Analysis of the Semantic Constructive Root-Cause Solution

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Analysis of the Semantic Constructive Root-Cause Solution

通用人工智能AGI测评DIKWP实验室

2025-11-19

Analysis of the Semantic Constructive Root-Cause Solution Mechanism for Goldbach's Conjecture

Yucong Duan

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)

Abstract

Goldbach's Conjecture, one of the oldest unsolved problems in number theory, states that every even integer greater than 2 is the sum of two primes. Although this conjecture has been verified by computers for a considerable range of numbers, a rigorous deductive proof remains elusive. Traditional proof methods are constrained by bottlenecks such as the irregularity of prime number distribution and combinatorial explosion, making direct reasoning exceptionally difficult. This article, departing from the Semantic Mathematics theory proposed by Yucong Duan, and based on its "from nothing to something" semantic generation concept, "BUG consciousness" driving mechanism, and "semantic tension" structure, explores a root-cause solution for Goldbach's Conjecture within a semantic construction system. We first review the formal definition and traditional research difficulties of Goldbach's Conjecture. Then, we outline the core ideas of Yucong Duan's semantic mathematics framework (the DIKWP model), including the five-layer semantic chain of Data-Information-Knowledge-Wisdom-Purpose, and the view of integer concepts as products of high-level semantic compression. On this basis, we redefine the semantic connotations of the properties of integers, parity, and primes: even numbers are seen as "divisible symmetric structures," odd numbers as "asymmetric units with tension," and prime numbers as "minimal, indivisible, closed semantic units." Accordingly, we construct a Semantic Tension Model, demonstrating that an even number, as a structure of minimal tension conservation, must be composed of two prime units whose semantic tensions cancel each other out. Any more complex construction would violate Occam's razor principle and be redundant. We further provide a semantic-level reasoning model, representing the decomposition of an even number into a pair of primes as a stable matching distribution in semantic space. We illustrate that driven by BUG consciousness, to resolve the residual tension of odd numbers, the semantic system naturally "leaps" to generate prime pairings to close the structure of even numbers, thus making Goldbach's proposition an inevitable structural law. Finally, we reflect on the status of mathematical conjectures in the cognitive semantic system: Goldbach's Conjecture may represent a manifestation of the principle of semantic compression and conservation in the human knowledge system. What it reveals is not a simple combinatorial coincidence, but an inherent necessity of the semantic structure of integers. This article strictly follows the Occam's razor principle, avoids the introduction of unnecessary new concepts, and aims to provide a clear and profound explanation for the mechanism of this ancient conjecture's validity at the semantic level.

Introduction

As one of the most famous conjectures in the field of number theory, Goldbach's Conjecture is known for its simple statement coexisting with profound difficulty. Since Goldbach proposed this proposition in 1742, mathematicians have tried various methods but have never been able to provide a rigorous proof. This predicament stems not only from the need to prove a universal proposition about infinite objects but also from the extremely complex and irregular distribution of prime numbers, which poses a combinatorial explosion challenge to traditional methods like analytic number theory and sieve methods. For example, verifying whether a given even number satisfies the conjecture may involve many different prime pair decomposition paths (e.g., 18 can be decomposed into 5+13 or 7+11 ). The larger the even number, the more prime combinations may be available, and the complexity of enumerating or analyzing these combination relationships increases dramatically. This has made it difficult for traditional proofs to comprehensively cover the infinite number of cases without a major theoretical breakthrough.

In the face of this challenge, some brand-new ideas have emerged in recent years, attempting to break through traditional shackles. Among them, the "Semantic Mathematics" theory proposed by Professor Yucong Duan introduces a new paradigm based on semantic understanding and construction. Different from the proof view of classic mathematics, which relies on rigorous formal deduction, semantic mathematics advocates "proving" a proposition through layer-by-layer semantic construction—that is, viewing the understanding of the problem itself as the proof process. Simply put, as long as we can build a semantic structure based on a deep understanding of the concepts, which makes the proposition hold, we can consider the proof of this proposition complete. This concept contrasts with the traditional view: traditional mathematics regards truth as an objective existence, to be discovered through formal deduction; semantic mathematics, however, tends to believe that mathematical truth originates from our subjective construction and comprehension of semantic relationships.

