Semantic Structure Reconstruction of the Four Color Theorem
通用人工智能AGI测评DIKWP实验室
Semantic Structure Reconstruction of the Four Color Theorem: A Semantic Graph Coloring Model Based on DIKWP and Yucong Duan's Theory
International Standardization Committee of Networked DIKWPfor Artificial Intelligence Evaluation(DIKWP-SC)
World Academy for Artificial Consciousness(WAAC)
World Artificial Consciousness CIC(WAC)
World Conference on Artificial Consciousness(WCAC)
(Email: duanyucong@hotmail.com)
This paper conducts a semantic structure reconstruction study revolving around the Four Color Theorem. By introducing the DIKWP semantic model and theoretical viewpoints such as Semantic Consistency, Information Tension, DIK Minimum Chain, and BUG Consciousness proposed by Yucong Duan, we explore a new interpretation path for the Four Color Problem different from traditional graph theory proofs. Traditional proofs of the Four Color Theorem rely on computer enumeration, which is controversial philosophically and technically. This paper attempts to view map coloring as a process of semantic compression mapping and constructs a new semantic graph coloring model with the aid of semantic hierarchy analysis. We first briefly describe the content of the Four Color Theorem and the limitations of traditional proofs, then introduce the DIKWP model to decompose the cognitive process of map coloring: from regional division in the Data Layer, to adjacency tension in the Information Layer, to the semantic graph structure in the Knowledge Layer, and then to achieving the cognitive distinction goal of the Purpose Layer through coloring decisions in the Wisdom Layer. Integrating Yucong Duan's semantic theory, we emphasize introducing "BUG Consciousness" in the cognitive mapping of regional distinction to identify conflicts, and gradually increasing color dimensions through the DIK Minimum Tension Chain to achieve closed-loop control of "semantic coloring tension." Based on this, this paper constructs a semantic reconstruction model of the Four Color Problem: using visualized adjacency tension graphs and "Semantic Color Tension Tensor" structures to represent independent distinction dimensions between regions, viewing coloring behavior as a minimum heterogeneous structure driven by the W-P chain (satisfying adjacent heterogeneity constraints with as few colors as possible). We further propose implementation ideas to simulate this model, including developing a DIKWP semantic graph generator, tension-adjustable coloring algorithms, and using Graph Neural Networks to simulate the propagation of semantic tension. Finally, we expand the discussion of the Four Color Theorem from the perspectives of philosophy and topological cognition: Does four-color sufficiency reflect the minimum complete cardinality of semantic tension compression? Starting from the semantic distinctiveness of planar regions, how does the complication of topological structures affect the evolutionary trend of the required number of colors? This paper aims to prove that the four-color property can be seen as a natural result of semantic compression mapping, revealing the cognitive semantic motivation behind mathematical theorems, and providing a new paradigm of semantic analysis for proving complex problems.
The Four Color Theorem is a famous proposition in mathematics: for any finite number of mutually adjacent regions divided on a plane, no more than four colors are needed to fill all regions such that no two adjacent regions have the same color. This proposition was first conjectured by Francis Guthrie in 1852. After more than a century of attempts and failures, the Four Color Conjecture was formally established as the Four Color Theorem in 1976 by Appel and Haken with the help of computers. The Four Color Theorem is remarkable for its intuitive and simple statement and unexpected difficulty of proof: coloring a map seems like an approachable problem, but rigorously proving that at most four colors are sufficient for any map is extremely difficult. More controversially, the final proof of the Four Color Theorem relies heavily on the exhaustive inspection of a large number of cases by computers—this is the first theorem proof in the history of mathematics primarily completed with the aid of computers. Since computer verification involves massive case analysis beyond the ability of the human brain to review one by one, some mathematicians did not recognize this proof method initially. Although computer-assisted proofs have gradually been accepted and the credibility of this theorem has been widely recognized, some scholars still expect to find a concise proof that does not rely on computers or to understand from a more intuitive perspective why the number "four" is sufficient and necessary in planar map coloring. As mathematician Seymour Papert lamented: "I would be happier if there were a proof of the Four Color Problem that did not depend on computers, but since there is no choice, I have to accept the computer proof." It is evident that at the level of mathematical philosophy, there is a discussion about "what constitutes a satisfactory proof": Does a proof relying purely on machine verification truly explain the reason why the theorem holds? Does the Four Color Theorem contain deeper mathematical structures or cognitive principles?
Bearing the above questions, this paper attempts to step out of the framework of existing graph theory methods and re-examine the Four Color Theorem from the perspectives of semantics and cognition. Our starting point is to view a map as a cognitive expression that conveys information about regional differences. Coloring a map can be regarded as a Semantic Compression Mapping: that is, under the premise of reducing color types as much as possible, ensuring that the color distribution after mapping still distinguishes all adjacent regional semantics in the original map. This suggests that the Four Color Theorem may not be an isolated combinatorial mathematical coincidence, but reflects a Limit Compression of Cognitive Semantic Tension—under the planar topological structure, there is an upper limit to the independent signal types required to transmit adjacent distinction information, and this upper limit is exactly 4. To verify this idea, we need a model capable of characterizing the "Map-Cognition-Semantics" relationship. To this end, this paper introduces the DIKWP Semantic Model (Data-Information-Knowledge-Wisdom-Purpose) and relevant semantic theories proposed by Professor Yucong Duan in recent years to decompose the cognitive process of the map coloring problem hierarchically and analyze the inevitability of the Four Color Theorem under this framework.
