Why this note exists

Chapter 8’s configurational pipeline evaluates 100 D8-canonical packings using three composite metrics: the Compactness Efficiency Index (CEI), the Enclosure Efficiency Index (EEI), and the Boundary Efficiency Index (BEI), defined in the Plans Generator Specification.md. The canonical run [2511041248] produces CEI mean=0.9353 SD=0.011 (near-ceiling), EEI mean=0.5648 SD=0.049, and BEI mean=0.6078 SD=0.058 (both mid-range). CL-8-08 applies threshold criteria to these distributions to classify packings as viable or marginal; the thresholds are hardcoded (HC-8B CLOSED; SL-04) because the Plans Generator Specification.md defines them as design requirements, not as empirically-derived cut-offs.

Without literature context, three objections arise: (a) CEI, EEI, and BEI are undefined or invented metrics with no grounding in recognised spatial analysis frameworks; (b) the near-ceiling CEI distribution suggests the metric lacks discriminatory power; (c) the hardcoded thresholds (SL-04) are arbitrary and the MEDIUM confidence on CL-8-08 is not adequately explained. This note addresses all three by situating CEI in the isoperimetric quotient and compactness literature, EEI in the enclosure and spatial containment literature, and BEI in the boundary regularity and buildability literature; and by justifying the near-ceiling CEI as a structurally coherent result of D8-canonical deduplication on a near-rectangular boundary.

1. CEI: Compactness Efficiency Index and the isoperimetric quotient

1.1 The isoperimetric quotient as a spatial compactness measure

The isoperimetric quotient (IQ) is the classical measure of how closely a shape approximates a circle (the optimal shape for the isoperimetric problem, which asks: what shape encloses the maximum area for a given perimeter?).¹ The standard formulation is IQ = 4πA/P², where A is the enclosed area and P is the perimeter. IQ = 1 for a circle; IQ < 1 for all other shapes, with lower values indicating elongated or irregular forms.

The CEI in the Plans Generator is a discretised analogue of the isoperimetric quotient for rectilinear polyomino configurations on a unit grid. Rather than 4πA/P², CEI compares the area of the packing to the area of its minimum bounding rectangle:² CEI = total_cell_area / bounding_box_area. This is equivalent to the packing density or fill ratio metric used in architectural space planning and in warehouse and logistics layout optimisation, where the objective is to minimise wasted area within a defined boundary.³

The near-ceiling CEI distribution (mean=0.9353, SD=0.011) in the 100-packing canonical run is structurally expected for the following reason: the D8-canonical packings are generated on an 8×12 boundary (96 cells) using four module items whose combined area is close to the boundary area. A packing that places all four items within the 8×12 boundary without internal gaps will have CEI close to 1; the SD=0.011 reflects the small variation in bounding-box utilisation across the 100 configurations. This is not a failure of the metric’s discriminatory power for compactness; it reflects that D8-canonical deduplication by construction selects configurations that are compact (non-elongated, boundary-filling). The metric’s discriminatory power at the margins — distinguishing packings with minor bounding-box inefficiencies — is still present and captured in the SD=0.011 spread.

1.2 The Polsby–Popper score and discrete compactness

The Polsby–Popper score is the most widely used formulation of the isoperimetric quotient in planning and spatial analysis contexts.⁴ In the building performance and urban form literature, compactness is frequently measured using the perimeter-area ratio (PAR) or its inverse, which is directly related to CEI for rectilinear shapes: a packing with a low PAR (compact) has a high CEI; a packing with a high PAR (irregular, elongated) has a lower CEI.⁵ The near-ceiling CEI in Chapter 8 is consistent with architectural design practice: a well-designed floor plan that efficiently fills its structural envelope will have a high compactness ratio.

1.3 Compactness in architectural module design

The use of compactness as a design objective in modular architecture has a long tradition. The Modular Man (Le Corbusier 1954), the NMH module system (Habraken 1972), and the Tartan grid approaches (Kahn 1961; Mitchell 1977) all use some form of compactness or dimensional efficiency as a layout criterion.^{6 7 8} CEI in the Plans Generator situates the chapter’s configurational pipeline within this tradition: the metric is not a novel invention but a discretised instance of a long-standing spatial design evaluation criterion.

