Compact buildings lose less heat—until the envelope changes

Tested hypothesis · Mathematical and model findings · 30 October 2026

Incoprea Research · 30 October 2026 · Mathematical/model test

Finding: Compactness has an exact role in conductive heat loss, but it cannot by itself rank buildings with different envelopes. In an idealized equal-floor-area comparison, a nine-storey form loses 56.8% less heat than a single-storey form at identical glazing fraction. Increasing glazing can erase or reverse the advantage. This supports the mechanism in Building Energy Is a Shape Ratio, while challenging its stronger suggestion that shape necessarily dominates glazing.

This is a transparent heat-balance calculation, not measured energy use, an annual building simulation or a recommendation for a specific design. The question comes from the Incoprea Research hypothesis collection. The underlying heat-transfer law is established; no discovery or external peer review is claimed.

A controlled geometry test

Each hypothetical building has 10,000 square metres of total floor area and three-metre storey height, giving the same 30,000 cubic metres of conditioned volume. The frozen plan covers 1, 4, 9, 16 and 25 storeys; footprint aspect ratios 1, 2, 4 and 8; and wall glazing fractions 20%, 40%, 60% and 80%. That produces 80 declared combinations, not selected examples from real buildings.

Opaque elements have U = 0.25 W/(m² K); glazing has U = 1.6 W/(m² K). These are illustrative assumptions, not product specifications. Every envelope surface is assigned the same 20 K inside-to-outside temperature difference. Roof and lowest floor are ideal exposed surfaces with the opaque U-value. Treating the lowest floor that way deliberately omits real ground coupling.

The PNNL Building America heat-flow explanation gives the standard relationship among area, temperature difference and thermal resistance. We apply its equivalent U-value form: Q = DeltaT × sum(U_i A_i). No fitting to the hypothesis is involved.

What shape does determine

If all surfaces share one U-value, dividing by volume gives the identity Q/V = U × DeltaT × A_envelope/V. Under those controlled conditions, the area-to-volume ratio exactly predicts conductive loss per volume. Since these cases also share storey height, conductive loss per floor area is equally well related to envelope area divided by total floor area.

With a square footprint, the single-storey example has envelope-to-volume ratio 0.70667 per metre; the nine-storey example has 0.19407 per metre. At 20% wall glazing, their conductive heating loads are 11.248 and 4.855 W per square metre of floor, respectively. The latter is 56.8% lower. This is a physically consistent explanation of why shape can matter at fixed floor area.

It does not establish a collapse of whole-building energy intensity onto one geometric predictor. Even this simple model needs the area-weighted conductance when materials differ.

Where glazing changes the answer

For the nine-storey form, raising wall glazing from 20% to 80% increases conductive load from 4.855 to 10.687 W/m² of floor. It still narrowly beats the single-storey 20%-glazed case. The initial analysis expected this particular comparison to reverse; it did not. That outcome is retained, and no input parameter was changed.

An exact crossover calculation puts the nine-storey threshold at about 85.77% wall glazing against that specific single-storey reference. The threshold lies above the declared 80% grid maximum. It is a mathematical boundary for this ideal model, not an architectural rule.

A separate combination already present in the complete declared grid illustrates an actual reversal: the 16-storey square building gives 5.617 W/m² at 20% glazing but 13.393 W/m² at 80%. At high glazing it exceeds the sprawling reference's 11.248 W/m² despite having a much smaller area-to-volume ratio. The failed preselected example and the whole-grid counterexample are both disclosed to prevent an overly tidy account.

What this means for the original claim

The calculation supports compactness as a controlled determinant of conduction. It rejects an unconditional inference that glazing must matter less than shape. Nor does “taller” automatically mean “more compact”: a sufficiently tall, thin form adds wall area while reducing roof and floor area.

Actual energy use includes ventilation, infiltration, solar gain, thermal bridges, orientation, thermal storage, occupant loads, schedules, climate and heating/cooling efficiency. The official EnergyPlus engineering reference treats multiple coupled heat-balance terms. Our calculation deliberately isolates one component and cannot rank actual buildings.

The useful next empirical test is to ask whether compactness improves prediction after envelope properties and operation are controlled. The current finding is already reproducible: geometry changes conductive demand, but sum(U_i A_i) shows exactly why geometry alone is insufficient.

Reproduce this result

The calculations below use Python 3 and its standard library. Save the code as reproduce.py in an empty folder and run python3 reproduce.py. It writes the result files there. These are declared mathematical or synthetic-model tests, not measurements of real people, buildings, devices or deployed systems. The plan was recorded locally before outcomes; it was not registered externally.