The DIKWP model was originally extended from the classic "Data-Information-Knowledge-Wisdom (DIKW)" framework, adding a "Purpose" layer to emphasize the intention and goal in the cognitive process. Yucong Duan and others applied this model to the research of "Understanding-based Proof," believing that the proof of mathematical propositions can be seen as a process of gradually Constructing Understanding at various semantic levels. In their view, as long as a reasoning structure that makes the proposition true can be constructed based on the semantic understanding of the problem, it can be considered that the proof of the proposition has been completed. This concept of proof differs from traditional formal deduction and emphasizes "explaining why the proposition is true" through insight into concepts and semantic relationships. This paper draws on this idea to transform the proof of the Four Color Theorem into a semantic analysis task: We hope to show that when we understand the essential reason "why planar maps do not need more than four colors" at the semantic level, we have actually obtained an understanding-based proof of the Four Color Theorem. Next, we will first review the traditional definition and proof points of the Four Color Theorem, then explain the DIKWP semantic model and Yucong Duan's relevant theories in detail, and finally construct a new model for semantic graph coloring and discuss its significance.
Traditional Review: The Four Color Theorem and Proof Dilemma
Content of the Four Color Theorem: The Four Color Theorem, also known as the Four Color Map Theorem, is informally stated as: Given any map without enclaves on a plane, we can always color the various countries or regions on the map with no more than four colors, and ensure that any two adjacent regions (regions sharing a common boundary) use different colors. "Adjacent regions" specifically refer to two regions sharing a segment of common boundary, not just touching at a point. For example, in a graphical example, the red region and the green region share a boundary and are thus mutually adjacent, while the yellow region only touches the red region at a point, so they are not considered adjacent. Figure 1 gives an example map that satisfies the requirements of the Four Color Theorem.
Figure 1: Four Color Map Example. Even if the map regions have complex shapes and intricate relationships, coloring can be completed with four or fewer colors, and adjacent regions have different colors. The classic Four Color Theorem asserts that no matter how finite adjacent regions are divided, such a four-color coloring scheme always exists.
Formally, map coloring can be equivalently transformed into a vertex coloring problem of a planar graph. For a map, we correspond each region to a vertex. If two regions are adjacent, we connect an edge between the corresponding vertices, obtaining a planar graph without crossings. The Four Color Theorem is equivalent to asserting: For any planar graph, there exists a scheme to color its vertices using no more than four colors such that vertices at both ends of every edge have different colors. In other words, the upper bound of the vertex chromatic number of a planar graph is 4. This theorem has a looser version—the Five Color Theorem, for which a concise proof was given as early as the nineteenth century: any planar map can always be colored using five colors. But reducing the number of colors from 5 to 4 sharply increases the complexity of the problem, exceeding the capabilities of general induction and simple configuration arguments. Over more than a hundred years, numerous mathematicians proposed proofs or counterexamples claiming to solve the Four Color Problem, all of which were found to have loopholes without exception. It was not until 1976 that Kenneth Appel and Wolfgang Haken, using a computer at the University of Illinois, verified a set of unavoidable reducible configuration sets, exhausted all cases of the Four Color Theorem from the perspective of Enumeration, and finally proved that there are no special maps requiring a fifth color.
Limitations and Controversies of Traditional Proof: The computer-assisted proof by Appel and Haken solved the Four Color Conjecture but also triggered a series of controversies and reflections. First, the proof itself is overly complex, relying on the classification and computer verification of nearly two thousand "irreducible configurations," a degree of tediousness far beyond what manual verification can afford. Many mathematicians were initially skeptical, believing this was not a "human-checkable proof." Second, could there be errors in the computer program and data itself? Although subsequent teams simplified and re-verified the proof (reducing the number of irreducible configurations to 633 and improving the correctness of the program), questions at the level of mathematical philosophy remain: Has a theorem truly been "understood" if our certainty comes only from machine calculation results? Precisely because of this, the mathematical community continued to discuss the nature of the proof of the Four Color Theorem for a long time after the proof came out. Many people expect to find a proof that is more concise, elegant, and independent of computers, revealing a "penetrating" principle for the Four Color Theorem. In fact, the Four Color Problem has a profound topological background—for surfaces of higher complexity, the number of colors required for coloring follows the Heawood formula. For example, a torus (genus
k=1
) requires at most 7 colors. The Heawood conjecture (which has been proved) states: The maximum number of colors required on an orientable surface is
7+
1+48
k
2
/
2
, where
k
is the genus of the surface. When
k=0
(plane or equivalent sphere), this formula gives 4 colors, which is exactly the content of the Four Color Theorem. This result implies that the four-color phenomenon is not a coincidence but an inevitable result of the interaction between topology and combinatorial structure. However, traditional proofs have not directly clarified "why the plane is exactly 4 independent colors," that is, the deep reason why one less color won't do and one more is wasteful. Computer proof solved the problem of "how to prove," but did not fully satisfy people's curiosity about "why it holds."
In summary, from the perspective of traditional graph theory, the Four Color Theorem is viewed as a discrete mathematics proposition, and the proof process mainly examines the exclusion of minimal counterexamples and the enumeration of complex configurations. Although this methodology is effective, it lacks intuitive insight. We hope to change the line of thought: viewing map coloring as a problem of information transmission and semantic distinction, and understanding the inevitability of the Four Color Theorem from cognitive structures. In the next section, we will introduce the DIKWP semantic model to decompose the semantic elements of each layer of the map coloring process. This model prepares for the introduction of cognitive semantic theory and will help us reconstruct the proof idea of the Four Color Problem.
DIKWP Semantic Structure Model
To analyze the Four Color Theorem from a semantic perspective, we first need a framework to describe information processing and cognitive goals in the map coloring process. The DIKWP model provides such a five-level structure. DIKWP is an abbreviation for "Data-Information-Knowledge-Wisdom-Purpose," where the first four layers are derived from the classic DIKW model, and the fifth layer "Purpose" is an extension added by Yucong Duan and others to emphasize cognitive intent. In the map coloring task, we can divide the entire process into the following levels according to DIKWP:
Data Layer (D): The basic data of the map, i.e., the regional division itself and basic attributes. This includes the shape and location of each "country" or region on the map, and the topological structure they form (which boundary lines divide which regions). Each region can be seen as a data point, and each regional boundary as a line separating data. This layer focuses on objectively given facts: how many regions there are, which regions border each other, etc. For example, a specific map can list regions A, B, C..., and their adjacency relationships as input for the Data Layer.