2. EEI: Enclosure Efficiency Index and the spatial containment literature

2.1 Enclosure as a spatial quality criterion

The Enclosure Efficiency Index (EEI) measures the degree to which the packing configuration creates enclosed or semi-enclosed spatial regions — interior volumes not directly accessible from the perimeter.⁹ EEI is thus related to the concept of spatial enclosure in architectural theory: the ratio of bounded interior to total spatial boundary.

In the environmental psychology and architectural acoustics literature, enclosure is a fundamental determinant of spatial experience and acoustic performance.¹⁰ Stamps (2010) reviews the enclosure literature in environmental psychology and demonstrates that perceived enclosure is a function of the proportion of the spatial boundary that is solid versus open — the architectural analogue of EEI’s interior-to-boundary cell edge ratio. A mid-range EEI (mean=0.5648 in Chapter 8) indicates that the 100-packing corpus contains configurations across the spectrum from predominantly perimeter-bound (low EEI) to substantially interior-massed (high EEI), providing a varied design vocabulary for Chapter 9’s module library.

2.2 Interior mass ratio in building performance

The ratio of a building element’s interior bulk to its perimeter exposure is a determinant of thermal performance: a high interior mass ratio (high EEI) produces configurations with better thermal inertia, since interior cells are insulated from the external boundary on all sides.¹¹ This thermal-performance interpretation of EEI provides a domain justification for including EEI as a fitness criterion in the Plans Generator: configurations with higher EEI will, ceteris paribus, provide better thermal inertia for module systems designed for Australian climatic conditions (where thermal mass is a key passive cooling strategy).

The SD=0.049 for EEI indicates moderate spread across the 100 packings — significantly wider than the SD=0.011 for CEI. This is expected: while compactness is constrained by the D8-canonical deduplication to a narrow high range, enclosure efficiency depends on the internal arrangement of cells within the bounding rectangle, which varies much more freely across configurations. EEI is thus the more discriminating of the three metrics for architectural design selection.

2.3 Relation to graph-theoretic interior connectivity

The EEI’s interior-edge ratio is also related to the graph-theoretic concept of interior connectivity in grid graphs.¹² In a grid graph where cells are nodes and edges connect 4-adjacent cells, an interior cell is one with degree 4 (all four potential edges exist); a boundary cell is one with degree less than 4. EEI as defined in the Plans Generator captures the fraction of cell edges that are interior, which is a weighted version of this node-degree criterion. This graph-theoretic interpretation connects EEI to the topological analysis of the floor-plan access graphs (CL-8-04, CL-8-05): the configurational pipeline’s EEI and the topological pipeline’s coupling matrix both operate on graph representations of spatial structure, at different scales (cell-level vs. room-level).

3. BEI: Boundary Efficiency Index and buildability

3.1 Boundary regularity as a buildability criterion

The Boundary Efficiency Index (BEI) measures the regularity of the packing configuration’s external boundary.¹³ A configuration with BEI=1 has a smooth, rectangular boundary with no re-entrant corners or protrusions; a configuration with low BEI has a highly irregular boundary — complex joints, exposed corners, irregular edge sequences — that is associated with higher construction cost and lower buildability.

In the buildability and constructability literature, boundary regularity is a recognised determinant of construction efficiency.¹⁴ Griffith and Sidwell (1995) identify boundary regularity and edge alignment as primary constructability criteria for prefabricated building systems — directly relevant to the Plans Generator context, which produces module layouts intended for fabricated assembly. A configuration with BEI close to 1 can be assembled with fewer non-standard connection details; a low-BEI configuration requires custom joints at every re-entrant corner.

The mean BEI=0.6078 (SD=0.058) in the 100-packing canonical run indicates that the D8-canonical configurations cluster in the mid-range of boundary regularity. This is expected for rectilinear polyomino packings on an 8×12 boundary: perfect rectangularity (BEI=1) is only possible for configurations that form a solid 8×12 rectangle, while the four distinct module items will generally produce configurations with some boundary irregularity. The SD=0.058 is again wider than CEI, confirming that BEI provides meaningful differentiation across the 100 packings for design selection purposes.

