diff --git a/backend/documents_parser/tests/test_summary_pdf_mapper_chain.py b/backend/documents_parser/tests/test_summary_pdf_mapper_chain.py
index 3faae0aa..790dd889 100644
--- a/backend/documents_parser/tests/test_summary_pdf_mapper_chain.py
+++ b/backend/documents_parser/tests/test_summary_pdf_mapper_chain.py
@@ -1650,6 +1650,48 @@ def test_summary_000565_section_12_1_extracts_mechanical_extract_decentralised_m
     )
 
 
+def test_summary_000565_detailed_rr_residual_area_closes_total_external_area_per_rdsap_10_section_3_10_1() -> None:
+    # Arrange — RdSAP 10 §3.10.1 (PDF p.24) "Default U-values of the
+    # roof rooms":
+    #   "The residual area (area of roof less the floor area of room(s)-
+    #    in-roof) has a U-value from Table 16 : Roof U-values when loft
+    #    insulation thickness is known according to its insulation
+    #    thickness if at least half the area concerned is accessible,
+    #    otherwise it is the default for the age band of the original
+    #    property or extension."
+    # Worksheet pattern (cert 000565 BP[0]): "Roof room Main remaining
+    # area" 43.97 m² × U=0.35 (Table 18 col 4 age H default).
+    # Pre-slice S0380.95 the cascade computed residual area ONLY for
+    # Simplified RR mode (via `rr_a_rr − rr_common − rr_gable` in
+    # `_part_geometry`); the Detailed-RR branch in `heat_transmission`
+    # iterated `rir.detailed_surfaces` and missed the residual entirely.
+    # Cert 000565 routes all 5 BPs through Detailed mode (mapper
+    # translates Simplified-Summary lodgements to `SapRoomInRoofSurface`
+    # records), so cascade total_external_element_area_m2 was 779.27 m²
+    # vs worksheet (31) = 857.64 m² (Δ −78.37 m² → thermal_bridging
+    # under by ~−11.76 W/K).
+    pages = _summary_pdf_to_textract_style_pages(_SUMMARY_000565_PDF)
+    site_notes = ElmhurstSiteNotesExtractor(pages).extract()
+    epc = EpcPropertyDataMapper.from_elmhurst_site_notes(site_notes)
+
+    # Act
+    from domain.sap10_calculator.worksheet.heat_transmission import (
+        heat_transmission_from_cert,
+    )
+    ht = heat_transmission_from_cert(epc, door_count=epc.door_count or 0)
+
+    # Assert — cascade closes to within ±10 m² of worksheet (31). The
+    # residual sums roughly to BP[0]'s 43.97 m² + BP[1]'s ~22 m² +
+    # BP[3]'s ~17 m² + BP[4]'s small contribution; remaining residual
+    # (BP[1] ~+3.7 m² over) traces to the spec's ambiguous Detailed-
+    # mode residual formula for extensions with multi-storey heights.
+    assert ht.total_external_element_area_m2 >= 845.0, (
+        f"cascade total_external_element_area_m2={ht.total_external_element_area_m2:.4f}; "
+        f"expected ≥845 m² after §3.10.1 Detailed-RR residual area closure "
+        f"(pre-slice was 779.27 m² vs worksheet 857.64)"
+    )
+
+
 def test_summary_000565_ext2_stud_wall_2_extracts_400_plus_mm_pur_or_pir_lodgement() -> None:
     # Arrange — cert 000565 Summary §8.1 BP[2] Ext2 (Detailed) lodges
     # "Stud Wall 2: 2.00 × 2.00, 400+ mm, PUR or PIR" with Default
diff --git a/domain/sap10_calculator/worksheet/heat_transmission.py b/domain/sap10_calculator/worksheet/heat_transmission.py
index fc3f391e..dc8670d8 100644
--- a/domain/sap10_calculator/worksheet/heat_transmission.py
+++ b/domain/sap10_calculator/worksheet/heat_transmission.py
@@ -845,7 +845,19 @@ def heat_transmission_from_cert(
             # line (30)); gable_wall routes to party_walls at U=0.25
             # (worksheet line (32), per Table 4 "as common wall").
             rir = part.sap_room_in_roof
-            for surf in rir.detailed_surfaces:
+            # RdSAP 10 §3.10.1 (PDF p.24) "residual area (area of roof
+            # less the floor area of room(s)-in-roof) has a U-value
+            # from Table 16 ... otherwise it is the default for the age
+            # band". Wall-going RIR surfaces (gable_wall, gable_wall_
+            # external, common_wall) deduct from the simplified A_RR
+            # to leave the residual area, mirroring the Simplified
+            # branch's `a_rr_final = rr_a_rr - rr_common - rr_gable`.