Show the complete runnable code
"""Ideal steady conduction only, standard library. Not an annual-energy building model."""
import csv,hashlib,json,math
from pathlib import Path
P=Path(__file__).resolve().parent;PLAN_SHA256='fcd7c0e5f5be65a13b43dc9c1543cdf09affb30c79bd789fc13e3f1dab31a1a7';p={'U_opaque': 0.25, 'U_window': 1.6, 'window_fraction': [0.2, 0.4, 0.6, 0.8], 'deltaT': 20, 'floor_area': 10000, 'floor_height': 3};
if (P/'PLAN.json').exists():assert hashlib.sha256((P/'PLAN.json').read_bytes()).hexdigest()==PLAN_SHA256
A=p['floor_area'];h=p['floor_height'];V=A*h;u=p['U_opaque'];ug=p['U_window'];dt=p['deltaT'];rows=[]
def geometry(storeys,aspect=1):
 footprint=A/storeys;L=math.sqrt(footprint*aspect);W=math.sqrt(footprint/aspect);height=h*storeys;wall=2*(L+W)*height;other=2*footprint
 return wall,other
for s in [1,4,9,16,25]:
 for a in [1,2,4,8]:
  wall,other=geometry(s,a);env=wall+other
  for f in p['window_fraction']:
   H=u*(other+(1-f)*wall)+ug*f*wall;uniform=u*env
   assert math.isclose(uniform/V,u*(env/V),rel_tol=1e-14)
   rows.append({'storeys':s,'aspect_ratio':a,'window_fraction':f,'wall_m2':wall,'other_envelope_m2':other,'A_over_V_per_m':env/V,'conductance_W_per_K':H,'heating_W_per_floor_m2':H*dt/A,'uniform_U_heating_W_per_floor_m2':uniform*dt/A})
sprawl=next(x for x in rows if x['storeys']==1 and x['aspect_ratio']==1 and x['window_fraction']==.2)
compact=next(x for x in rows if x['storeys']==9 and x['aspect_ratio']==1 and x['window_fraction']==.2)
compact_glass=next(x for x in rows if x['storeys']==9 and x['aspect_ratio']==1 and x['window_fraction']==.8)
wall,other=geometry(9); crossover=(sprawl['conductance_W_per_K']-u*(wall+other))/((ug-u)*wall)
assert 0<crossover<1
assert compact['heating_W_per_floor_m2']<sprawl['heating_W_per_floor_m2']
# The planned nine-storey/80% example did NOT reverse ordering; retain that result.
# The whole declared grid supplies a valid crossover at sixteen storeys.
compact16=next(x for x in rows if x['storeys']==16 and x['aspect_ratio']==1 and x['window_fraction']==.2)
compact16glass=next(x for x in rows if x['storeys']==16 and x['aspect_ratio']==1 and x['window_fraction']==.8)
assert compact16['heating_W_per_floor_m2']<sprawl['heating_W_per_floor_m2']<compact16glass['heating_W_per_floor_m2']
with (P/'grid.csv').open('w',newline='') as f:
 w=csv.DictWriter(f,fieldnames=list(rows[0]));w.writeheader();w.writerows(rows)
r={'scope':'Steady conduction for equal floor area, same temperature difference and specified U values; no empirical or annual energy claim. Ground/floor modeled as same exposed temperature boundary, deliberately idealized.','sprawl':sprawl,'compact':compact,'compact_high_glazing':compact_glass,'compact16':compact16,'compact16_high_glazing':compact16glass,'analysis_deviation':'Initial assertion expected 9-storey/80-percent glazing to exceed single-storey/20-percent; it did not. No parameters changed. Both results retained; declared 16-storey grid supplies crossover illustration.','compact_saving_fraction_at_equal_glazing':1-compact['heating_W_per_floor_m2']/sprawl['heating_W_per_floor_m2'],'compact_9_floor_window_fraction_crossover_against_1_floor_20percent':crossover,'model_cases':len(rows),'checks_passed':True,'plan_sha256':PLAN_SHA256,'code_sha256':hashlib.sha256(Path(__file__).read_bytes()).hexdigest()}
(P/'results.json').write_text(json.dumps(r,indent=2)+'\n');print(json.dumps(r,indent=2))
Show the frozen analysis plan
{
  "claim": "building-shape-ratio",
  "volume": 8,
  "model": "Steady uniform temperature conduction H=sum(U_i*A_i). Equal total floor area 10000 m2, floor height 3m. Compare square footprint 1,4,9,16,25 storeys and aspect ratios 1,2,4,8 at each. Roof/floor are exposed ideal surfaces, same U as opaque walls; windows only walls. Ignore gains, air exchange, thermal bridging and HVAC. Q=H*DeltaT.",
  "parameters": {
    "U_opaque": 0.25,
    "U_window": 1.6,
