diff --git a/backend/documents_parser/tests/test_heating_systems_corpus.py b/backend/documents_parser/tests/test_heating_systems_corpus.py
index 777f409f..507962cb 100644
--- a/backend/documents_parser/tests/test_heating_systems_corpus.py
+++ b/backend/documents_parser/tests/test_heating_systems_corpus.py
@@ -371,6 +371,21 @@ class _CorpusExpectation:
 # one slice. All 25 cascade-OK variants now SAP / cost / CO2 / PE
 # EXACT vs worksheet (except solid fuel 2 which carries a separate
 # S0380.154 summer-immersion-blend CO2/PE artifact).
+#
+# Slice S0380.164 closed the last open variant in the cascade-OK tier:
+# `solid fuel 2`. The §12.4.4 back-boiler HW blend (S0380.154) had
+# computed summer immersion CO2/PE at the spec-literal Table 12d/12e
+# monthly cascade only. Per-line worksheet walk back-solved the (264)
+# and (278) factors as `W × anth_annual + S × (monthly_summer_avg +
+# Table 12 annual electric)` — i.e. the same Elmhurst-mirror that
+# S0380.163 introduced for full-electric HW, but ADDITIVE rather than
+# substitutive, applied on top of the monthly cascade for the summer-
+# immersion portion of the §12.4.4 blend. The new gate fires on dual-
+# rate tariffs (7-hour / 10-hour / 18-hour / 24-hour). Closure: SF2
+# ΔCO2 −93.10 → ±0.0000 EXACT, ΔPE −1027.51 → ±0.0000 EXACT. All 25
+# cascade-OK variants now SAP / cost / CO2 / PE EXACT on every metric.
+# Documented at `SAP_CALCULATOR.md §8.2` with the explicit single-cert
+# caveat (heating-systems corpus has only one §12.4.4 fixture).
 _EXPECTATIONS: tuple[_CorpusExpectation, ...] = (
     _CorpusExpectation(variant='ashp', block='11a', expected_sap_resid=+0.0000, expected_cost_resid_gbp=+0.0000, expected_co2_resid_kg=+0.0000, expected_pe_resid_kwh=-0.0000),
     _CorpusExpectation(variant='electric 1', block='11a', expected_sap_resid=-0.0000, expected_cost_resid_gbp=-0.0000, expected_co2_resid_kg=-0.0000, expected_pe_resid_kwh=+0.0000),
@@ -394,7 +409,7 @@ _EXPECTATIONS: tuple[_CorpusExpectation, ...] = (
     # cost / CO2 / PE all route via the correct Table 32 fuel code.
     # Remaining residuals are likely heating-system efficiency or
     # control-type gaps — separate slices.
-    _CorpusExpectation(variant='solid fuel 2', block='11a', expected_sap_resid=-0.0000, expected_cost_resid_gbp=-0.0000, expected_co2_resid_kg=-93.0988, expected_pe_resid_kwh=-1027.5099),
+    _CorpusExpectation(variant='solid fuel 2', block='11a', expected_sap_resid=-0.0000, expected_cost_resid_gbp=-0.0000, expected_co2_resid_kg=+0.0000, expected_pe_resid_kwh=+0.0000),
     _CorpusExpectation(variant='solid fuel 3', block='11a', expected_sap_resid=-0.0000, expected_cost_resid_gbp=-0.0000, expected_co2_resid_kg=+0.0000, expected_pe_resid_kwh=-0.0000),
     _CorpusExpectation(variant='solid fuel 4', block='11a', expected_sap_resid=-0.0000, expected_cost_resid_gbp=+0.0000, expected_co2_resid_kg=+0.0000, expected_pe_resid_kwh=-0.0000),
     _CorpusExpectation(variant='solid fuel 5', block='11a', expected_sap_resid=+0.0000, expected_cost_resid_gbp=+0.0000, expected_co2_resid_kg=-0.0000, expected_pe_resid_kwh=-0.0000),
diff --git a/domain/sap10_calculator/docs/SAP_CALCULATOR.md b/domain/sap10_calculator/docs/SAP_CALCULATOR.md
index 61c6e6da..cbb1d1df 100644
--- a/domain/sap10_calculator/docs/SAP_CALCULATOR.md
+++ b/domain/sap10_calculator/docs/SAP_CALCULATOR.md
@@ -444,3 +444,54 @@ fuel 4/5/6/7/8/9/10/11, ashp, gshp). Each variant moved from PE +25.51
 or +48.66 → ±0.0000, CO2 +6.31 or +11.95 → ±0.0000. Cohort-1 ASHP certs
 (STANDARD tariff) and the 6 Elmhurst U985 fixtures (gas combi, STANDARD
 tariff) are unaffected — they continue to use the monthly cascade.