Information Layer (I): Patterns and relationship information refined from the Data Layer, i.e., the tension relationship formed by regional adjacency. "Tension" here refers to the need for distinction brought by adjacency relationships: when two regions share a boundary, they need to be treated differently in cognition, otherwise they merge into one region as a whole. The core of the Information Layer is the adjacency graph structure of the map, where each node represents a region and each edge represents a constraint that needs to be distinguished between adjacent regions. Semantically, we can view each adjacency edge as a Distinction Requirement or Semantic Boundary Tension: it requires two connected regions to differ in some explicit feature. This level refines the pattern of the map: "which regions need to be distinguished from each other." For example, if region A is adjacent to B, this is an information unit indicating a semantic tension of "must distinguish" exists between A and B.
Knowledge Layer (K): Elevating relationships in the Information Layer to an overall structure and constraint rules, i.e., forming a Regional Semantic Integration Graph. In the Knowledge Layer, we integrate all adjacency relationships to obtain the overall topological structure of the map (usually representable as a planar graph or adjacency matrix). More importantly, we establish Universal Coloring Constraint Laws at this layer: adjacent regions must have different colors. This is actually the embodiment of the proposition of the Four Color Problem itself on the current specific map, i.e., a constraint set. The semantic structure of the Knowledge Layer can be seen as "general knowledge about this map": it contains all distinction conditions that must be met, equivalent to giving a complete constraint description of the problem. In traditional graph theory, this corresponds to the structural qualification of planar graphs, such as "this is a cubic graph," "this graph does not have substructures requiring five colors," etc. If we step out of a specific map, this layer also covers the Universal Proposition that the Four Color Theorem aims to prove: "Any map has a coloring scheme within four colors that complies with the adjacent heterogeneity rule."
Wisdom Layer (W): Strategy and choices oriented towards execution, i.e., the Behavioral Layer of Regional Expression/Coloring. At this layer, we actually Assign Colors to various regions (or decide what means to use to distinguish adjacent regions) based on constraints from the Knowledge Layer. The Wisdom Layer includes Formulation and Adjustment of Coloring Schemes: how to choose a specific color configuration for the map so that all adjacency constraints are met. This requires comprehensive judgment and strategy: such as what order to color the regions, how to adjust if conflicts occur, how to use as few colors as possible while satisfying constraints, etc. It can be said that the Wisdom Layer embodies active decision-making and optimization processes, semantically corresponding to Manifestation of Semantic Signals, i.e., the process of transforming internal requirements (difference) into external symbols (colors). An effective coloring scheme is precisely the result output by the Wisdom Layer, satisfying all distinction conditions prescribed by the Knowledge Layer.
Purpose Layer (P): The cognitive goal of the entire process, i.e., the Ultimate Purpose of Map as Cognitive Expression. In the Four Color Coloring Problem, the Purpose Layer can be defined as "enabling the cognizer to easily distinguish different regions on the map." In other words, we pursue a Minimalist and Unambiguous Regional Identification scheme for the observer. Colors play the role of signals here: by painting regions with different colors, the viewer can know which regions belong to different entities without effort. The Purpose Layer also implies an optimization pursuit: using as few color types as possible while clearly distinguishing regions, achieving a balance between symbol economy and cognitive efficiency. It can be considered that the Four Color Theorem itself embodies a kind of purpose optimization: using the minimum types of marking signals to distinguish and identify any complex regional distribution, and "four" is the upper limit of this signal type.
The above DIKWP hierarchical structure characterizes the information flow and cognitive decision chain from the map itself to successful coloring. In an uncolored map, the Data Layer provides the "factual basis" of regional division, and the Information Layer reveals the adjacency relationships that need distinction. The Knowledge Layer summarizes the universal constraint laws to be followed for "single map coloring." The Wisdom Layer makes actual color assignments based on these constraints (i.e., trying to construct a coloring scheme that meets conditions), and the Purpose Layer verifies whether the scheme achieves the goal of cognitive distinction. This flow can be viewed as a closed loop from Semantic Generation to Semantic Satisfaction: the map provides semantic objects to be distinguished, and the coloring scheme provides symbolic means to achieve distinction. The DIKWP model allows us to introduce semantic analysis tools at each layer to understand different aspects of the Four Color Problem: for example, the Information Layer corresponds to the Semantic Tension Graph of the map, the Wisdom Layer corresponds to Coloring Algorithms, and the Purpose Layer corresponds to Effect Evaluation and Optimization.
It is worth noting that in this hierarchical process, there are Tension Transmission and Information Compression phenomena between different levels: each adjacency tension in the Information Layer requires the Wisdom Layer to use at least two different colors between the corresponding two regions (to relieve tension); at the same time, the Wisdom Layer tries to reuse limited colors to compress the signal types satisfying all tensions. The core of the Four Color Problem lies in that all tension relationships of planar maps can be compressed into four or fewer color signals, satisfying them without omission. This will be explained below through the integration of semantic theories.
Integration and Analysis of Yucong Duan's Semantic Theory
Under the DIKWP framework established above, we introduce several semantic theoretical concepts proposed by Yucong Duan to deeply analyze the cognitive semantic essence of the Four Color Problem. Professor Yucong Duan's "Semantic Mathematics" research involves ideas such as Concept Semantic Consistency, Information Tension and Logical Jumping, DIK Minimum Chain, and Consciousness BUG Theory. These concepts were originally used to analyze the semantic consistency and reasoning process of AI-generated content. Here we analogize them to the semantic process of map coloring, hoping to reveal the semantic inevitability of the Four Color Theorem.
