover structure. Axis share
 is the Gabriel inertia of the first two leftover-map axes (ADR 0148).
+Each pair also names unexplained leftover ``U = R − R̂`` after two-axis
+Gabriel reconstruction ``R̂ = ξ_{1:2} · ζ_{1:2}`` so the leftover cell
+the map does not reconstruct is not confused with leftover residual
+``R`` or leftover-map distance ``d``. Reconstruction is computed
+internally and is not persisted.
 """
 
 from __future__ import annotations
@@ -40,6 +45,7 @@ class LeftoverPair:
     observed_response: float
     expected_response: float
     leftover_map_rank: int
+    leftover_map_unexplained: float | None = None
 
 
 @dataclass(frozen=True)
@@ -66,7 +72,11 @@ def leftover_pairs_from_residual(
     stable closest/farthest pair so seed is not empty and does not
     invent a leftover score. Stored residual equals observed ``Y`` minus
     expected ``E[Y|θ, item]``. Stored leftover-map rank is the number
-    of Gabriel singular values above the floor.
+    of Gabriel singular values above the floor. When Gabriel coordinates
+    exist, unexplained leftover ``U = R − R̂`` names the leftover cell
+    the two-axis map does not reconstruct; ``R̂`` stays internal and is
+    never persisted. Fallback pairs (no complete-case map) omit
+    unexplained leftover rather than fabricating one.
     """
     pairs, _axes = leftover_map_from_residual(post_ids, item_codes, matrix, expected)
     return pairs
@@ -114,7 +124,7 @@ def leftover_map_from_residual(
     )
     axes = leftover_map_axes_from_singular(singular)
     leftover_map_rank = int(singular.size)
-    candidates: list[tuple[float, str, str, float, float, float]] = []
+    candidates: list[tuple[float, str, str, float, float, float, float | None]] = []
     if person_pos is not None and item_pos is not None:
         person_index = np.flatnonzero(keep_person)
         item_index = np.flatnonzero(keep_item)
@@ -130,8 +140,22 @@ def leftover_map_from_residual(
             )
             if not np.isfinite(distance):
                 continue
+            reconstruction = float(
+                np.dot(person_xy[local_person[person]], item_xy[local_item[item]])
+            )
+            unexplained = _unexplained_leftover(float(residual[person, item]), reconstruction)
             candidates.append(
-                _candidate_row(post_ids, item_codes, matrix, expected, residual, person, item, distance)
+                _candidate_row(
+                    post_ids,
+                    item_codes,
+                    matrix,
+                    expected,
+                    residual,
+                    person,
+                    item,
+                    distance,
+                    unexplained,
+                )
             )
     if not candidates:
         leftover_map_rank = 0
@@ -147,6 +171,7 @@ def leftover_map_from_residual(
                     person,
                     item,
                     max(distance, 0.0),
+                    None,
                 )
             )
     closest = min(candidates, key=lambda row: (row[0], row[1], row[2]))
@@ -158,6 +183,16 @@ def leftover_map_from_residual(
     return pairs, axes
 
 
+def _unexplained_leftover(residual: float, reconstruction: float) -> float | None:
+    """Return ``U = R − R̂`` when both terms are finite; otherwise omit."""
+    if not np.isfinite(reconstruction):
+        return None
+    unexplained = residual - reconstruction
+    if not np.isfinite(unexplained):
+        return None
+    return float(unexplained)
+
+
 def _candidate_row(
     post_ids: list[str],
     item_codes: tuple[str, ...],
@@ -167,8 +202,9 @@ def _candidate_row(
     person: int,
     item: int,
     distance: float,
-) -> tuple[float, str, str, float, float, float]:
-    """One observed leftover cell: distance, ids, residual, Y, E."""
+    leftover_map_unexplained: float | None,
+) -> tuple[float, str, str, float, float, float, float | None]:
+    """One observed leftover cell: distance, ids, residual, Y, E, unexplained U."""
     leftover_residual = float(residual[person, item])
     observed_response = float(matrix[person, item])
     expected_response = float(expected[person, item])
@@ -181,12 +217,13 @@ def _candidate_row(
         leftover_residual,
         observed_response,
         expected_response,
+        leftover_map_unexplained,
     )
 