+
+### 8.2 §12.4.4 back-boiler summer-immersion CO2/PE doubles the summer term
+
+**Slice:** S0380.164.
+**Code:**
+[`_section_12_4_4_hw_blend`](../rdsap/cert_to_inputs.py).
+**Tests:**
+`test_section_12_4_4_hw_blend_mirrors_elmhurst_summer_annual_pe_co2_double_count`,
+`test_section_12_4_4_hw_blend_standard_tariff_keeps_spec_literal_monthly_cascade`.
+
+SAP 10.2 §12.4.4 (PDF p.36-37) routes DHW through the boiler Oct-May and
+an electric immersion Jun-Sep for back-boiler combos (Table 4a codes
+156 + 158). The spec-literal CO2/PE formula multiplies summer-immersion
+fuel by the Table 12d / 12e monthly cascade (per Table 12 footnotes
+(s)/(t)). The BRE-approved Elmhurst engine adds a SECOND term —
+`summer_fuel × Table 12 ANNUAL electric factor` — on top of the
+monthly cascade for the (264) HW CO2 and (278) HW PE worksheet lines on
+dual-rate tariffs. Same shape as §8.1 / S0380.163 but additive rather
+than substitutive.
+
+**Cascade rule (post-S0380.164):**
+
+| Tariff | §12.4.4 winter CO2 / PE | §12.4.4 summer immersion CO2 / PE |
+|---|---|---|
+| STANDARD | `W_fuel × boiler_annual_factor` | `Σ wh_summer_m × Table 12d/e monthly` (spec literal) |
+| 7-hour / 10-hour / 18-hour / 24-hour | `W_fuel × boiler_annual_factor` | `Σ wh_summer_m × Table 12d/e monthly` **+ `S_fuel × Table 12 annual electric`** (Elmhurst mirror) |
+
+Cost is computed cleanly per spec (`W_fuel × boiler_price + S_fuel ×
+off_peak_low_price`) — the double-count quirk only affects the CO2 and
+PE factor lines.
+
+### Cohort impact
+
+The heating-systems corpus has exactly one §12.4.4 fixture: `solid fuel 2`
+(Table 4a code 158, anthracite, 18-hour tariff, 110 L cylinder + cyl
+thermostat). Pre-slice the cascade carried ΔCO2 = −93.10 kg/yr / ΔPE
+= −1027.51 kWh/yr — matching `684.55 kWh × 0.136 CO2` and
+`684.55 kWh × 1.501 PE` to within rounding. Post-slice closes to
+±0.0000 on all four metrics, completing the cohort closure at 25/25
+cascade-OK variants EXACT vs the Elmhurst worksheet.
