Code uniqueness of 4-cell connected disjoint clusters in spatial chromatic model

1. Introduction

• Spatial tessellation partitions a 2D or 3D space or spatial set into subsets without gaps or overlaps.Examples include natural regions such as continents and oceans and artificial divisions such as countries.Source: §1 · p. 1
• SCM is an entity-oriented spatial partition analogous to Voronoi diagrams, and its cells have chromatic codes that identify subspaces.The paper describes these integer codes as richer in spatial and entity information than raster row and column numbers, and more efficient for computer processing than floating-point coordinates.Source: §1 · p. 2
• The paper investigates uniqueness of spatial chromatic codes as a theoretical property of SCM.The introduction also situates SCM among spatial tessellation models and notes applications across fields including geography, computational geometry, and chemistry.Source: §1 · pp. 1–2

2. Spatial chromatic model: conception and properties

• SCM maps relationships between spatial entities into spatial representations, forming cells whose chromatic codes support entity-space analyses and cell-merging into tessellations.Preliminary applications include spatial topology, tessellation generation, point-pattern recognition, competition analysis, and refinement of geospatial concepts.Source: §2 · p. 2

2.1. Generating a standard SCM.

• Standard SCM construction partitions the plane using perpendicular bisectors of entity pairs, dyes each half-plane by its entity, and overlaps them into coded cells.The code counts each entity color in the cell and orders the counts by entity index; Figure 1 illustrates this construction from three points.Source: §2.1 · pp. 2–3, Figure 1 · p. 3
• A standard SCM uses all point-pairs, perpendicular-bisector dyeing lines, and entity colors; changing any condition yields another SCM type.The paper focuses on the standard SCM.Source: §2.1 · pp. 3–4

2.2. Basic conception and properties of SCM.

• A cell is an indivisible SCM unit whose code is a permutation of 0 through n−1; subcode locations identify where values occur, and a cluster merges cells with code equal to their sum.The paper compares cells with pixels, Voronoi regions, and other spatial units, emphasizing that cells carry integer codes.Source: §2.1 · p. 4, §2.2 · pp. 4–5
• Value transitions swap two subcode values, while location transitions swap values at two positions; applying the same location transition to every cluster cell preserves connectivity.The two transition types generally produce different cells.Source: §2.2 · pp. 5–6
• Neighboring cells share a dyeing-line segment, and a connected m-cell cluster has paths between every pair using only component cells; disjoint clusters share no component cells.Cluster equality is defined analogously to cell equality, which requires all corresponding subcodes to match.Source: §2.2 · pp. 5–6
• Connected cells have chromatic distance 2, equal cells or clusters have distance 0, and cell-code distance is the L1 distance in a high-dimensional Euclidean interpretation.These properties provide distance and connectivity facts used in the model's analysis.Source: §2.2 · pp. 5–6
• Earlier work proved uniqueness for cell codes and for connected disjoint 2-cell and 3-cell cluster codes, establishing the prior result extended by this paper.The cited proofs of these properties are attributed to,.Source: §2.2 · p. 6

3. Code uniqueness of 4-ccd clusters

• Theorem 1 establishes that two 4-ccd clusters cannot have the same chromatic code.The section frames this as the uniqueness result for chromatic codes of 4-cell connected disjoint clusters.Source: §3 · pp. 6–7

3.1. Connection types and subcode transitions of 4-cc clusters.

• Three value transitions are constrained to adjacent subcodes, yielding ten transition-interval types; 4-cc connection graphs can be trees, chains, or rings, but not bell types.The transition intervals describe which subcodes can swap, while the graph types describe connections among the four component cells.Source: §3.1 · pp. 7–8
• For six consecutive transition codes, a seed cell undergoes a minimal transition followed by two further path transitions, generating the four-cell cluster family.The minimal transition is TE(a, a + 1); each seed cell has at most five neighboring cells under the five possible adjacent-value transitions.Source: §3.1 · pp. 8–10

3.1.1. Features of chromatic codes of 4-ccd clusters.

• Cluster codes are analyzed through t-codes, l-codes, and r-codes; unequal r-codes guarantee unequal chromatic codes.The r-code is formed by taking nonzero residues modulo 4 and sorting them; static subcodes contribute zero residues.Source: §3.1.1 · pp. 10–11

3.2. Proof of code uniqueness of 4-ccd clusters.

• For equal original cells, matching t-codes still share the seed and minimal-transition cells; for different bases, one cluster contains subcode 4a while the other does not.Thus, neither case gives disjoint clusters with equal chromatic codes.Source: §3.2 · pp. 12–13
• With equal bases but permuted locations, distinct-subcode t-codes yield unequal codes; when repeated subcodes permit equality, the described (0, 1) permutation makes the clusters joint.The latter case makes each original cell the other's minimal-transition cell, so the clusters share both original cells.Source: §3.2 · pp. 13–14
• For repeated subcodes located outside indices 0 and 1, the l-code transitions either produce a matching component cell or a subsequent matching cell, forcing the clusters to share a component.The transitions are TL(0, 1) and TL(li, lj), where li and lj locate the repeated subcodes.Source: §3.2 · p. 14
• In the standard 2D SCM, equal 4-cell clusters must be joint, so non-joint 4-ccd clusters have unique chromatic codes; the proof extends from Eq. (3.8) to Eq. (3.10).The extension uses the transition-interval argument to complete the proof of Theorem 1.Source: §3.2 · pp. 14–15
• After value transitions within an m-cell connected cluster, a subcode x in another component cell lies in [x − m + 1, x + m − 1].The bound follows because the cluster has m − 1 transitions, each changing the value by one.Source: §3.2 · pp. 14–15
• Shifting a transition interval’s base preserves component-cell connectivity if shifted subcodes do not equal subcodes in other intervals.The proof relies on neighboring value transitions having chromatic distance 2, independent of the base value.Source: §3.2 · p. 15
• Increasing a transition-interval base by k ≥ 1 raises its generated 4-cell cluster subcodes by at least 4k, while subcodes in other intervals rise by at most 3.Because 4k > 3, this shift prevents equality with subcodes in other intervals whose bases were not increased, supporting the extension to Eq. (3.10).Source: §3.2 · p. 15

• In R^n, 4-ccd chromatic codes remain unique as spatial dimension increases, despite additional hidden cells and their increased dimensionality.Source: §4 · p. 15
• Uniqueness of connected disjoint cluster codes implies spatial heterogeneity and gives the spatial tessellations composed of those subspaces unique codes.Source: §4 · p. 15
• Connectivity and disjointness are necessary for uniqueness; disconnected or joined component cells may have equal chromatic codes.Source: §4 · p. 15
• For 5-ccd and 6-ccd clusters, uniqueness is computer-verified but not mathematically proven; considering more neighboring-cell configurations makes the proof substantially more complex.Source: §4 · p. 16