 
 def _pair_from_candidate(
     pair_kind: str,
-    row: tuple[float, str, str, float, float, float],
+    row: tuple[float, str, str, float, float, float, float | None],
     leftover_map_rank: int,
 ) -> LeftoverPair:
     """Build a leftover pair from a candidate row."""
@@ -201,6 +238,7 @@ def _pair_from_candidate(
         observed_response=row[4],
         expected_response=row[5],
         leftover_map_rank=leftover_map_rank,
+        leftover_map_unexplained=row[6],
     )
 
 
@@ -289,7 +327,13 @@ def _leftover_map_positions(
 
 
 def _pad_map_axes(positions: np.ndarray) -> np.ndarray:
-    """Pad or truncate Gabriel coordinates to two leftover-map axes."""
+    """Pad or truncate Gabriel coordinates to two leftover-map axes.
+
+    Unused axes pad with zero rather than inventing a second component.
+    Hidden SVD axes after the second are dropped so reconstruction is
+    ``ξ_{1:2} · ζ_{1:2}``, not the full-rank inner product. That
+    reconstruction stays internal; only unexplained leftover is named.
+    """
     padded = np.zeros((positions.shape[0], _LEFTOVER_MAP_AXES), dtype=np.float64)
     width = min(_LEFTOVER_MAP_AXES, positions.shape[1])
     padded[:, :width] = positions[:, :width]
diff --git a/migrations/0182_report_leftover_map_unexplained.sql b/migrations/0182_report_leftover_map_unexplained.sql
new file mode 100644
index 000000000..9dfc250ed
--- /dev/null
+++ b/migrations/0182_report_leftover_map_unexplained.sql
@@ -0,0 +1,10 @@
+-- ADR 0182: persist unexplained leftover U = R − R̂ after two-axis
+-- leftover-map reconstruction R̂ = ξ_{1:2} · ζ_{1:2}. Distance stays
+-- Euclidean leftover-map d. Reconstruction is computed internally and
+-- is not persisted. Upgrade column is nullable so older leftover rows
+-- keep distance and residual without fabricating unexplained leftover.
+-- 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_unexplained numeric;
diff --git a/migrations/rollback/0182_report_leftover_map_unexplained.sql b/migrations/rollback/0182_report_leftover_map_unexplained.sql
new file mode 100644
index 000000000..2352e3054
--- /dev/null
+++ b/migrations/rollback/0182_report_leftover_map_unexplained.sql
@@ -0,0 +1,4 @@
+-- Reverse 0182. Leftover distance and residual stay on the pair row.
+
+alter table report_leftover_pair
+    drop column if exists leftover_map_unexplained;
diff --git a/pyproject.toml b/pyproject.toml
index 8b01c9850..c5ffda1bf 100644
--- a/pyproject.toml
+++ b/pyproject.toml
@@ -1,6 +1,6 @@
 [project]
 name = "lineageweave"
-version = "2.12.19"
+version = "2.12.26"
 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/scripts/seed_demo_data.py b/scripts/seed_demo_data.py
index 514ae2768..62249b9e8 100644
--- a/scripts/seed_demo_data.py
+++ b/scripts/seed_demo_data.py
@@ -119,6 +119,7 @@ def seed(
             cur.execute((migrations / "0012_report_leftover_pair.sql").read_text())
             cur.execute((migrations / "0163_report_leftover_observed_expected.sql").read_text())
             cur.execute((migrations / "0164_report_leftover_map_rank.sql").read_text())
+            cur.execute((migrations / "0182_report_leftover_map_unexplained.sql").read_text())
             cur.execute((migrations / "0060_role_responsibility_agent_type.sql").read_text())
             cur.execute((migrations / "0013_person_job_title.sql").read_text())
             cur.execute((migrations / "0014_role_responsibility_team_actor_type.sql").read_text())
@@ -1198,8 +1199,9 @@ def _persist_seed_period_report(
             "insert into report_leftover_pair ("
             "grouping_kind, grouping_key, period_code, rubric_version, "
             "pair_kind, post_id, criterion_code, leftover_distance, leftover_residual, "
-            "observed_response, expected_response, leftover_map_rank"
-            ") values (%s,%s,%s,%s,%s,%s,%s,%s,%s,%s,%s,%s)",
+            "observed_response, expected_response, leftover_map_rank, "
+            "leftover_map_unexplained"
+            ") values (%s,%s,%s,%s,%s,%s,%s,%s,%s,%s,%s,%s,%s)",
             (
                 grouping_kind,
                 grouping_key,
@@ -1213,6 +1215,7 @@ def _persist_seed_period_report(
                 pair.observed_response,
                 pair.expected_response,
                 pair.leftover_map_rank,
+                pair.leftover_map_unexplained,
             ),
         )
     for axis in report.leftover_map_axes:
diff --git a/tests/test_leftover_pairs.py b/tests/test_leftover_pairs.py
index 2e6881801..135ddcd2b 100644
--- a/tests/test_leftover_pairs.py
+++ b/tests/test_leftover_pairs.py
@@ -1,6 +1,7 @@
 """Leftover post–criterion pairs after the main-effect IRT.
 