+
+### ⚠ Single-cert evidence
+
+The §12.4.4 divergence is documented here on **one** worksheet (SF2)
+because the corpus has no second §12.4.4 fixture (`solid fuel 1` =
+code 156 is an empty folder). The math nonetheless matches the
+worksheet to within rounding and aligns with §8.1's S0380.163 mirror
+shape (Table 12 annual where spec literal says monthly), so the gate
+is implemented under the same `dual-rate → annual on top of monthly`
+discipline. If a second §12.4.4-eligible cert worksheet diverges from
+this rule it should be raised against this row before re-tuning.
diff --git a/domain/sap10_calculator/rdsap/cert_to_inputs.py b/domain/sap10_calculator/rdsap/cert_to_inputs.py
index 5816b3de..540cabfd 100644
--- a/domain/sap10_calculator/rdsap/cert_to_inputs.py
+++ b/domain/sap10_calculator/rdsap/cert_to_inputs.py
@@ -4932,7 +4932,15 @@ def _section_12_4_4_hw_blend(
 
     # CO2: boiler fuel at its Table 12 annual factor (winter) + electric
     # at the summer-month-weighted Table 12d cascade (per Table 12d
-    # header — "monthly factors instead the annual average").
+    # header — "monthly factors instead the annual average"). On dual-
+    # rate tariffs the BRE-approved Elmhurst engine applies an
+    # *additional* `summer_fuel × Table 12 annual electric CO2` term on
+    # top of the Table 12d monthly cascade — same shape as the S0380.163
+    # Elmhurst-mirror for the (264) HW CO2 line, here added rather than
+    # substituted. See SAP_CALCULATOR.md §8.2 for the single-cert
+    # worksheet evidence (SF2 (264) factor 0.371 = W×0.395 + S×(0.116
+    # monthly_summer + 0.136 annual) / total). STANDARD tariff keeps the
+    # spec-literal monthly-only cascade.
     boiler_co2 = (
         co2_factor_kg_per_kwh(boiler_fuel_code)
         if boiler_fuel_code is not None else 0.0
@@ -4940,16 +4948,29 @@ def _section_12_4_4_hw_blend(
     elec_co2_monthly = co2_monthly_factors_kg_per_kwh(
         _SECTION_12_4_4_IMMERSION_FUEL_CODE_TABLE_12
     )
-    summer_co2_kg = (
+    summer_co2_kg_monthly = (
         sum(
             wh_output_monthly_kwh[i] * elec_co2_monthly[i]
             for i in _SECTION_12_4_4_SUMMER_MONTH_INDICES
         )
         if elec_co2_monthly is not None else 0.0
     )
-    blended_co2 = (winter_fuel * boiler_co2 + summer_co2_kg) / total_fuel
+    elec_co2_annual = co2_factor_kg_per_kwh(
+        _SECTION_12_4_4_IMMERSION_FUEL_CODE_TABLE_12
+    )
+    summer_co2_kg_annual_mirror = (
+        summer_fuel * elec_co2_annual if tariff is not Tariff.STANDARD else 0.0
+    )
+    blended_co2 = (
+        winter_fuel * boiler_co2
+        + summer_co2_kg_monthly
+        + summer_co2_kg_annual_mirror
+    ) / total_fuel
 
-    # PE: same shape (Table 12e monthly cascade for summer electric).
+    # PE: same shape (Table 12e monthly cascade for summer electric)
+    # with the same Elmhurst-mirror `summer_fuel × Table 12 annual` term
+    # on dual-rate tariffs. SF2 (278) factor 1.3771 = W×1.064 + S×(1.429
+    # monthly_summer + 1.501 annual) / total.
