Slice 102f-prep.4: Equation N5 zone-mean blending leaf

SAP 10.2 Appendix N3.5 Equation N5 (PDF p.107):

    T = [N24,9 × Th + N16,9 × T_uni + (Nm − N16,9 − N24,9) × T_bi] / Nm

`extended_zone_mean_temperature_c` is the pure-math leaf: takes
pre-computed bimodal (9-hour heating, two off periods) and unimodal
(16-hour heating, one 8-hour off period per Table N7 footnote b)
zone temperatures and the per-month day allocations, blends across
the three heating patterns (Th for 24-hour days, T_uni for 16-hour,
T_bi for the standard 9-hour SAP schedule).

Pinned against cert 0380's January living-area MIT: Th=21, T_bi
=18.5551 (worksheet "Living" row), T_uni back-solved from (87)
= 19.6153, N24=3, N16=28, Nm=31 → 19.7493 (worksheet (87) Jan).

Collapses cleanly: N24=N16=0 → T_bi (warm months / non-HP certs);
N24=Nm → Th (full 24-hour heating).

Co-Authored-By: Claude Opus 4.7 <noreply@anthropic.com>

domain/sap10_calculator/worksheet/mean_internal_temperature.py

@@ -146,6 +146,45 @@ def allocate_extended_heating_days_to_months(
     return tuple(allocations)
 
 
+def extended_zone_mean_temperature_c(
+    *,
+    heating_temperature_c: float,
+    t_bimodal_c: float,
+    t_unimodal_c: float,
+    n24_9_m: int,
+    n16_9_m: int,
+    days_in_month: int,
+) -> float:
+    """SAP 10.2 Appendix N3.5 Equation N5 (PDF p.107) — blend a zone's
+    monthly mean temperature across three heating patterns:
+
+        T = [N24,9 × Th + N16,9 × T_uni + (Nm − N16,9 − N24,9) × T_bi] / Nm
+
+    The three patterns differ in their daily off-period structure:
+      - 24-hour day: zero off periods → T = Th
+      - Unimodal (16-hour day): one off period of 8h (0700-2300 heating
+        period per Table N7 footnote b) → T = Th − u1(8h)
+      - Bimodal (9-hour day): two off periods (7+8h for living-area and
+        elsewhere control-type 1/2; 9+8h for elsewhere control-type 3
+        per Table N7) → T = Th − u1 − u2
+
+    The caller passes the pre-computed `t_bimodal_c` and `t_unimodal_c`
+    (already reduced from Th by the relevant Table 9b u terms); this
+    helper just does the day-weighted blend.
+
+    When `n24_9_m = n16_9_m = 0` the formula collapses to `t_bimodal_c`,
+    so non-HP certs and warm months flow through unchanged.
+    """
+    if days_in_month <= 0:
+        return t_bimodal_c
+    bimodal_days = days_in_month - n16_9_m - n24_9_m
+    return (
+        n24_9_m * heating_temperature_c
+        + n16_9_m * t_unimodal_c
+        + bimodal_days * t_bimodal_c
+    ) / days_in_month
+
+
 def elsewhere_heating_temperature_c(
     *,
     heat_loss_parameter: float,

domain/sap10_calculator/worksheet/tests/test_mean_internal_temperature.py

@@ -20,6 +20,7 @@ from domain.sap10_calculator.worksheet.mean_internal_temperature import (
     allocate_extended_heating_days_to_months,
     elsewhere_heating_temperature_c,
     extended_heating_days_from_psr_variable,
+    extended_zone_mean_temperature_c,
     mean_internal_temperature_monthly,
     off_period_temperature_reduction_c,
 )
@@ -431,6 +432,73 @@ def test_allocate_extended_heating_days_zero_is_all_zero() -> None:
     assert monthly == ((0, 0),) * 12
 
 
+def test_extended_zone_mean_temperature_matches_cert_0380_january_living() -> None:
+    """SAP 10.2 Appendix N3.5 Equation N5 (PDF p.107):
+
+        T = [N24,9 × Th + N16,9 × T_uni + (Nm − N16,9 − N24,9) × T_bi] / Nm
+
+    Cert 0380 January (Mitsubishi PUZ-WM50VHA, PSR 1.43):
+      Living-area bimodal "Living" row = 18.5551 (worksheet)
+      N24,9 = 3, N16,9 = 28 (Jan), Nm = 31, Th = 21
+
+    Back-solving the worksheet's MIT_living(87) = 19.7493:
+        31 × 19.7493 = 612.228
+        612.228 = 3 × 21 + 28 × T_uni + 0 × 18.5551
+        T_uni = (612.228 − 63) / 28 = 19.6153
+
+    So this leaf, given Th=21, T_bi=18.5551, T_uni=19.6153, N24=3,
+    N16=28, Nm=31, must return 19.7493.
+    """
+    # Arrange / Act
+    t = extended_zone_mean_temperature_c(
+        heating_temperature_c=21.0,
+        t_bimodal_c=18.5551,
+        t_unimodal_c=19.6153,
+        n24_9_m=3,
+        n16_9_m=28,
+        days_in_month=31,
+    )
+
+    # Assert
+    assert abs(t - 19.7493) < 1e-4
+
+
+def test_extended_zone_mean_temperature_collapses_to_bimodal_when_zero_extension() -> None:
+    """When N24,9_m = N16,9_m = 0, Equation N5 reduces to T_bi — i.e.
+    the standard SAP heating schedule is unchanged. This guards the
+    "no extended heating" path so that warm months (Jun..Sep) and
+    non-HP certs flow through the legacy bimodal calculation."""
+    # Arrange / Act
+    t = extended_zone_mean_temperature_c(
+        heating_temperature_c=21.0,
+        t_bimodal_c=18.0,
+        t_unimodal_c=19.5,  # Should be ignored when n16_9_m = 0
+        n24_9_m=0,
+        n16_9_m=0,
+        days_in_month=30,
+    )
+
+    # Assert
+    assert t == 18.0
+
+
+def test_extended_zone_mean_temperature_collapses_to_th_for_full_24h_month() -> None:
+    """When N24,9_m = days_in_month, Equation N5 reduces to Th — every
+    day operates at the heating temperature with no off period."""
+    # Arrange / Act
+    t = extended_zone_mean_temperature_c(
+        heating_temperature_c=21.0,
+        t_bimodal_c=18.0,
+        t_unimodal_c=19.5,
+        n24_9_m=31,
+        n16_9_m=0,
+        days_in_month=31,
+    )
+
+    # Assert
+    assert t == 21.0
+
+
 def test_allocate_extended_heating_days_variable_year_totals_are_preserved() -> None:
     """The helper's invariant for the Variable case: every input
     (N24,9, N16,9) day from Table N5 must land in some cold month