-Covers ADR 0048 as amended by ADR 0119, ADR 0148, ADR 0163, and ADR 0164.
+Covers ADR 0048 as amended by ADR 0119, ADR 0148, ADR 0163, ADR 0164,
+and ADR 0182.
 
 Uses a constructed residual matrix so the closest and farthest pair
 are known without calling ``fit_polytomous``. Loads
@@ -99,6 +100,9 @@ def test_leftover_residual_biplot_separates_aligned_and_opposed_cells() -> None:
     assert farthest.observed_response == pytest.approx(-2.0)
     assert farthest.expected_response == pytest.approx(0.0)
     assert farthest.leftover_distance == pytest.approx(2.0 * np.sqrt(2.0), rel=1e-6)
+    assert closest.leftover_map_unexplained == pytest.approx(0.0, abs=1e-6)
+    assert farthest.leftover_map_unexplained == pytest.approx(0.0, abs=1e-6)
+    assert not hasattr(closest, "leftover_map_reconstruction")
     for pair in pairs:
         _assert_residual_reconciles(pair)
         assert pair.leftover_map_rank == 1
@@ -120,6 +124,8 @@ def test_zero_residual_still_emits_stable_leftover_pairs() -> None:
     assert pairs[0].expected_response == pytest.approx(1.0)
     assert pairs[1].post_id == "beta-post"
     assert pairs[1].criterion_code == "item_two"
+    assert pairs[0].leftover_map_unexplained == pytest.approx(0.0)
+    assert pairs[1].leftover_map_unexplained == pytest.approx(0.0)
     for pair in pairs:
         _assert_residual_reconciles(pair)
         assert pair.leftover_map_rank == 0
@@ -152,6 +158,7 @@ def test_partial_observation_does_not_treat_missing_as_zero_residual() -> None:
     for pair in pairs:
         _assert_residual_reconciles(pair)
         assert pair.leftover_map_rank == 1
+        assert pair.leftover_map_unexplained == pytest.approx(0.0, abs=1e-6)
 
 
 def test_leftover_is_empty_without_observed_cells() -> None:
@@ -208,6 +215,7 @@ def test_leftover_residual_rejects_database_tolerance_boundary() -> None:
             0,
             0,
             0.0,
+            None,
         )
 
 
@@ -308,6 +316,56 @@ def test_rank_four_pair_distances_match_two_dimensional_gabriel_coords() -> None
     assert (post_index[farthest.post_id], item_index[farthest.criterion_code]) == farthest_map
 