     boiler_pe = (
         primary_energy_factor(boiler_fuel_code)
         if boiler_fuel_code is not None else 0.0
@@ -4957,14 +4978,24 @@ def _section_12_4_4_hw_blend(
     elec_pe_monthly = pe_monthly_factors_kwh_per_kwh(
         _SECTION_12_4_4_IMMERSION_FUEL_CODE_TABLE_12
     )
-    summer_pe_kwh = (
+    summer_pe_kwh_monthly = (
         sum(
             wh_output_monthly_kwh[i] * elec_pe_monthly[i]
             for i in _SECTION_12_4_4_SUMMER_MONTH_INDICES
         )
         if elec_pe_monthly is not None else 0.0
     )
-    blended_pe = (winter_fuel * boiler_pe + summer_pe_kwh) / total_fuel
+    elec_pe_annual = primary_energy_factor(
+        _SECTION_12_4_4_IMMERSION_FUEL_CODE_TABLE_12
+    )
+    summer_pe_kwh_annual_mirror = (
+        summer_fuel * elec_pe_annual if tariff is not Tariff.STANDARD else 0.0
+    )
+    blended_pe = (
+        winter_fuel * boiler_pe
+        + summer_pe_kwh_monthly
+        + summer_pe_kwh_annual_mirror
+    ) / total_fuel
 
     # Standing charges: Table 12 note (a) adds the off-peak electric
     # standing when HW uses off-peak electricity. The §12.4.4 summer
diff --git a/domain/sap10_calculator/rdsap/tests/test_cert_to_inputs.py b/domain/sap10_calculator/rdsap/tests/test_cert_to_inputs.py
index 7d29d20f..856e79c1 100644
--- a/domain/sap10_calculator/rdsap/tests/test_cert_to_inputs.py
+++ b/domain/sap10_calculator/rdsap/tests/test_cert_to_inputs.py
@@ -1845,6 +1845,190 @@ def test_section_12_4_4_summer_immersion_applies_to_back_boiler_combos() -> None
     ) is False
 
 
+def test_section_12_4_4_hw_blend_mirrors_elmhurst_summer_annual_pe_co2_double_count() -> None:
+    # Arrange — SAP 10.2 §12.4.4 back-boiler combos with the boiler heating
+    # the cylinder Oct-May + an electric immersion Jun-Sep. The spec-literal
+    # CO2/PE formula multiplies summer immersion fuel by Table 12d / 12e
+    # monthly factors (footnotes (s)/(t)). The BRE-approved Elmhurst engine
+    # adds a SECOND term: `summer_fuel × Table 12 ANNUAL electric factor`
+    # for dual-rate tariffs — same Elmhurst-mirror shape as S0380.163 for
+    # the (278) "Water heating (low-rate cost)" line, here applied on TOP
+    # of the spec-literal monthly cascade (not as a replacement).
+    #
+    # Worksheet evidence (heating-systems corpus property 001431,
+    # `solid fuel 2` — Table 4a code 158 closed-room-heater + back boiler,
+    # 65 % winter η + 100 % summer η, anthracite, 18-hour off-peak tariff,
+    # 110 L cylinder + cylinder thermostat lodged):
+    #
+    #   (62)m heat       303.12, 268.95, 287.76, 257.67, 252.99,
+    #                    168.95, 166.27, 173.42, 175.91,
+    #                    261.69, 273.22, 300.40 kWh
+    #   winter fuel (W)  = 2205.80 / 0.65          = 3393.51 kWh anthracite
+    #   summer fuel (S)  = 684.55 / 1.00           =  684.55 kWh immersion
+    #   total fuel       = (219) = 4078.06 kWh
+    #
+    #   (264) HW CO2 = 4078.06 × 0.3710 = 1513.15 kg/yr
+    #     decomposes as W × 0.395 (anth annual) + S × 0.116 (Jun-Sep Table
+    #     12d cascade) + S × 0.136 (Table 12 annual electricity)
+    #     = 1340.43 + 79.61 + 93.10 = 1513.14 ✓ within rounding
+    #
+    #   (278) HW PE = 4078.06 × 1.3771 = 5616.04 kWh/yr
+    #     decomposes as W × 1.064 (anth annual) + S × 1.429 (Jun-Sep Table
+    #     12e cascade) + S × 1.501 (Table 12 annual electricity)
+    #     = 3610.69 + 977.84 + 1027.51 = 5616.04 ✓ exact
+    #
+    # Pre-slice the blend helper returned the spec-literal sum (W × anth +
+    # S × monthly) and the SF2 corpus pin carried ΔCO2 = −93.10 / ΔPE =
+    # −1027.51, matching the missing summer × annual term exactly. Per
+    # [[feedback-software-no-special-handling]] mirror the Elmhurst engine.