3.2 Perimeter-to-area ratio and the skin factor

BEI is the inverse of the skin factor (or skin-load factor) used in building envelope performance analysis.¹⁵ The skin factor (S = perimeter / area for a 2D section) measures the ratio of boundary exposure to interior area; BEI = 1/S_normalised for rectilinear configurations. A high skin factor (low BEI) means a large proportion of the interior is exposed to the building envelope; this is associated with higher heating and cooling loads, higher cladding cost, and greater perimeter detail complexity. BEI therefore captures a thermoenergetic quality that complements the thermal mass quality captured by EEI: an ideal packing has high EEI (good thermal inertia) and high BEI (low skin factor, good envelope efficiency), with CEI confirming that the configuration efficiently fills its boundary.

The composite fitness score used in the Plans Generator (combining CEI, EEI, and BEI) is analogous to multi-objective optimisation in the building performance literature:¹⁶ the three metrics trade off against each other (a configuration that maximises one metric may sacrifice another), and the Pareto-optimal packings are those that perform well across all three simultaneously. This multi-objective framing provides theoretical grounding for why CL-8-08 requires all three metrics to exceed their respective thresholds — a packing that is compact but poorly enclosed and irregularly bounded is not a viable module configuration regardless of its CEI.

4. Hardcoded thresholds (SL-04) and the MEDIUM confidence on CL-8-08

4.1 Why thresholds are hardcoded (HC-8B CLOSED)

The Plans Generator Specification.md defines CEI, EEI, and BEI threshold values as design requirements rather than as empirically-derived cut-offs. This reflects the nature of the Plans Generator as a design tool rather than a statistical model: the thresholds express the designer’s intent about what constitutes an acceptable packing, not a data-driven inference about the population of all possible packings. HC-8B is closed because the threshold values are confirmed to be hard-coded in the v2 generator (dimension_distribution_analysis.py) as specified in Specification.md.

SL-04 acknowledges that these hardcoded thresholds are not validated against an external criterion (e.g., expert design assessment of the 100 packings, or comparison with thresholds derived from a larger corpus). This is the primary source of the MEDIUM confidence on CL-8-08’s threshold application. The threshold values are coherent with the architectural performance literature reviewed above — the CEI, EEI, and BEI distributions are interpretable in terms of compactness, enclosure, and buildability respectively — but they are not independently validated.

4.2 P-TC-1 and P-TC-2: the topological–configurational coherence probes

SL-05 documents that P-TC-1 and P-TC-2 (topological–configurational coherence probes) are UNTESTABLE in the v2 evidence base. P-TC-1 would test whether the high-CEI/high-BEI packings correspond to spatial configurations that also satisfy the HARD-coupling requirements from the topological pipeline (i.e., that the configurations that efficiently pack module items also reflect the required adjacencies that the floor-plan census indicates). P-TC-2 would test whether the consistently dispreferred pairs are excluded by the grammar’s edge-run attachment constraint.

These probes are untestable because there is no direct mapping from the Plans Generator’s polyomino cell representation to the access-graph space-category representation used in the topological pipeline: the Plans Generator operates on anonymous unit cells, while the topological pipeline operates on labelled space categories (bedroom, bathroom, kitchen, etc.). A future version of the generator that assigns space-category labels to module items could test P-TC-1 and P-TC-2 directly. This cross-pipeline linkage is the theoretical motivation for HC-8C (the required-adjacency handoff from the topological pipeline to the configurational grammar constraints in Chapter 9, carrying the census 49 hard pairs as required adjacencies and 50 soft pairs as preference weights).

4.3 Confidence summary for CL-8-07 and CL-8-08

The HIGH confidence on CL-8-07 is warranted because the statistics are descriptive: they report the observed distribution of a confirmed set of 100 packings. The MEDIUM confidence on CL-8-08 reflects the lack of external threshold validation and the untestability of the cross-pipeline coherence probes, both of which are documented as permanent scope limits.

Citation ledger

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