    "window_fraction": [
      0.2,
      0.4,
      0.6,
      0.8
    ],
    "deltaT": 20,
    "floor_area": 10000,
    "floor_height": 3
  },
  "checks": [
    "At fixed effective U and uniform U, Q/V proportional A/V exactly",
    "Compare 1 vs 9-storey at equal window fractions, then 20 percent vs80 percent to show glazing crossover",
    "Compute crossover window fraction analytically; do not assume compactness dominates"
  ],
  "source_file": "[source path recorded in saved evidence package]",
  "source_sha256": "1fe9cd6085de9ab1f5988a82c3251b50eef8c2baf68aea9693afb47d42f7e875",
  "plan_frozen_utc": "2026-10-30T17:04:41.306310+00:00",
  "classification": "Exact mathematics and synthetic model check; not measured data, novelty or external peer review"
}
Show the complete numerical results
{
  "scope": "Steady conduction for equal floor area, same temperature difference and specified U values; no empirical or annual energy claim. Ground/floor modeled as same exposed temperature boundary, deliberately idealized.",
  "sprawl": {
    "storeys": 1,
    "aspect_ratio": 1,
    "window_fraction": 0.2,
    "wall_m2": 1200.0,
    "other_envelope_m2": 20000.0,
    "A_over_V_per_m": 0.7066666666666667,
    "conductance_W_per_K": 5624.0,
    "heating_W_per_floor_m2": 11.248,
    "uniform_U_heating_W_per_floor_m2": 10.6
  },
  "compact": {
    "storeys": 9,
    "aspect_ratio": 1,
    "window_fraction": 0.2,
    "wall_m2": 3600.0000000000005,
    "other_envelope_m2": 2222.222222222222,
    "A_over_V_per_m": 0.1940740740740741,
    "conductance_W_per_K": 2427.555555555556,
    "heating_W_per_floor_m2": 4.855111111111112,
    "uniform_U_heating_W_per_floor_m2": 2.9111111111111114
  },
  "compact_high_glazing": {
    "storeys": 9,
    "aspect_ratio": 1,
    "window_fraction": 0.8,
    "wall_m2": 3600.0000000000005,
    "other_envelope_m2": 2222.222222222222,
    "A_over_V_per_m": 0.1940740740740741,
    "conductance_W_per_K": 5343.5555555555575,
    "heating_W_per_floor_m2": 10.687111111111115,
    "uniform_U_heating_W_per_floor_m2": 2.9111111111111114
  },
  "compact16": {
    "storeys": 16,
    "aspect_ratio": 1,
    "window_fraction": 0.2,
    "wall_m2": 4800.0,
    "other_envelope_m2": 1250.0,
    "A_over_V_per_m": 0.20166666666666666,
    "conductance_W_per_K": 2808.5,
    "heating_W_per_floor_m2": 5.617,
    "uniform_U_heating_W_per_floor_m2": 3.025
  },
  "compact16_high_glazing": {
    "storeys": 16,
    "aspect_ratio": 1,
    "window_fraction": 0.8,
    "wall_m2": 4800.0,
    "other_envelope_m2": 1250.0,
    "A_over_V_per_m": 0.20166666666666666,
    "conductance_W_per_K": 6696.500000000001,
    "heating_W_per_floor_m2": 13.393000000000002,
    "uniform_U_heating_W_per_floor_m2": 3.025
  },
  "analysis_deviation": "Initial assertion expected 9-storey/80-percent glazing to exceed single-storey/20-percent; it did not. No parameters changed. Both results retained; declared 16-storey grid supplies crossover illustration.",
  "compact_saving_fraction_at_equal_glazing": 0.568357831515726,
  "compact_9_floor_window_fraction_crossover_against_1_floor_20percent": 0.8577046181984451,
  "model_cases": 80,
  "checks_passed": true,
  "plan_sha256": "fcd7c0e5f5be65a13b43dc9c1543cdf09affb30c79bd789fc13e3f1dab31a1a7",
  "code_sha256": "135870cbcb3e66c1355e0a969a9870605ca12febe1d5380025b0921e89e31109"
}

Original hypothesis: DeepSeek / Incoprea. Analysis, code, figures and report: Codex. Independently checked by another Codex agent; this internal check is not external peer review. Established mathematics is not claimed as a new discovery. Return to Incoprea Research · Read the unchanged original hypothesis.

Publication code SHA-256: 135870cbcb3e66c1355e0a969a9870605ca12febe1d5380025b0921e89e31109
Saved result SHA-256: 70c6e835ed13682fe546b87be279e1c64a3d401f18ea74bece266646ea58e46b

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