+            # Roof-going surfaces (slope / flat_ceiling / stud_wall)
+            # do NOT deduct — they sit inside the RR shell rather than
+            # forming its perimeter walls.
+            rr_walls_in_a_rr_area = 0.0
+            detailed_surfaces = rir.detailed_surfaces or []
+            for surf in detailed_surfaces:
                 kind = surf.kind
                 # RdSAP10 §15 — RR detailed sub-area rounded to 2 d.p.
                 area = _round_half_up(surf.area_m2, _AREA_ROUND_DP)
@@ -878,6 +890,7 @@ def heat_transmission_from_cert(
                     )
                 elif kind == "gable_wall":
                     party += 0.25 * area
+                    rr_walls_in_a_rr_area += area
                 elif kind == "gable_wall_external":
                     # RdSAP10 Table 4 (p.22) row 1: exposed gable U = "as
                     # common wall" — i.e. the main-wall U of the storey
@@ -887,6 +900,7 @@ def heat_transmission_from_cert(
                     u_gable = surf.u_value if surf.u_value is not None else uw
                     rr_detailed_area += area
                     walls += u_gable * area
+                    rr_walls_in_a_rr_area += area
                 elif kind == "common_wall":
                     # RdSAP 10 §3.9.2 Simplified Type 2 + Table 4 p.22
                     # "Common wall": billed as external wall at the
@@ -899,6 +913,36 @@ def heat_transmission_from_cert(
                     u_common = surf.u_value if surf.u_value is not None else uw
                     rr_detailed_area += area
                     walls += u_common * area
+                    rr_walls_in_a_rr_area += area
+            # RdSAP 10 §3.10.1 residual area = simplified A_RR shell
+            # minus the wall surfaces just enumerated. Uses the same
+            # §3.9.1 `12.5 × √(A_RR_floor / 1.5)` formula as the
+            # Simplified branch since the worksheet applies it
+            # consistently when the Summary is Simplified-mode but the
+            # mapper has translated wall lodgements to
+            # `detailed_surfaces` records (cert 000565 BP[0]:
+            # 12.5 × √30 − 24.5 = 43.97 ✓ matches worksheet's "Roof
+            # room Main remaining area"). The residual gets the
+            # `u_rr_default_all_elements` value (Table 18 col 4).
+            #
+            # Discriminator: only fire when the lodged set contains
+            # WALL-going surfaces but no ROOF-going surfaces (slope,
+            # flat_ceiling, stud_wall). True Detailed-mode lodgements
+            # (e.g. cohort fixture 000516, where the assessor lodges
+            # every slope / flat_ceiling / stud_wall) lodge the roof
+            # shell explicitly — there is no residual to add.
+            kinds = {s.kind for s in detailed_surfaces}
+            roof_kinds = {"slope", "flat_ceiling", "stud_wall"}
+            has_roof_lodgement = bool(kinds & roof_kinds)
+            rr_floor_for_a_rr = float(rir.floor_area)
+            if not has_roof_lodgement and rr_floor_for_a_rr > 0.0:
+                a_rr_shell = 12.5 * sqrt(rr_floor_for_a_rr / 1.5)
+                residual_area = max(0.0, a_rr_shell - rr_walls_in_a_rr_area)
+                if residual_area > 0.0:
+                    rr_detailed_area += residual_area
+                    roof += residual_area * u_rr_default_all_elements(
+                        country=country, age_band=rir.construction_age_band,
+                    )
         floor += uf * floor_area_total
         # RdSAP "first floor over passageway" cantilever — only fires
         # for houses (property_type=0); see `_part_geometry` filters.