 
+def test_unexplained_equals_residual_minus_two_axis_reconstruction() -> None:
+    """Unexplained leftover U is R − R̂, not leftover residual R, not leftover-map distance d.
+
+    Uses the same rank-4 matrix as the two-axis distance proof above: R̂ is
+    the two-axis Gabriel reconstruction ``person_map @ item_map.T``, built
+    from the same padded coordinates ``leftover_distance`` already uses.
+    """
+    post_ids = ["post-a", "post-b", "post-c", "post-d"]
+    item_codes = ("item-a", "item-b", "item-c", "item-d")
+    matrix = np.array(
+        [
+            [4.0, 1.0, 0.0, -1.0],
+            [0.0, 3.0, 1.0, -2.0],
+            [-2.0, 0.0, 2.0, 1.0],
+            [1.0, -1.0, 0.0, 4.0],
+        ],
+        dtype=np.float64,
+    )
+    expected = np.zeros_like(matrix)
+    filled = matrix - float(np.mean(matrix))
+    person_full, item_full = _gabriel_positions(filled)
+    assert person_full.shape[1] == 4
+    person_map = _pad_map_axes(person_full)
+    item_map = _pad_map_axes(item_full)
+    reconstruction = person_map @ item_map.T
+    full_inner = person_full @ item_full.T
+    assert float(np.max(np.abs(reconstruction - filled))) > 1e-6
+    assert float(np.max(np.abs(reconstruction - full_inner))) > 1e-6
+
+    pairs = leftover_pairs_from_residual(post_ids, item_codes, matrix, expected)
+    assert [pair.pair_kind for pair in pairs] == [PAIR_KIND_CLOSEST, PAIR_KIND_FARTHEST]
+    post_index = {post_id: index for index, post_id in enumerate(post_ids)}
+    item_index = {code: index for index, code in enumerate(item_codes)}
+    for pair in pairs:
+        person = post_index[pair.post_id]
+        item = item_index[pair.criterion_code]
+        expected_unexplained = float(pair.leftover_residual) - float(reconstruction[person, item])
+        assert pair.leftover_map_unexplained == pytest.approx(expected_unexplained)
+        assert pair.leftover_map_unexplained != pytest.approx(pair.leftover_residual)
+        assert pair.leftover_map_unexplained != pytest.approx(pair.leftover_distance)
+        assert not hasattr(pair, "leftover_map_reconstruction")
+
+
+def test_pad_map_axes_truncates_hidden_svd_components() -> None:
+    """Axes after the second leftover-map axis do not enter reconstruction."""
+    padded = leftover._pad_map_axes(np.array([[1.0, 2.0, 9.0]], dtype=np.float64))
+    assert padded.shape == (1, 2)
+    assert padded[0].tolist() == pytest.approx([1.0, 2.0])
+
+
 def test_rejects_response_and_expectation_shape_mismatches() -> None:
     """Scientific inputs must match their declared post and criterion axes."""
     with pytest.raises(ValueError, match="matrix shape"):
@@ -343,6 +401,27 @@ def test_sparse_residual_uses_only_observed_cells_for_fallback_distance() -> Non
     assert [pair.leftover_map_rank for pair in pairs] == [0, 0]
 