+    # SAP_CALCULATOR.md §8.2 documents the divergence (single-cert
+    # evidence — pending re-verification when a second §12.4.4 fixture
+    # lands; the math is the same shape as §8.1 / S0380.163's Table 12
+    # annual mirror for dual-rate HW).
+
+    from domain.sap10_calculator.rdsap.cert_to_inputs import (
+        _section_12_4_4_hw_blend,  # pyright: ignore[reportPrivateUsage]
+    )
+    from domain.sap10_calculator.tables.table_12 import (
+        co2_factor_kg_per_kwh,
+        co2_monthly_factors_kg_per_kwh,
+        pe_monthly_factors_kwh_per_kwh,
+        primary_energy_factor,
+    )
+    from domain.sap10_calculator.tables.table_12a import Tariff
+
+    sf2_monthly_heat = (
+        303.1182, 268.9461, 287.7571, 257.6651, 252.9865,
+        168.9471, 166.2739, 173.4236, 175.9056,
+        261.6901, 273.2184, 300.4020,
+    )
+    boiler_eff_pct = 65.0
+    anthracite_code = 15  # Table 12 anthracite
+
+    # Act — invoke the §12.4.4 blend on SF2-shape inputs with the 18-hour
+    # off-peak tariff Elmhurst lodges for this fixture.
+    total_fuel, _cost, blended_co2, blended_pe, _standing = (
+        _section_12_4_4_hw_blend(
+            wh_output_monthly_kwh=sf2_monthly_heat,
+            boiler_efficiency_pct=boiler_eff_pct,
+            boiler_fuel_code=anthracite_code,
+            tariff=Tariff.EIGHTEEN_HOUR,
+            prices=SAP_10_2_SPEC_PRICES,
+        )
+    )
+
+    # Assert — total_fuel sums the two-fuel split; CO2 and PE reproduce the
+    # Elmhurst-mirror "annual on top of monthly" double-count.
+    summer_indices = frozenset({5, 6, 7, 8})  # Jun-Sep
+    winter_heat = sum(
+        h for i, h in enumerate(sf2_monthly_heat) if i not in summer_indices
+    )
+    summer_heat = sum(
+        h for i, h in enumerate(sf2_monthly_heat) if i in summer_indices
+    )
+    expected_winter_fuel = winter_heat / (boiler_eff_pct / 100.0)
+    expected_summer_fuel = summer_heat / 1.0
+    expected_total_fuel = expected_winter_fuel + expected_summer_fuel
+    assert abs(total_fuel - expected_total_fuel) <= 1e-9
+
+    anth_co2 = co2_factor_kg_per_kwh(anthracite_code)
+    anth_pe = primary_energy_factor(anthracite_code)
+    elec_co2_monthly = co2_monthly_factors_kg_per_kwh(30)
+    elec_pe_monthly = pe_monthly_factors_kwh_per_kwh(30)
+    assert elec_co2_monthly is not None and elec_pe_monthly is not None
+    elec_co2_annual = co2_factor_kg_per_kwh(30)
+    elec_pe_annual = primary_energy_factor(30)
+
+    expected_co2 = (
+        expected_winter_fuel * anth_co2
+        + sum(sf2_monthly_heat[i] * elec_co2_monthly[i] for i in summer_indices)
+        + expected_summer_fuel * elec_co2_annual
+    ) / expected_total_fuel
+    expected_pe = (
+        expected_winter_fuel * anth_pe
+        + sum(sf2_monthly_heat[i] * elec_pe_monthly[i] for i in summer_indices)
+        + expected_summer_fuel * elec_pe_annual
+    ) / expected_total_fuel
+
+    assert abs(blended_co2 - expected_co2) <= 1e-9
+    assert abs(blended_pe - expected_pe) <= 1e-9
+
+
+def test_section_12_4_4_hw_blend_standard_tariff_keeps_spec_literal_monthly_cascade() -> None:
+    # Arrange — the Elmhurst-mirror "summer × annual" addition is gated on
+    # dual-rate tariffs (mirroring S0380.163's STANDARD ↔ off-peak split
+    # for the (278) "Water heating (low-rate cost)" line). On STANDARD
+    # tariff the spec-literal monthly cascade is the canonical form and
+    # the blend stays at `W × anth_annual + S × monthly_summer_avg`.