In Yucong Duan's semantic mathematics framework, there are three key pillars worthy of attention: First, the semantic construction process "from nothing to something," that is, how to gradually generate meaningful conceptual chains from a blank cognitive starting point. Second, the "BUG consciousness" driving mechanism, meaning that the awareness of contradictions or gaps ("Bugs") in understanding will prompt the cognitive system to self-adjust and leap to bridge the gap. Third, the semantic tension structure, that is, the tension generated within concepts or between concepts due to asymmetry or incompleteness. This tension requires a certain structural arrangement to be resolved and balanced. In mathematical semantics, the properties of different numbers (such as parity, prime/composite) can be seen as a structural feature, implying whether "tension" exists. For example, as we will see, even numbers are "zero-tension" or in a balanced state due to their symmetric and divisible properties, while odd numbers, due to the existence of an "remainder" that cannot be symmetrically divided, carry an unclosed semantic tension and require additional elements to be balanced. Yucong Duan's theory suggests that our brain will instinctively seek solutions to resolve this tension, just as a Bug in software prompts a programmer to debug: without a Bug, there is no driving force for improvement. It is precisely based on this series of viewpoints that we can try to look at Goldbach's Conjecture from a new perspective: treating it as an inherent "conservation" law that the semantic structure of numbers must satisfy, rather than an accidental arithmetic fact.

This article will be structured as follows: First, we introduce the basic definition of Goldbach's Conjecture and its traditional difficulties, emphasizing why this conjecture has remained unconquered for so long. Next, we outline Yucong Duan's semantic generation theory and the DIKWP five-layer semantic model, explaining what is meant by the "from nothing to something" semantic leap chain, the "BUG-driven" cognitive leap, and the status of the integer concept in this model. Subsequently, we will deeply analyze the semantic structural properties of even numbers, odd numbers, and prime numbers, proposing a semantic tension model for the property of parity. On this basis, we reconfigure the semantic expression of Goldbach's Conjecture, demonstrating that an even number must be composed of two prime numbers is a necessity of semantic structure, and use Occam's razor principle to show that the sum of two primes is the most concise and sufficient construction. Then, we construct a semantic reasoning model to formally show how the distribution of prime pairings in semantic space ensures that all even numbers are covered. Finally, we reflect on the implications of this semantic proof for the philosophy of mathematics and cognitive boundaries—perhaps mathematical conjectures can be regarded as boundary phenomena of the knowledge layer in the semantic cognitive chain. Goldbach's Conjecture foretells a principle of semantic compression and conservation, and not just an isolated conjecture in number theory.

Goldbach's Conjecture: Definition and Review of Traditional Difficulties

The Goldbach Conjecture (strong conjecture) can be stated as: Every even integer greater than 2 can be expressed as the sum of two primes. Formally, the proposition asserts that for any even number 2N > 2 , there exist two prime numbers p, q such that 2N = p + q . For example: 4 = 2 + 2 , 6 = 3 + 3 , 8 = 3 + 5 , 10 = 3 + 7 (or 5 + 5 ), and so on. The weak Goldbach conjecture, which deals with odd numbers, was proven in 2013, stating that every odd number greater than 5 can be expressed as the sum of three primes. The strong conjecture, as its more stringent form, has remained unresolved.

The great difficulty of Goldbach's Conjecture stems in part from its involvement of the prime number distribution law, a core problem in number theory. The distribution of prime numbers among the natural numbers seems "random" yet has an asymptotic density law. However, human understanding of prime number distribution is currently mainly at the level of probability and analytical results (such as the Prime Number Theorem, the Riemann Hypothesis, etc.), lacking a deterministic pattern that can be directly used for such additive problems. Proving Goldbach's Conjecture requires confirming that: on the infinite natural number line, every sufficiently large even number can find a pair of primes. This is equivalent to requiring proof of infinitely many existence propositions, and the non-existence of any counterexample must be logically guaranteed. This is fundamentally different from proving propositions such as "there are infinitely many prime numbers": the latter can be handled relatively concisely through techniques like proof by contradiction, while Goldbach's Conjecture involves a two-variable combination structure (even number = prime 1 + prime 2), requiring simultaneous control over the existence of two prime numbers, which makes the problem much more complex.