 
+def test_leftover_fallback_omits_unexplained_without_complete_case_map() -> None:
+    """No complete-case rectangle: persist distance from |R − center|, omit U."""
+    post_ids = ["sparse-a", "sparse-b"]
+    item_codes = ("item_near", "item_far")
+    matrix = np.array(
+        [
+            [2.0, np.nan],
+            [np.nan, -2.0],
+        ],
+        dtype=np.float64,
+    )
+    expected = np.zeros_like(matrix)
+    pairs = leftover_pairs_from_residual(post_ids, item_codes, matrix, expected)
+    assert [pair.pair_kind for pair in pairs] == [PAIR_KIND_CLOSEST, PAIR_KIND_FARTHEST]
+    assert {pair.post_id for pair in pairs} == {"sparse-a", "sparse-b"}
+    for pair in pairs:
+        assert pair.leftover_map_unexplained is None
+        assert pair.leftover_distance >= 0.0
+        assert not hasattr(pair, "leftover_map_reconstruction")
+
+
 def test_nonfinite_map_distance_falls_back_to_centered_residual(
     monkeypatch: pytest.MonkeyPatch,
 ) -> None:
@@ -388,6 +467,6 @@ def test_leftover_map_rank_rejects_negative_rank() -> None:
     with pytest.raises(ValueError, match="non-negative integer"):
         leftover._pair_from_candidate(
             PAIR_KIND_CLOSEST,
-            (0.0, "public-post", "sales_lead_specificity", 0.0, 1.0, 1.0),
+            (0.0, "public-post", "sales_lead_specificity", 0.0, 1.0, 1.0, None),
             -1,
         )
diff --git a/tests/test_period_report.py b/tests/test_period_report.py
index f5cee497b..943321394 100644
--- a/tests/test_period_report.py
+++ b/tests/test_period_report.py
@@ -270,6 +270,9 @@ def test_calibrated_report_attaches_leftover_pairs() -> None:
             pair.observed_response - pair.expected_response, abs=1e-6
         )
         assert pair.leftover_map_rank >= 0
+        if pair.leftover_map_unexplained is not None:
+            assert np.isfinite(pair.leftover_map_unexplained)
+        assert not hasattr(pair, "leftover_map_reconstruction")
     assert [axis.axis_index for axis in report.leftover_map_axes] == [1, 2]
     for axis in report.leftover_map_axes:
         assert axis.leftover_singular_value >= 0.0
diff --git a/tests/test_schema.py b/tests/test_schema.py
index 253ec3a84..f2a295cc6 100644
--- a/tests/test_schema.py
+++ b/tests/test_schema.py
@@ -58,6 +58,11 @@
     / "migrations"
     / "0169_report_leftover_map_axis.sql"
 )
+_LEFTOVER_MAP_UNEXPLAINED_MIGRATION = (
+    Path(__file__).resolve().parents[1]
+    / "migrations"
+    / "0182_report_leftover_map_unexplained.sql"
+)
 
 
 def _postgres_available() -> bool:
@@ -97,6 +102,7 @@ def schema_db():
                 cur.execute(_LEFTOVER_OBSERVED_EXPECTED_MIGRATION.read_text())
                 cur.execute(_LEFTOVER_MAP_RANK_MIGRATION.read_text())
                 cur.execute(_LEFTOVER_MAP_AXIS_MIGRATION.read_text())
+                cur.execute(_LEFTOVER_MAP_UNEXPLAINED_MIGRATION.read_text())
             conn.commit()
             yield conn
         finally:
@@ -227,6 +233,23 @@ def test_leftover_pair_names_leftover_map_rank_column(schema_db) -> None:
     assert columns["leftover_map_rank"] == "YES"
 
 
+def test_leftover_pair_names_nullable_unexplained_column(schema_db) -> None:
+    """Every install path preserves legacy pairs while naming unexplained leftover."""
+    with schema_db.cursor() as cur:
+        cur.execute(
+            """
+            select column_name, is_nullable
+            from information_schema.columns
+            where table_name = 'report_leftover_pair'
+            """
+        )
+        columns = dict(cur.fetchall())
+    assert columns["leftover_map_unexplained"] == "YES"
+    assert columns["leftover_residual"] == "NO"
+    assert columns["leftover_distance"] == "NO"
+    assert "leftover_map_reconstruction" not in columns
+
+
 def test_leftover_map_axis_references_period_score(schema_db) -> None:
     """Axis share is report-level; it must cascade with the period score."""
     with schema_db.cursor() as cur:
diff --git a/uv.lock b/uv.lock
index 095fa6696..8bf26bf75 100644
--- a/uv.lock
+++ b/uv.lock
@@ -454,7 +454,7 @@ wheels = [
 
 [[package]]
 name = "lineageweave"
-version = "2.12.19"
+version = "2.12.26"
 source = { editable = "." }
 dependencies = [
     { name = "certifi" },