Semantic Consistency and Concept Consistency: Semantic Consistency (Semantic Coherence) refers to the consistency and lack of ambiguity of concepts during use. In the context of map coloring, semantic consistency is embodied as: the meaning we assign to colors must be stable and unique, that is, a color can only represent one category of regions in the same map and cannot be confused. Similarly, each region can only have one color mark, and cannot be contradictory. This seems obvious, but it is what ensures the logical consistency of the coloring scheme in the Knowledge Layer. If semantics are not consistent, for example, if the same color sometimes represents region A and sometimes represents neighboring region B, it will cause confusion for the cognizer and violate the distinction goal of the Purpose Layer. Therefore, semantic consistency is a basic requirement for coloring schemes, ensuring that each color category has a clear referential domain and each region has a clear and unique color identity. From the perspective of proof, this is equivalent to acknowledging that the Four Color Problem solves for a Correct Color Assignment rather than arbitrary coloring—our solution space is limited to the set of semantically consistent schemes.
Information Tension and Minimum Chain: Yucong Duan calls the span in the reasoning process "Information Tension," meaning the degree of semantic pulling between conclusion and premise. If many intermediate steps need to be crossed or many hidden assumptions filled, the information tension is high. Accordingly, DIK Minimum Chain can be understood as establishing a reasoning chain with as small a step size as possible between Data-Info-Knowledge layers, avoiding large jumps to keep reasoning continuous and smooth. Applied to the Four Color Theorem, we hope to construct an argument that gradually increases map complexity, introducing only Minimal New Information (such as adding a boundary line) at each step and immediately dealing with the ensuing tension, rather than facing all complex constraints of a complete map from the start. If this can be achieved, the proof will be closer to "continuous deformation" rather than "discrete jumping." In fact, Yucong Duan adopts this strategy in his semantic argument for the Four Color Theorem: he starts from a plane with 0 boundaries (one region) and adds boundary lines one by one, analyzing how the number of colors changes accordingly. This step-by-step approach makes the information tension at each step small and solvable locally, thereby avoiding the huge tension of "a complex whole right from the start" in traditional proofs. The DIK Minimum Chain idea ensures that the transition from Data Layer to Knowledge Layer is gradual: Adding Boundary (Data Layer) -> Generating New Adjacency Tension (Information Layer) -> Need to Add a Color to Satisfy Tension (Knowledge Layer Rule Update). By analogy, every link on the chain is closely connected and tension is controllable. This phased proof idea is semantically more comprehensible and matches the process of gradually deepening human cognition.
BUG Consciousness and Tension Monitoring: The "Consciousness BUG Theory" proposed by Yucong Duan stems from the detection of potential contradictions in AI outputs. He believes that when the subconscious process of a system cannot resolve a contradiction or defect, a "BUG" is produced, requiring higher-level consciousness to intervene and correct. Corresponding to map coloring, we can view Adjacent Regions with Same Color as a "Semantic BUG." When the Wisdom Layer colors the map, if a pair of adjacent regions is assigned the same color, this violates the constraints of the Knowledge Layer, equivalent to introducing a semantic contradiction—for the cognizer, these two regions that should be distinguished suddenly become indistinguishable. This is obviously an "error" in the coloring process. Therefore, a coloring algorithm with "BUG Consciousness" will be immediately alert to this situation and take measures to correct it (such as changing the color of one region). This consciousness BUG mechanism can be seen as a feedback established between the Wisdom Layer and the Knowledge Layer: when the output of the Wisdom Layer violates the Knowledge Layer constraints (i.e., a BUG occurs), the system detects that the tension has not been relieved and needs to readjust the color allocation until the contradiction is eliminated. With BUG Consciousness, our model becomes a Closed-Loop Control System: continuously monitoring the satisfaction of adjacent heterogeneity constraints during the coloring process, treating any conflict as a "Bug" and repairing it. This process continues until there are no adjacent conflicts, at which point the output scheme meets the consistency requirements of the Knowledge Layer. It can be said that BUG Consciousness ensures the final correctness of the coloring scheme. It implements the requirement of the Purpose Layer "to make adjacent regions easy to distinguish" into local checks and adjustments for every adjacency edge. It is worth mentioning that a smart "conscious" algorithm will not only passively correct errors but also actively avoid Bugs—for example, when coloring a new region, try to avoid colors already used by neighbors, thereby reducing the possibility of conflict. This reflects an Active Semantic Modeling idea: the algorithm considers global constraints and future impacts at every step of building the scheme, striving to get it right the first time or minimize rework. This is actually the experience humans follow when coloring manually: whenever choosing a color for a new region, always check what colors the surrounding neighbors have used, then pick a different one. This experience can be viewed as humanity's "Subconscious Bug Prevention Mechanism."
Semantic Tension Closed Loop and Coloring Strategy: Synthesizing the above ideas, we view the map coloring process as a Closed Loop of Semantic Tension Perception and Release. Each adjacency relationship in the Information Layer generates one unit of "tension," requiring the Wisdom Layer to use a different color to release it. If the Wisdom Layer misses releasing even one tension (i.e., adjacent same-color Bug not eliminated), the identification goal of the Purpose Layer cannot be fully achieved. Therefore, the Wisdom Layer needs to continuously encode and solve all tensions in the output. When all tensions are released (no adjacent conflicts), the closed loop reaches stable equilibrium. It is worth noting that while releasing tension, the Wisdom Layer also undertakes the task of Semantic Compression—it wants to use as few colors as possible to cover as many distinction needs, which is an optimization process. This optimization cannot be easily achieved by mechanical enumeration but requires strategy. Under Yucong Duan's framework, we can use "Semantic Coloring Tension Closed Loop" to visually understand strategy formation: every time the Wisdom Layer adds a new color, it will try to solve as many adjacency tensions as possible; the Purpose Layer inspects whether current tensions are all relieved; if not relieved, identify the remaining "stubborn tensions"—these actually correspond to a new independent distinction dimension in the map, requiring the introduction of a new color to solve; thus the Wisdom Layer introduces a new color to release tension on that dimension. This repeats until tensions in all dimensions find corresponding color release channels, then the closed loop converges, and the output scheme is completed. This process reflects the correspondence between color types and independent tension dimensions, which is exactly where the semantic substance of the Four Color Problem lies—we will elaborate on this in the next section.
Through the above analysis, we refine the semantic key of the Four Color Problem as: The distinction tension dimensions introduced by map adjacency relationships are finite and can be covered by a few colors. Yucong Duan's theory provides a cognitive reason: the structure of planar maps determines that in the process of gradually dividing regions, adding a boundary line introduces at most one new independent tension source, and humans can solve all tensions with minimum new colors through active BUG consciousness error correction and tension closed-loop optimization, maintaining semantic consistency and information continuity. This lays the foundation of cognitive semantics for understanding the Four Color Theorem. Next, based on this, we construct a specific Semantic Reconstruction Model of the Four Color Problem, giving an intuitive explanation of the four-color property as a result of semantic compression mapping.
Semantic Reconstruction Model of the Four Color Problem
With the support of the DIKWP framework and semantic theory, we now reconstruct the proof idea of the Four Color Theorem from a semantic perspective. The core lies in viewing regional adjacency as semantic tension and analyzing the evolution law of tension as map complexity increases. In this process, we will introduce the concept of "Semantic Color Tension Tensor" to describe the multi-dimensional structure of regional distinctions and explain why these dimensions accumulate to at most 4 on a plane.
Visualization of Adjacency Tension: Imagine a scenario: gradually adding boundary lines on a plane, constantly dividing out new regions. Every time a new boundary line is drawn, it is like imposing a "constraint" on the plane, requiring the regions separated by this line to use different colors. This line thus introduces a semantic tension—if two separated regions have the same color, the existence meaning of this line disappears, and the two regions will visually merge into one, contrary to our original intention of zoning by drawing lines. To visualize this tension, we can construct a Regional Adjacency Tension Graph: nodes represent regions, and lines between nodes represent adjacency tension relationships. Whenever a pair of regions are adjacent (sharing a boundary), we connect an edge in the graph to indicate "these two regions need different colors." Initially, without boundaries, there is only one region, no tension exists; after adding the first boundary line, there are two adjacent regions, generating a tension edge between them, requiring different colors to distinguish. As boundaries increase, this tension graph may become complex. But the key is that this tension relationship graph has a special structure under Planar conditions: it corresponds to the Dual Graph of a Planar Graph (the regional adjacency graph itself is essentially the face adjacency relationship of a planar graph, which can be approximated as a planar graph), thus possessing certain sparsity and colorability. More importantly, we find that every newly added boundary (in general position) Does Not increase the required color types indefinitely, but tends to reuse existing colors, or add at most one new color to satisfy new tension. This indicates that each new boundary can be seen as providing a new "distinction capability," but this capability often produces Redundancy with existing combinations: some newly divided regions can share colors with existing regions without being adjacent, thus avoiding the introduction of completely new colors.
Semantic Color Tension Tensor Structure: To describe the above phenomenon more deeply, we introduce the concept of "Semantic Color Tension Tensor." Intuitively, this can be understood as a high-dimensional matrix representing regional distinction conditions, where each dimension corresponds to an independent "distinction semantic." In the Four Color Problem, we might as well view each independent boundary line as a distinction semantic dimension because it divides the plane into two categories on either side of the line: left side and right side. For example, assume there are two intersecting boundary lines
l
1
and
l
2
dividing the plane into 4 regions. Superficially, these 4 regions are mutually adjacent, seeming to require 4 different colors; but we notice that the regional differences can be broken down into two independent dimensions: left or right relative to
l
1
, upper or lower relative to
l
2
(assuming a coordinate system). Each region can be identified by a combination of these two binary attributes. For example, regions can be described as four combinations: "Left of
l
1
& Below
l
2
", "Left of
l
1
& Above
l
2
", "Right of
l
1
& Below
l
2
", "Right of
l
1
& Above
l
2
". However, from a semantic perspective, the distinction of each dimension only requires two colors: for example, dimension
l
1
can use color A to mark "left side" and color B to mark "right side", and dimension
l
2
can introduce color C to mark distinction on the existing basis. In this way, we actually use 3 colors to give a coloring scheme for the above 4 regions, where two diagonal regions (e.g., both Left of
l
1
& Above
l
2
vs. both Right of
l
1
& Below
l
2
) share a color, and they are not adjacent. This shows that the color demand tensor of these 4 regions is 2-dimensional (
2
×
2
combination), and its rank is 3 instead of 4, because the combination redundancy of one dimension can be covered by existing colors. Generally, each new boundary line adds a binary distinction dimension. Theoretically, the number of regional combinations doubles, but due to planar topological limitations, new combinations always partially do not constitute adjacency relationships and can be reused by existing colors. We can abstract the entire plane division process as building a high-dimensional "Distinction Tensor": initially 0-dimensional (no boundary), one element (one region, requires 1 color); every time a dimension is added (introducing a line), the number of elements increases according to some rule, but the rank of the tensor (independent color number) increases slower, existing an upper bound.