+    #
+    # No corpus fixture exercises STANDARD-tariff §12.4.4 — all 41
+    # heating-systems corpus variants lodge `meter_type='18 Hour'`. This
+    # test guards the gate so a future STANDARD-tariff §12.4.4 fixture
+    # cascades through the spec-literal path.
+
+    from domain.sap10_calculator.rdsap.cert_to_inputs import (
+        _section_12_4_4_hw_blend,  # pyright: ignore[reportPrivateUsage]
+    )
+    from domain.sap10_calculator.tables.table_12 import (
+        co2_factor_kg_per_kwh,
+        co2_monthly_factors_kg_per_kwh,
+        pe_monthly_factors_kwh_per_kwh,
+        primary_energy_factor,
+    )
+    from domain.sap10_calculator.tables.table_12a import Tariff
+
+    sf2_monthly_heat = (
+        303.1182, 268.9461, 287.7571, 257.6651, 252.9865,
+        168.9471, 166.2739, 173.4236, 175.9056,
+        261.6901, 273.2184, 300.4020,
+    )
+
+    # Act
+    _total, _cost, blended_co2, blended_pe, _standing = (
+        _section_12_4_4_hw_blend(
+            wh_output_monthly_kwh=sf2_monthly_heat,
+            boiler_efficiency_pct=65.0,
+            boiler_fuel_code=15,
+            tariff=Tariff.STANDARD,
+            prices=SAP_10_2_SPEC_PRICES,
+        )
+    )
+
+    # Assert — STANDARD tariff matches spec-literal (no Elmhurst-mirror
+    # summer×annual term).
+    summer_indices = frozenset({5, 6, 7, 8})
+    winter_heat = sum(
+        h for i, h in enumerate(sf2_monthly_heat) if i not in summer_indices
+    )
+    summer_heat = sum(
+        h for i, h in enumerate(sf2_monthly_heat) if i in summer_indices
+    )
+    winter_fuel = winter_heat / 0.65
+    summer_fuel = summer_heat
+    total_fuel = winter_fuel + summer_fuel
+    anth_co2 = co2_factor_kg_per_kwh(15)
+    anth_pe = primary_energy_factor(15)
+    elec_co2_monthly = co2_monthly_factors_kg_per_kwh(30)
+    elec_pe_monthly = pe_monthly_factors_kwh_per_kwh(30)
+    assert elec_co2_monthly is not None and elec_pe_monthly is not None
+    expected_co2_literal = (
+        winter_fuel * anth_co2
+        + sum(sf2_monthly_heat[i] * elec_co2_monthly[i] for i in summer_indices)
+    ) / total_fuel
+    expected_pe_literal = (
+        winter_fuel * anth_pe
+        + sum(sf2_monthly_heat[i] * elec_pe_monthly[i] for i in summer_indices)
+    ) / total_fuel
+    assert abs(blended_co2 - expected_co2_literal) <= 1e-9
+    assert abs(blended_pe - expected_pe_literal) <= 1e-9
+    # Spec-literal stays strictly below the Elmhurst-mirror dual-rate
+    # value (the +annual term is additive).
+    elmhurst_mirror_co2 = expected_co2_literal + (summer_fuel * 0.136) / total_fuel
+    elmhurst_mirror_pe = expected_pe_literal + (summer_fuel * 1.501) / total_fuel
+    assert blended_co2 < elmhurst_mirror_co2
+    assert blended_pe < elmhurst_mirror_pe
+
+
 def test_separately_timed_dhw_solid_fuel_boiler_codes_per_sap_10_2_table_3() -> None:
     # Arrange — SAP 10.2 Table 3 (PDF p.160) gives three primary-loss
     # rows keyed off the DHW timing arrangement: