ContextualWisdomLab · seonghobae · Aug 24, 2026 · Aug 24, 2026 · Aug 24, 2026 · Aug 24, 2026
diff --git a/AGENTS.md b/AGENTS.md
@@ -259,23 +259,24 @@ 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) are computed in `lineageweave/leftover_pairs.py` from the
+0163 / 0164 / 0182 / 0186) 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 with the leftover criterion current
-in Post quality (ADR 0158). 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. Complete-case coverage (ADR 0168) persists to
+`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, plus two-axis
+leftover-map reconstruction `R̂` of centered leftover, plus unexplained
+leftover `U = R − R̂` next to leftover-map distance `d`. They sit above
+the member list so a click opens that post with the leftover criterion
+current in Post quality (ADR 0158). 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.
+Complete-case coverage (ADR 0168) persists to
 `report_leftover_map_coverage` and captions the pair list with how
-many scored posts entered the map.
+many scored posts entered the map; without a complete-case rectangle
+there is no pair to caption, only coverage counts.
 
 Period leftover pairs (ADR 0017 / 0018 / 0048 / 0049 / 0149) are computed in
 `lineageweave/leftover_pairs.py` from the residual after a real

diff --git a/ARCHITECTURE.md b/ARCHITECTURE.md
@@ -596,11 +596,12 @@ 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 / 0049 / 0119 / 0158 / 0162 / 0163 / 0164 / 0182) persist to
+ADR 0017 / 0048 / 0049 / 0119 / 0158 / 0162 / 0163 / 0164 / 0182 / 0186) persist to
 `report_leftover_pair` with signed residual `R`, observed `Y`, expected
-`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
+`E[Y|θ, item]`, full leftover-map rank, two-axis leftover-map
+reconstruction `R̂`, 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`. Complete-case leftover-map coverage (ADR
 0168) persists to `report_leftover_map_coverage` so readers see how
 many scored posts entered the factorization. Results persist to
@@ -617,7 +618,8 @@ through opens those DAG posts. Report members include the earliest
 open ticket title, status lookup label, and due date when one exists. The home page renders
 the actual mean θ, the FIPC delta, the CAT-selected item, leftover
 closest/farthest pairs (signed residual `R`, observed `Y`, expected
-`E`, full rank, and two-axis leftover-map distance `d` after IRT main
+`E`, full rank, two-axis leftover-map reconstruction `R̂`, and
+two-axis leftover-map distance `d` after IRT main
 effects) above the member list, leftover-map axis share for residual
 SVD axes 1 and 2, and complete-case coverage captions (map used N of M
 scored posts), plus the

diff --git a/CHANGELOG.d/2.12.30-leftover-map-reconstruction.md b/CHANGELOG.d/2.12.30-leftover-map-reconstruction.md
@@ -0,0 +1,8 @@
+## 2.12.30 — Leftover-map reconstruction
+
+- Persist two-axis leftover-map reconstruction `R̂_c = ξ_{1:2} · ζ_{1:2}`
+  of centered leftover on leftover post–criterion pairs (ADR 0186).
+  After `make seed`, closest and farthest leftover pairs sit above the
+  member list with `R̂ {value}` next to leftover-map distance `d`; click
+  opens that post. Fallback pairs omit reconstruction. Rank-0 origin
+  maps store `R̂ = 0`. Never invent a leftover score or a theta.
diff --git a/CHANGELOG.md b/CHANGELOG.md
@@ -107,15 +107,24 @@ All notable changes to this project are documented here. Format follows
   is unwired, with `이 범위의 일정을 아직 받을 수 없습니다`. Weekly VOC and
   newspaper stay on the board.
 
+## [2.12.30] - 2026-08-24
+
+### Added
+
+- Period leftover pair rows now name two-axis leftover-map reconstruction
+  `R̂` of centered leftover next to leftover-map distance `d`, then open
+  that post (Gabriel, 1971; Jeon et al., 2021, eq. 3; ADR 0186). Fallback
+  pairs omit the reconstruction rather than inventing one. Rank-0 origin
+  maps store `R̂ = 0`. Never invent a leftover score or a theta.
+
 ## [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.
+  omits the badge rather than inventing a leftover score.
 
 ## [2.12.19] - 2026-08-24
 

diff --git a/CLAUDE.md b/CLAUDE.md
@@ -48,7 +48,7 @@ cutoff.
 