In Yucong Duan's semantic proof, he strictly proved the above law:
0 Lines: 1 region, requires 1 color.
1 Line: Divides region into 2 adjacent regions, requires 2 colors.
2 Lines: If two lines are parallel, get 3 regions; if intersecting, get 4 regions. In either case, 3 colors are sufficient. The introduction of the second line adds At Most 1 type of color, not the superficial 2 types, because the newly appearing regional combinations did not introduce completely independent distinction dimensions, and there are two regions that can use the same color without being adjacent.
3 Lines: In general position, three mutually intersecting lines can produce up to 7 regions (in fact, 3 lines cannot produce 8 mutually adjacent regions). Yucong Duan's analysis shows that the third line adds a distinction dimension again, but this is the Last Time increasing, i.e., requiring the number of colors to increase to 4 is enough to satisfy all adjacency constraints. Even if 7 regions appear, they can be colored with 4 colors without adjacent same colors.
More than 3 Lines: Any new line after the fourth one No Longer Increases the required color types. Yucong Duan proved a "Semantic Saturation Principle": Recursively, introducing new parallel lines in the plane will not increase the required essential color number, and the distinction semantics provided by newly introduced intersecting lines can also be represented by combinations of the existing 4 colors. Intuitively, multiple lines crisscrossing form finer regions, but the adjacency patterns of these regions are similar to the previous small region patterns, which can be satisfied by Alternating Repetition of existing colors without introducing new colors. Therefore, once 4 colors are reached, no matter how complex the regional division is, a fifth color is not needed.
Summarizing the above process, we obtain a clear law: As the complexity of plane division increases, the required number of colors gradually increases from 1, but stabilizes at most at 4, and adding more divisions does not require a 5th color. This is precisely the embodiment of the Four Color Theorem from a semantic perspective. Through this calculation of Existence Semantics of Conceptual Region (EXCR) and analysis of Essential Semantics of Conceptual Region (ESCR), Yucong Duan fundamentally explained why planar map coloring does not require more than four colors. He actually constructed a semantic topological proof idea: each added boundary (map complexity increase) introduces at most one new independent "Color Distinction Semantic" dimension; the plane initially has only one color semantic dimension, gradually increasing, but the Planar Geometric Structure can only provide at most 4 independent distinction dimensions, so four colors are sufficient. This proof avoids tedious case analysis or computer verification, but gives an intuitive explanation, called "Semantic Space Explanation" of the Four Color Theorem by the author.
To visually demonstrate this semantic reasoning path, we can draw a Semantic Topology Schematic:
The starting point is a node representing "unbounded planar region," marked needing 1 color.
After adding the first line, two nodes (Region 1 and Region 2) are drawn, they are adjacent, marked with two different colors (e.g., Red, Green).
After adding the second line, four regions appear. Dashed lines are used in the diagram to connect a pair of diagonal regions to indicate they can share a color (e.g., still using Red), and the other two regions use the previous Green and a newly added Blue respectively, forming a 3-color scheme.
After adding the third line, regions are further subdivided, reaching up to seven. But by continuing to use dashed lines in the diagram to indicate color reuse of certain regions, we see that introducing a fourth color Yellow can satisfy the need, and a 4-color scheme is sufficient to cover the heterochromatic requirements of up to 7 regions.
Adding more lines, the color types in the schematic no longer increase, only the existing Red, Green, Blue, and Yellow appear alternately and repeatedly in the more complex regional grid.
Through this topology diagram, it can be intuitively seen that Semantic Distinction Dimensions reach saturation when accumulated to 4 dimensions, and how color marks are reused in the new division structure. This reflects our previous analysis exactly: the essential dimensions of distinction between regions are finite, so the required color types are finite and have an upper bound. In summary, we reconstructed the argument of the Four Color Theorem from a semantic perspective: the adjacency relationship of planar maps corresponds to limited independent distinction dimensions, and the gradual division process reflects the gradual increase of distinction dimensions and semantic compression, finally reaching completeness in the fourth dimension; therefore, no more than four colors are needed to satisfy all adjacency constraints. From this perspective, the four-color property is understood as a natural result of Semantic Compression Mapping: planar topology allows us to compress infinitely possible regional configurations into four types of color signals for marking without losing any adjacent distinction information.
Simulation Model Construction Suggestions
To further verify and apply the above semantic reconstruction model, we propose several suggestions for simulation implementation and technical approaches in this section. These suggestions aim to ground the theoretical framework into algorithms or systems, demonstrating the effectiveness of the Four Color Semantic Model on one hand, and providing ideas for future expansion to more general graph coloring and cognitive distinction problems on the other.
Develop DIKWP Semantic Graph Generator: Establish a programmatic map generation and coloring verification platform working according to the DIKWP hierarchy. Specifically, this generator can generate Map Data Layer (i.e., plane division) randomly or according to set rules, then automatically extract Adjacency Information Layer (build Regional Adjacency Tension Graph), and then form the constraint representation of the Knowledge Layer (list of all adjacent pairs). Next, the system can attempt to execute coloring in the Wisdom Layer: for example, using heuristic algorithms to assign colors to each region, with a built-in "BUG Consciousness" detection mechanism ensuring no adjacent same-color conflicts occur. Finally, output the scheme and verify correctness (all adjacent heterogeneity constraints met) and optimization degree (whether color usage is compressed to minimum) by the Purpose Layer. Such a generator is equivalent to a Four Color Problem Sandbox: we can use it to simulate the coloring process under different topological situations, verifying that 4 colors are sufficient on various maps, and observing whether the relationship between semantic tension release and new color addition conforms to theoretical expectations when approaching boundary cases (such as complex figures requiring exactly 4 colors). This platform can also help research situations outside four colors, such as changing topology (generating adjacency relationships on surfaces like tori) to check if the required color types conform to Heawood numbers. When the DIKWP generator repeatedly verifies a large number of instances, it can further refine experience, corroborate or correct semantic theoretical assumptions.
Implement "Tension-Adjustable" Intelligent Coloring Algorithm: Consider designing a coloring algorithm simulating physical processes, viewing adjacency constraints as spring-like "tensions." Initially, all regions are uncolored or randomly colored, then let the system iteratively adjust: if adjacent same colors are found (tension not released), just like a spring under pressure needs to expand, change one of the regions to a different color to relieve that tension. We can define a global "Energy Function," for example, let every pair of adjacent regions with the same color contribute a certain energy, and excessive color types also contribute a certain energy penalty. The goal of the algorithm is to decrease the total energy to zero by changing regional colors. In implementation, heuristic search or simulated annealing methods can be used to optimize color configuration. This "Coloring-Annealing" process is essentially forming a closed-loop feedback between the Wisdom Layer and the Purpose Layer, constantly reducing the number of unsatisfied constraints while trying to reduce color types. By adjusting penalty weights, we can control the algorithm to bias more towards satisfying constraints or compressing color numbers, thereby observing algorithm behavior, such as whether a feasible solution can be found when the color type limit is 3 (if not found, energy cannot be zero). This algorithm draws on physical intuition, analogizing semantic tension to physical tension, solving by energy minimization, which may have more intuitive explanatory power than traditional backtracking algorithms. In addition, this algorithm can also be extended to other similar problems, such as minimizing constraint conflicts (definition of conflict can vary) in graphs.
Graph Neural Network Assisted Modeling: In recent years, Graph Neural Networks (GNN) have shown potential in dealing with combinatorial optimization problems such as graph coloring. We can explore using GNN to learn patterns of four-color coloring, simulating the propagation and resolution process of semantic tension. The specific approach is to input the map adjacency graph into a GNN model, treating coloring as a multi-classification labeling problem for nodes. Unsupervised strategies can be adopted during training, such as setting the optimization goal to satisfy adjacent heterogeneity constraints and use minimum color numbers, which is similar to solving the energy minimization of the Potts model. After training with a large number of random maps, GNN might Spontaneously Learn some human-like coloring strategies, such as prioritizing assigning unique colors to highly adjacent regions, or learning to detect and avoid "conflict chains." It is worth noting that we can try to endow the GNN model with an explicit "Attention" mechanism to simulate BUG consciousness—for example, designing a message passing method that enables nodes to recognize color conflicts with neighbors and adjust accordingly. This is equivalent to embedding the idea of semantic closed-loop regulation into machine learning. Preliminary research has already used graph coloring as a case for GNN verification, proving that GNN can find near-optimal coloring schemes on a certain scale. By introducing our semantic perspective, we might improve the explainability of the network: We not only hope for the black box network to give results but also hope to understand the internal "Semantic Tension" distribution of the network and how it eliminates conflicts through iterative updates. This will help combine purely data-driven methods with semantic models to create smarter and more efficient map coloring solutions.
The core of the above suggestions lies in implementing the semantic interpretation of the Four Color Problem to verify theory and inspire new ideas. The DIKWP generator allows us to automatically deduce the semantic layering process, the tension-adjustable algorithm provides a simulated dynamic perspective, and GNN brings data-driven learning capabilities. These tools not only serve the research of the Four Color Theorem itself but can also be extended to general Graph Partitioning and Labeling Problems, such as community detection (requiring different community nodes to have different marks), frequency allocation (adjacent nodes have different frequencies), etc., all having similar constraint structures. Through integrating semantic levels and cognitive feedback, these traditional problems might find new solutions and analytical dimensions.
Philosophy and Topological Cognitive Expansion
The Four Color Theorem is not only a combinatorial topology problem but also triggers many thoughts at the levels of mathematical philosophy and cognitive science. In this section, we discuss the significance of the four-color property from a broader perspective and explore trends that may appear when expanding its semantic interpretation to other situations.
Essence of Four-Colorness: Minimum Complete Cardinality of Semantic Tension Compression? In the semantic model, we see that the adjacency tension of planar maps can ultimately be fully expressed by 4 basic color signals, and 4 is the minimal number to achieve complete distinction. This discovery can be understood as: For a plane, 4 independent symbols are sufficient to characterize arbitrarily complex adjacency relationships, any more is redundant, any less results in information loss. Philosophically, this suggests that the Four Color Theorem reflects the existence of a "Semantic Cardinality." We cannot help but ask: Why exactly 4? Do spatial dimensions and planar connectivity determine that 4 basic symbols are needed to achieve full distinction? Or from another angle, if our cognitive system can only distinguish three colors, what map distinction information would we be unable to capture? The Four Color Theorem indicates that there indeed exist some levels of planar division complexity where any 3-color scheme cannot satisfy all its adjacency distinctions (i.e., the tension dimension of these maps is 4). Therefore, 4 plays a Critical Point in planar semantic compression: below 4 cannot carry certain information, above 4 there is no new information to carry. To some extent, this is similar to the coding limit in Shannon's information theory, except that the Four Color Problem discusses the limit of information transmission under spatial topological constraints. From the perspective of human cognition, color is just a symbol, and our brain can actually distinguish any given number of categories (although presenting too many at the same time may lead to confusion, in principle, there is no upper limit on symbol types). However, the Four Color Theorem implies that the objective spatial structure itself sets an upper limit for information mapping—even if the cognizer has an infinite palette, the independent categories required by planar topology do not exceed 4. This is quite thought-provoking: it means that when we cognize a regional distribution on a plane, we only need to pay attention to at most four types of attributes, providing evidence for Cognitive Simplicity. It might be said that the plane is "four-dimensionally recognizable": we can assign a 4-bit binary code (corresponding to left/right attributes of 4 dimensions) to each region, then the codes of any two adjacent regions must differ in at least one bit. This coding method is the embodiment of the Semantic Tension Tensor. If someone seeks a purely mathematical new proof of the Four Color Theorem in the future, it is likely to be based on finding some ingenious coding such that the 5-color hypothesis leads to a contradiction. Our semantic perspective provides an inspiration, namely to find those 4 "Essential Distinction Dimensions."
Topological Complexity and Evolution of Color Cardinality: When topology extends from the plane to surfaces of higher genus, the required number of colors increases (as mentioned earlier, torus needs 7 colors, double torus needs 8 colors, according to Heawood formula). Semantically, this can be understood as: on surfaces with holes, regional adjacency relationships can be entangled more complexly, introducing more independent distinction dimensions. Taking the torus as an example, a surface of genus 1 makes certain adjacency relationship patterns possible that cannot appear on a plane, thus independent sources of semantic tension increase, requiring more colors. That is to say, the more complex the topological structure, the more "Semantic Tension Dimensions" it can accommodate, and the higher the upper limit of the corresponding color cardinality. This reflects an Evolutionary Trend from Semantic Distinctiveness to Topological Closed Tension: when spatial connectivity introduces new loops or complex connections, relying on a limited few dimensions alone can no longer describe all adjacent distinctions, and new distinction signals must be introduced to maintain cognitive complete distinction. In other words, every "hole" added topologically adds a combination possibility of distinction information, making the previously complete 4-color system incomplete and needing expansion. Cognitively, this is similar to adding a dimension to a classification task: on a 2D plane, 4 categories are enough to distinguish any regional adjacency pattern, but on a surface with a loop, a situation requiring a fifth category may arise. It is worth pondering whether this pattern also exists in other systems. For example, in abstract graphs like social networks, without the limitation of planar topology, the chromatic number can be very high (general graph coloring is NP-hard, with no fixed upper limit unless graph size is limited). This shows that constraint structure has a huge impact on information compression capability: planar topology is a strong constraint, bringing the limit of information compressibility, while unconstrained general graphs do not have a universal upper limit like four colors. Our model highlights that Constraints (Topological Constraints/Semantic Constraints) Shape Symbol Needs. In more scenarios of human cognition, we also see similar phenomena: environments with many constraints are easier to describe (e.g., the physical world has unified laws, so finite laws can describe various phenomena), while unstructured situations require a lot of information. The Four Color Theorem can be seen as the embodiment of this law in the topological dimension: the high constraint of planar structure makes 4 marks sufficient.
Perspective of Fusion of Cognition and Mathematical Proof: Finally, we reflect on the significance of the above semantic reconstruction from the perspective of mathematical philosophy. Traditionally, mathematical proofs pursue rigor and formal correctness, while our semantic proof attempt on the Four Color Problem is more about explaining "Why it holds" rather than "How to derive." The two are not opposites but complementary. Semantic proof provides cognitive insight into the theorem, making people "understand" the true connotation of the theorem; formal proof ensures the reliability of the conclusion, no matter how people understand it, the conclusion holds logically. Yucong Duan's view reveals the aspect of Subjective Initiative in mathematical discovery: we do not just passively accept objective truths but can also actively "see" truths by reconstructing conceptual relationships. After the Four Color Theorem was proved by computers, people finally believed it was true, but still longed to "see why it is true." Semantic reconstruction provides such a perspective: integrating the proof process into the track of human understanding, making proof closer to intuition and intelligent behavior. In the case of the Four Color Theorem, we feel we have touched the skeleton behind the truth—the number 4 appears because the "Distinction Dimensions" of the plane are only 4. The satisfaction brought by this realization may be another value of mathematical proof. It can greatly Inspire Inspiration: for example, we might conjecture whether there is a duality principle connecting planar four colors with solid geometry or algebraic structures? Or, can we use the semantic principle of the Four Color Theorem algorithmically to improve the efficiency of solving planar partitioning problems? These new ideas all stem from our deeper understanding of the essence of the Four Color Theorem.
In summary, through semantic structure reconstruction, we have sublimated the Four Color Theorem from a computational puzzle to a cognitive proposition, revealing the semantic compression laws hidden behind it. This not only answers part of the "why four" doubt but also demonstrates the possibility of merging human cognitive modes with mathematical reasoning. The semantic proof of the Four Color Theorem may not replace the status of traditional proof in a strict sense, but it enriches our way of understanding mathematical truth. As shown in this paper, the Four Color Theorem can be seen as the inevitable result of Semantic Tension Compression Mapping—this provides a insightful explanatory path for the establishment of mathematical propositions and provides a vivid case for us to reflect on the relationship between mathematics, cognition, and computation. In the future, with the development of artificial intelligence and cognitive science, perhaps more mathematical puzzles will welcome similar semantic reconstruction, bridging the gap between rigorous proof and semantic understanding, opening up newer paths for human exploration of truth.
Four Color Theorem - Wikipedia, The Free Encyclopedia, https://zh.wikipedia.org/zh-hans/%E5%9B%9B%E8%89%B2%E5%AE%9A%E7%90%86
(PDF) Semantic Mathematics Argumentation: Goldbach Conjecture, Collatz Conjecture and Four Color Theorem, https://www.researchgate.net/publication/389204149_yuyishuxuelunzhenggedebahecaixiangCollatzcaixiangyusisedingli
(PDF) Deconstruction Analysis of AI "Thinking Graph" Reports under the Perspective of Artificial Consciousness Theory, https://www.researchgate.net/publication/393649792_rengongyishililunshiyuxia_AI_siweitubaodaojiegoufenxi
Graph coloring with physics-inspired graph neural networks | Phys. Rev. Research, https://journals.aps.org/prresearch/abstract/10.1103/PhysRevResearch.4.043131
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人工意识概论：以DIKWP模型剖析智能差异，借“BUG”理论揭示意识局限
人工智能通识 2025新版 段玉聪 朱绵茂 编著 党建读物出版社
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