 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), the text-channel embedding swap and cosine
+(ADR 0048–0164 / 0182 / 0186), the text-channel embedding swap and cosine
 clamp (ADR 0190), per-edge channel-score persistence (ADR 0195),
 migration replay (ADR 0166), docstring coverage, and the measurement
 boundary are all stated in [AGENTS.md](AGENTS.md) -- read it before

diff --git a/backend/app/report_ingestion.py b/backend/app/report_ingestion.py
@@ -445,8 +445,8 @@ async def persist_period_report(
                 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,
-                leftover_map_unexplained
-            ) values ($1,$2,$3,$4,$5,$6,$7,$8,$9,$10,$11,$12,$13)
+                leftover_map_reconstruction, leftover_map_unexplained
+            ) values ($1,$2,$3,$4,$5,$6,$7,$8,$9,$10,$11,$12,$13,$14)
             """,
             grouping_kind,
             grouping_key,
@@ -460,6 +460,7 @@ async def persist_period_report(
             pair.observed_response,
             pair.expected_response,
             pair.leftover_map_rank,
+            pair.leftover_map_reconstruction,
             pair.leftover_map_unexplained,
         )
     for axis in report.leftover_map_axes:
@@ -646,7 +647,7 @@ async def fetch_period_reports(
         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,
-               lp.leftover_map_unexplained, p.post_title,
+               lp.leftover_map_reconstruction, 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
@@ -785,6 +786,11 @@ async def fetch_period_reports(
                             if row["leftover_map_rank"] is None
                             else int(row["leftover_map_rank"])
                         ),
+                        "leftover_map_reconstruction": (
+                            None
+                            if row["leftover_map_reconstruction"] is None
+                            else float(row["leftover_map_reconstruction"])
+                        ),
                         "leftover_map_unexplained": (
                             None
                             if row["leftover_map_unexplained"] is None

diff --git a/backend/tests/test_api.py b/backend/tests/test_api.py
@@ -12,6 +12,7 @@
 
 from __future__ import annotations
 
+import math
 import os
 import uuid
 from contextlib import closing
@@ -162,6 +163,11 @@
     / "migrations"
     / "0182_report_leftover_map_unexplained.sql"
 )
+_LEFTOVER_MAP_RECONSTRUCTION_MIGRATION = (
+    Path(__file__).resolve().parents[2]
+    / "migrations"
+    / "0186_report_leftover_map_reconstruction.sql"
+)
 
 
 def _postgres_available() -> bool:
@@ -305,6 +311,7 @@ def seeded_db(demo_analyst_token):
             cur.execute(_GLOBAL_ASK_JOB_MIGRATION.read_text())
             cur.execute(_LEFTOVER_MAP_AXIS_MIGRATION.read_text())
             cur.execute(_LEFTOVER_MAP_UNEXPLAINED_MIGRATION.read_text())
+            cur.execute(_LEFTOVER_MAP_RECONSTRUCTION_MIGRATION.read_text())
             cur.execute(
                 "insert into common_lookup_value (lookup_category, lookup_code, lookup_label) values "
                 "('corporate_entity_level', 'group', 'Group'), "
@@ -5044,9 +5051,11 @@ 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
+        reconstruction = pair.get("leftover_map_reconstruction")
+        if reconstruction is not None:
+            assert math.isfinite(reconstruction)
         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
@@ -100,14 +100,14 @@ 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 / 0182): after
-   IRT main effects, persist closest and farthest post–criterion pairs
-   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.
+7. **Leftover-pair slice** (shipped in 0.71.2; ADR 0017 / 0018 / 0048 /
+   0049 / 0182, amended by ADR 0186): after IRT main effects, persist
+   closest and farthest post–criterion pairs from the residual leftover
+   map, including two-axis leftover-map reconstruction `R̂` and
+   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 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
    SVD. Rank-0 residuals emit two zero-share axes. Do not invent a

diff --git a/docs/adr/0048-persist-lsirm-leftover-pairs.md b/docs/adr/0048-persist-lsirm-leftover-pairs.md
@@ -5,7 +5,8 @@
 **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 0182](0182-leftover-map-unexplained.md) (unexplained leftover U)
+[ADR 0182](0182-leftover-map-unexplained.md) (unexplained leftover U);
+[ADR 0186](0186-leftover-map-reconstruction.md) (two-axis reconstruction R̂)
 
 ## Context
 
@@ -38,11 +39,11 @@ 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). 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.
+leftover row also names two-axis leftover-map reconstruction
+`R̂_c = ξ_{1:2} · ζ_{1:2}` of centered leftover (ADR 0186) and
+unexplained leftover `U = R − R̂_c` 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).
 
 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
@@ -6,7 +6,8 @@
 [ADR 0163](0163-leftover-observed-expected.md) (observed Y and expected E);
 [ADR 0164](0164-leftover-map-rank.md) (full map rank);
 [ADR 0182](0182-leftover-map-unexplained.md) (unexplained leftover U);
-[ADR 0158](0158-leftover-criterion-evaluation-landing.md) (criterion evaluation landing)
+[ADR 0158](0158-leftover-criterion-evaluation-landing.md) (criterion evaluation landing);
+[ADR 0186](0186-leftover-map-reconstruction.md) (two-axis reconstruction R̂)
 
 ## Context
 
@@ -23,8 +24,9 @@ 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`, expected `E` when finite, and
-unexplained leftover `U` when finite.
+distance, full map rank, leftover-map reconstruction `R̂` when finite,
+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, rank 0 explicitly names no
 leftover structure, and unexplained leftover names "leftover map leaves
@@ -35,8 +37,9 @@ Clicking the button opens that post with leftover focus so Post
 quality marks the named criterion current (ADR 0158). 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), unexplained leftover naming
-is [ADR 0182](0182-leftover-map-unexplained.md).
+is [ADR 0164](0164-leftover-map-rank.md), reconstruction naming is
+[ADR 0186](0186-leftover-map-reconstruction.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 with the leftover

diff --git a/docs/adr/0186-leftover-map-reconstruction.md b/docs/adr/0186-leftover-map-reconstruction.md
@@ -0,0 +1,93 @@
+# ADR 0186 — Name leftover-map reconstruction on leftover pairs
+
+**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 *centered* 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_c = ξ_{1:2} · ζ_{1:2}` therefore recovers centered leftover
+`R̃ = R − center`, not raw residual `R` and not leftover-map distance
+`d`. Hiding `R̂_c` lets a buyer read distance as if it named how much
+leftover the two leftover-map axes reconstruct.
+
+This increment does not persist leftover-map coordinates, does not
+persist leftover-map unexplained leftover `U` or `U_c`, does not
+persist leftover-map explained leftover share `e`, unexplained leftover
+share `s`, or leftover-map cross share `x`, does not name leftover-map
+cosine or 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 two-axis Euclidean (ADR 0119).
+
+The unprotected-stack reconstructions for neighbouring leftover facts
+use 0162–0185. This protected-main increment uses **0186** so it does
+not collide with leftover-map cross share (0185), leftover-map
+explained leftover share (0184), leftover-map unexplained leftover
+share (0183), leftover-map unexplained leftover (0182), leftover-map
+length (0181 on the length stack), leftover-map cosine (0180),
+leftover-map inner product (0179), leftover residual disclosure
+(0162 / 0178), leftover observed `Y` / expected `E` (0163 / 0170),
+leftover-map rank (0164 / 0172), two-axis leftover-map distance
+(0119 / 0166), leftover coverage (0165 / 0168), leftover-map axis
+share (0148), or leftover interaction-map persistence (0121).
+
+## Decision
+
+Each leftover pair names `leftover_map_reconstruction` — two-axis
+Gabriel reconstruction `R̂_c = ξ_{1:2} · ζ_{1:2}` of centered leftover
+after IRT main effects. Migration `0186` 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 a reconstruction. Fallback pairs that have no
+complete-case leftover map omit the value rather than inventing one.
+A rank-0 origin map stores `0.0` (`ξ = 0`, `ζ = 0`), not a missing
+value. A non-finite reconstruction stores null rather than inventing
+a leftover score. A finite negative reconstruction is stored; do not
+add a nonnegative CHECK. Do not persist
+`leftover_map_explained_share`, `leftover_map_unexplained_share`,
+`leftover_map_cross_share`, or leftover-map coordinates. The signed
+`leftover_map_unexplained` value `U = R - R̂` remains persisted under ADR
+0182; it is evidence, not one of the prohibited derived share aliases.
+
+The pair button shows `R̂ {reconstruction}` next to leftover-map
+distance `d` when the value is a finite number, including a signed
+negative reconstruction. Next action: two leftover-map axes
+reconstruct centered leftover `R̂` after IRT main effects; open this
+post. A missing or non-finite reconstruction omits the badge and
+keeps the existing closest/farthest next action. Do not invent a
+leftover score or a theta.
+
+## Consequences
+
+`GET /api/reports/{grouping}/{period}` returns
+`leftover_map_reconstruction`. After `make seed`, closest and farthest
+leftover pairs sit above the member list with leftover-map
+reconstruction; click opens that post. Hidden posts stay hidden.
+
+Independent of leftover interaction-map persistence, leftover-criterion
+evaluation landing, leftover residual UI extraction, two-axis leftover-map
+distance, leftover observed `Y` / expected `E`, leftover-map rank, and
+leftover-map share identities.
+
+## 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