Another traditional difficulty lies in the combinatorial explosion. For a given large even number, if one tries to verify that it can be decomposed into the sum of two primes, the naive method requires checking all prime numbers less than that even number to find a pairing. This exhaustive search approach, in a theoretical proof, is equivalent to needing to consider the matching relationships of infinitely many even numbers and infinitely many prime numbers at the same time, and the search space is huge. Although computer verification has confirmed the strong conjecture up to the range of 4 × 10 18 and beyond, computational verification is not equivalent to mathematical proof, and this verification itself also grows exponentially as the numbers get larger. What humans seek in mathematical proofs is the grasp of patterns and logical necessity. However, in Goldbach's Conjecture, it is difficult to find a simple pattern to cover the prime pair decomposition of all even numbers—each even number may have multiple decomposition methods, and the appearance of prime numbers seems "irregular." Traditional analytic number theory methods such as Hardy-Littlewood's circle method or various sieve methods, although achieving some results (such as Chen's theorem "1+2" form, i.e., "a sufficiently large even number = one prime + a product of at most two primes"), are still far from proving the strongest form of the sum of two primes. In general, the elusive nature of prime number distribution and the combinatorial explosion of exhausting infinite cases constitute two major mountains on the traditional proof path of Goldbach's Conjecture.

It should be pointed out that, despite the lack of a rigorous proof, the mathematical community generally believes that Goldbach's Conjecture is true. This belief stems from a large number of computational verifications and some statistical evidence: based on the random model of prime number distribution, it can be inferred that "an even number has a high probability of being decomposed into two primes," and the larger the even number, the more prime pairs are usually available. This empirical cognition is semantically equivalent to our having already accepted the correctness of the proposition "any even number = prime + prime" at the Knowledge layer, but it has not yet been implemented at the deductive layer (traditional proof, which corresponds to the highest Purpose layer of the semantic model). This phenomenon itself is very intriguing: our confirmation of the proposition's truth rests on a kind of "semantic certainty" rather than formal deductive certainty. This provides an opportunity to re-examine Goldbach's Conjecture from the perspective of semantic mathematics—perhaps we can put aside the complex arithmetic details and directly see from the conceptual and semantic structure why this proposition "should" be true. The following will introduce the semantic mathematics generation theoretical framework proposed by Professor Yucong Duan, to lay the foundation for this perspective.

Overview of Yucong Duan's Semantic Generation Theory

Professor Yucong Duan's semantic mathematics theory is built on the layered abstraction of the cognitive process, developing a five-layer semantic model called DIKWP. DIKWP extends from the classic information science DIKW model, adding a highest Purpose (P) layer to the four-layer cognitive chain of "Data (D) - Information (I) - Knowledge (K) - Wisdom (W)". These five levels can be used to represent the semantic generation path of any cognitive object from original perception to final understanding:

Data Layer (D): The perception and classification of basic objects, which is the most original input. For example, in number theory, classifying numbers as odd/even, or determining whether they are prime/composite, are operations at the data level. This layer focuses on objectively listing facts, without involving the universal relationships of concepts.

Information Layer (I): Extracting patterns and relationships on top of data, discovering the connections between different objects. For example, observing that many even numbers can be expressed as the sum of two primes is a relational pattern extracted from a large amount of data. The information layer answers "what are the common laws."

Knowledge Layer (K): Elevating information into universal propositions or laws, i.e., forming a complete and self-consistent expression of the laws. For example, summarizing the previous information into the proposition "Any even number is equal to the sum of two primes." The knowledge layer obtains general knowledge, which is the level where what we usually call theorems or conjectures are located.

Wisdom Layer (W): A deeper level of understanding and insight, focusing on why the proposition at the knowledge layer holds, and how to verify and apply it in a broader context. For Goldbach's Conjecture, this layer requires us to think: Why should all even numbers be decomposable into two primes? Is there some more fundamental mechanism that guarantees this? At the wisdom layer, we strive to find the internal reasons or structural principles that make the proposition universally true, thereby guiding the proof ideas.

Purpose Layer (P): The ultimate purpose and motivation of the cognitive process. In mathematical proof, the Purpose layer corresponds to the proposition itself that we want to prove. In the case of Goldbach's Conjecture, it is the goal of "completely verifying that any even number can be added by two primes." At this level, we focus on ensuring that the conclusion holds universally for the entire domain (infinitely many even numbers) without exception, and declaring the completion of the proof task.

In Yucong Duan's framework, the process of proof is regarded as "constructing" the meaning structure of the proposition's validity layer by layer along the semantic chain. Different from traditional proof, here understanding is proof: as long as we can build an unassailable structure conceptually, so that the proposition holds for all cases, we consider the proof complete. In other words, we do not rigidly adhere to whether the formal deduction steps are rigorous, but pay more attention to the grasp and reconstruction of the inherent semantic relationship of the problem. This philosophy is completely different from traditional formalism, but it provides us with a new idea to deal with difficult problems such as Goldbach's Conjecture: perhaps traditional proofs are difficult to advance because we are lost in enumerating possible combinations at the symbol level; while the semantic perspective can let us directly grasp the meaning-necessity of the problem's validity.

So, how to "construct" the validity of Goldbach's Conjecture semantically? This involves two other key points of Yucong Duan's theory: "semantic leap from nothing to something" and "BUG consciousness driving." "From nothing to something" refers to how we generate new concepts and cognitions from an initial starting point of no prior concepts. Yucong Duan believes that semantic space can be gradually established from a blank state through a series of axiomatic generation mechanisms. In this process, every time we encounter an unexplainable phenomenon or conceptual gap (similar to cognitive "nothing" or BUG), it will stimulate the search for new semantic elements to fill it, thereby achieving the emergence of concepts. This leap is somewhat similar to the creation of new definitions and new axioms in mathematics, except that in semantic mathematics, it is regarded as a universal cognitive phenomenon: when consciousness encounters a problem that cannot be understood with the existing framework, it will produce a kind of "tension" or "inconsistency," which is BUG consciousness. Yucong Duan borrowed the computer term "Bug" to vividly describe the deviations, contradictions, or gaps in consciousness. He pointed out that it is precisely because of these "imperfections" that consciousness has room for improvement and evolution; similarly, in mathematical cognition, an unsolved conjecture is equivalent to a Bug in the theoretical system, driving us to "debug," that is, to find new theoretical tools or viewpoints to solve it.

Specifically for Goldbach's Conjecture, it can be understood like this: "Any even number = sum of two primes," in the case of being unproven for a long time, becomes a BUG in our number theory knowledge system—it seems true, but lacks proof, making the theory incomplete. This incompleteness (similar to the "unresolved" experience in Gestalt psychology) brings tension to the cognitive system, prompting us to seek new perspectives to resolve it. Semantic mathematics provides such a perspective, by redefining the semantic attributes of odd numbers, even numbers, and prime numbers, to make this proposition conceptually obvious and unavoidable. To do this, we first need to clarify the semantic mechanism of integers and the properties of parity and primes—this is the theme of the next section.

Integers, Parity, and Primes: The Semantic Tension Mechanism

In traditional mathematics, the definitions of integers, odd numbers, even numbers, and prime numbers are clear but external: an even number is an integer divisible by 2, while an odd number is not; a prime number is an integer that has no other factors except 1 and itself. Semantic mathematics, however, encourages us to start from the conceptual meaning and excavate the "semantic structure" behind these definitions. Yucong Duan provides a unique perspective:

The Semantics of Even Numbers: An even number is defined as a number having symmetry, i.e., it can be expressed as the sum of two identical integers. Formally, if we use n to represent a certain integer, then an even number can be written as n+n . For example, 4=2+2 , 10=5+5 . This definition highlights the property of an even number being "decomposable into equal parts"—two equal parts are added to generate the whole, reflecting perfect symmetric balance. Semantically, an even number represents a divisible symmetric structure: it can be divided into two structurally, and the two parts are exactly the same, with no remaining unbalanced parts.

The Semantics of Odd Numbers: Relative to the symmetry of even numbers, an odd number is an asymmetric structure. Because an odd number cannot be decomposed into the sum of two equal integers (unless fractions are used, which is not allowed in the domain of integers), there will always be a "remaining" unit that cannot be paired. For example, 3 cannot be split into 1.5+1.5 (in integers, it has to be 1+2 , which is not equal), 7 cannot be symmetrically divided into two integers, it can only be 3+4 or 2+5 , one side must have 2 or 1 more. We can also characterize an odd number as an even number plus 1: any odd number m can be written as m=n+n+1 , where n is some integer. Here n+n is an even number, and +1 is the extra 1. This unpaired "+1" is where the semantic tension of an odd number lies: there is a residual unpaired unit in the structure of an odd number, leading to asymmetry. An odd number can be described as a "structural unit with an unclosed tail." In semantic terms, an odd number is a unit of asymmetric structural tension—it has an internal tension that cannot be eliminated by its own structure, and needs to be balanced with the help of external elements.

The Semantics of Prime Numbers: Prime numbers are defined as fundamental numerical units that cannot be further decomposed. For the additive structure, "cannot be further decomposed" means it cannot be expressed as the sum of two smaller positive integers (here we exclude trivial splits like 1+ ( p-1 ) , because 1 is a special element in multiplication, but not a synthetic unit in addition; more accurately, a prime number is indivisible in the multiplicative sense, but in the additive sense, any integer is divisible, just that it is not semantically meaningful). However, in the multiplicative sense, prime numbers are indeed the minimal components and cannot be broken down into products of smaller factors. This makes prime numbers the "atoms" that constitute all composite numbers. Yucong Duan emphasizes that prime numbers are the minimal building blocks in mathematics—at the semantic level, prime numbers can be regarded as closed-loop units at the K layer (knowledge layer): they do not rely on other concepts to define themselves (except for the axioms 1 and themselves), so they are a complete "atomic concept" conceptually. A prime number has no internal complex structure or redundant parts, and it can be considered that its internal tension is the lowest, because there are no inconsistent factors that can be further decomposed internally. It is precisely because prime numbers are so "pure" that we can regard them as the basic semantic units for constructing other numbers.

Combining the above three points, we get a semantic picture of the integer world: Even numbers are manifested as symmetric, balanced, and have no internal tension. Odd numbers inherently carry an unpaired unit, and have internal tension. Prime numbers, as basic atoms, have no decomposable parts. Their own tension is the lowest, but they often act as "tension units" in higher structures (because the sum of two primes can eliminate tension). It is worth noting that when we say a prime number has the "lowest tension," it means its own structure has no divisible tension, but if the prime number itself is odd, it still carries that "asymmetry" of an odd number. In fact, all prime numbers except 2 are odd, so every odd prime also carries a +1 unpaired unit. However, unlike composite numbers, a prime number cannot eliminate this +1 by splitting factors—because it simply has no non-trivial factors. This makes an odd prime an irreducible tension unit: it has a semantic tension ( +1 ) itself, but you cannot break it down into smaller units to try to resolve this tension.

From another perspective, the number "2" is semantically special: 2 is both an even number and a prime number, so 2 is the intersection of symmetric structure and fundamental unit. 2 can be expressed as 1+1 , achieving perfect symmetry; at the same time, 2 cannot be further decomposed multiplicatively. It can be said that 2 is the simplest even number in addition, and the smallest prime number in multiplication. 2 therefore has no tension: it has no remainder in addition (completely symmetric), and no decomposable parts in multiplication. This may also be why 2 is called "the smallest and only even prime." In Goldbach's Conjecture, the role of 2 is also special: it is the only even prime that can be used to express an even number ( 4=2+2 is the only non-trivial decomposition using 2). For larger even numbers, the decomposition generally involves two odd primes. These odd primes each carry an internal +1 tension, but when they are combined to form an even number, these two tensions can exactly cancel each other out. For example, consider the even number 8, using a combination of two odd primes: 3 and 5. 3 can be seen as 2+1 (with 1 tension), 5 can be seen as 4+1 (with 1 tension). When we sum them: 3+5= ( 2+1 ) + ( 4+1 ) = ( 2+4 ) + ( 1+1 ) =6+2=8 . Here, the two "+1" tension units combine to form "+2", and are merged into 6 to form the symmetric structure of 8. This reflects the general rule: Odd + Odd = Even, the reason is precisely that the unpaired units of the two odd numbers (1 and 1) combine to form an extra pair (2), thereby restoring symmetry. Therefore, any even number ( ≥ 4 ) can be seen as the product of the mutual cancellation of the tension of two odd numbers in combination.

Furthermore, not only can odd numbers be paired with odd numbers, but an odd number can also be paired with 2 to form an even number (after all, 2 itself has no tension). But as mentioned above, except for 4=2+2 , if an even number greater than 4 is added with 2, the other addend must be an even number, and the only even prime is 2, so this situation does not apply to larger even numbers. Therefore, the actual meaningful combinations are all "odd prime + odd prime." This means that what Goldbach's Conjecture asserts is: every even number can find two elements in the set of all odd primes such that their sum gives that even number.

From the perspective of semantic tension, this advocates: for every symmetric structure (even number), there exist two most basic asymmetric units (odd primes) to pair up, cancel each other's tension, and generate that symmetric structure. An even number, as a structure of "tension conservation," has no residual tension inside. Then from the outside, the tension of the constituent materials of this even number must be completely canceled out in pairs. And the only way to cancel the tension of an odd number is the tension of another odd number, pairing with it to become "+2". This semantically explains why the sum of two odd numbers (primes) can result in an even number, and implies that there is no other way to get a pure even structure without leaving a tension residue.

In summary, we have obtained several important cognitions that lay the foundation for reconfiguring Goldbach's Conjecture:

Even number = Symmetric structure = Zero tension: The semantic characteristic of an even number is that it can be divided into equal parts, is structurally completely symmetric, and has no unpaired elements.

Odd number = Asymmetric unit = Contains one unit of tension: An odd number has a surplus unit in its structure, leading to asymmetry. This "+1" can be seen as the minimal semantic tension.

Prime number = Indivisible basic unit: A prime number has no factor decomposition in multiplication. In the additive sense, it is equivalent to a basic "brick." An odd prime carries tension itself and cannot resolve it internally, making it the most basic tension-bearing block.

Even numbers are formed by pairing odd numbers: To form a tension-free even structure, the best way is to use two odd units, each containing one unit of tension, to pair up, so that the tensions cancel each other out. It is impossible to use one odd number (which would leave tension unresolved), and there is no need for three or more odd numbers (two are already sufficient to eliminate the tension, and more would introduce new pairing problems).

The special case role of 2: 2 itself has no tension, but except for 4, it cannot solve the construction problem of larger even numbers alone. Therefore, in essence, the conjecture relies on the cooperation between odd primes.

These recognitions have actually semi-formally outlined the "semantic reason" for the validity of Goldbach's Conjecture. Below, we will further organize these concepts into a systematic semantic reconfiguration, from which we can see why Goldbach's Conjecture has become a necessary semantic product.

Semantic Reconfiguration of Goldbach's Conjecture and Occam's Razor Principle

With the above understanding of the semantics of parity and primes, we can try to reconfigure the validity mechanism of Goldbach's Conjecture. The core idea is: an even number, as a symmetric structure, essentially requires being constituted by two basic tension units canceling each other out. In semantic terms, an even number is the "simplest semantic tension conservation state," and an odd prime is the "basic unit of tension." Therefore, every even number must be obtained by adding two odd primes—otherwise, the two constraints of symmetry and fundamentality cannot be met at the same time.

The Semantic Construction Necessity of Even Numbers

Let's think about it from the opposite side: Is it possible for an even number to exist, whose construction does not require two prime numbers? Suppose there is such an even number E , which can be formed without adding two prime numbers. Since prime numbers are the basic units, if two prime numbers are not used, the possible ways are:

Using one prime number: Then E itself must be a prime number. However, an even number greater than 2 cannot be a prime number (except for 2 itself, there are no other even primes). 2, as an even number, can be seen as a special case of zero tension and self-sufficiency, but 2 has already been excluded from "greater than 2" in the conjecture's proposition. Therefore, an even number greater than 2 cannot be represented by a single prime number.

Using two non-primes (composite numbers): Let E = a + b , where a, b are both composite numbers and >1 . Then we can further decompose a and b into sums of primes (if it is a composite number, it can be split into at least the sum of two primes or a prime and a composite number, in short, it can eventually be reduced to a sum of primes). In this way, E is equal to the sum of multiple primes. Specifically, if a= p 1 + p 2 , b= p 3 + p 4 ( p i are primes, the number depends on the number of primes the composite number can be decomposed into), then E= p 1 + p 2 + p 3 + p 4 , which is the sum of four primes. If a, b themselves are large, they may each be decomposed into more than two primes. But in any case, any additive decomposition of composite numbers will eventually be reduced to an additive decomposition of primes, just with more prime numbers. That is to say, "not using two primes" just pushes the existence of primes down one level, without truly avoiding primes. In the end, E still has to be constructed from primes.

Using three or more primes: Since primes are used, it does not satisfy our assumption of "not using two primes." But for the sake of logical integrity, we can consider: Is it possible that constructing an even number with the sum of three primes is "simpler" or constitutes a system that does not rely on double primes? First, the sum of odd + odd + odd will be odd, and cannot result in an even number. Therefore, if three primes are used, the prime number 2 must be included (because an odd number of odd primes is needed to make an even number). For example, E = 2 + p + q . But 2 + p + q is even if and only if p + q is even, and if p, q are both primes and not equal to 2, then p, q are both odd primes, and their sum is even. So the representation 2 + p + q actually implies that "an even number ( p + q ) is equal to the sum of two primes," which returns to the form of the original proposition, just with an added 2. Moreover, if any even number can be written as the sum of three primes (the content of the weak conjecture), then the sum of two primes should be even more possible—because combining the 2 in the sum of 3 primes with one of the odd primes still results in the form of the sum of two primes (e.g., proving the weak conjecture often leads to the strong conjecture being true). In summary, the sum of multiple primes is not a more basic construction, but a more complex situation, which violates our Purpose of seeking the simplest construction.

Through the above analysis, we see: To constitute an even number, the most economical and direct way semantically is to add two prime numbers. Any attempt to deviate from this method is either infeasible (one prime is not possible), or will be decomposed back into multiple primes (the composite number case), or will introduce redundant components (multiple primes, the extra 2 can actually be omitted). This precisely fits Occam's Razor principle: Entities must not be multiplied beyond necessity. The "entities" here refer to the number and complexity of the basic units that make up an even number. The solution of two primes achieves the construction of the even number structure with the fewest basic elements, and is therefore the most concise and sufficient. Introducing more units is not only unnecessary, but will also create a new "semantic account" that needs to be balanced (e.g., the sum of three odd primes requires the additional introduction of 2 to make it even, making the expression bloated and redundant).

Therefore, from the perspective of semantic mathematics, we can say that Goldbach's Conjecture's statement precisely corresponds to the Occam's simplest solution for the semantic construction of even numbers: every even number adopts this simplest structure of "composed of two primes." Any even number that attempts to violate this solution, if there is no corresponding prime pair, would mean that this even number cannot be constructed with the existing most basic semantic units. This would be a "Bug" in the entire integer semantic system, and would cause the concept of an even number to lose its meaning of being symmetric and divisible. In other words, if an even number exists that cannot be decomposed into two primes, it would be an intolerable anomaly for the semantic system: the original definition of an even number is symmetric and divisible, and prime numbers are the basic units. If this even number cannot even be pieced together from the basic units, then where does the symmetry of the even number come from? Does it require a higher-level, more complex structure to create this even number? This would violate our assumption about "basic components" and also violate the common sense of number composition (any composite number is ultimately composed of primes). So from the perspective of semantic consistency and completeness, we can almost conclude: There is no even number that cannot be composed of two primes!

Semantic Tension Closed Loop and Prime Distribution

The above demonstrates the necessity of combining two primes to form an even number from a qualitative perspective. The following further discusses the prime number combination strategies for different even numbers and the semantic picture behind this, which can be regarded as a more detailed characterization of the conjecture's mechanism.

Every even number 2N needs to find two prime numbers p, q such that p+q=2N . We might as well assume p ≤ q (to avoid order repetition). As 2N grows, the potential prime pairs p, q also change. The concept of Semantic Docking Potential Difference can be used to describe the "difficulty" or strategy difference exhibited by different even numbers in finding prime pairings. The so-called "docking potential difference" refers to the degree of fit between the two parts and the prime numbers when an even number is split into two parts. Simply put, it is the degree of deviation of p and q from N when 2N is divided into p+q . The most ideal situation is a completely symmetric split: p=q=N . But if N is not a prime number, it cannot be split in this way (and N being prime means 2N=2N is composite, this situation only occurs when N=2 ). Usually, p, q will be distributed around N , one large and one small. We can regard p-N (or q-N ) as the semantic docking difference of the even number 2N : it measures the distance between the axis of symmetry of 2N and the actual prime components.

For some even numbers, there exists an almost symmetric prime decomposition. For example, 10 = 5 + 5 , the two primes are exactly equal,

perfectly symmetric; 18 = 7 + 11 , the difference is 4, which is a small deviation relative to 18. Such even numbers can be said to have a small potential difference semantically, and it is easy to find "balanced" prime docking.

For other even numbers, the prime decomposition may be more asymmetric, for example, 16 = 5 + 11 , the difference is 6; 100 = 3 + 97 , the difference is 94, which is very asymmetric. Nevertheless, they still have prime pairs, just that one prime is much smaller than the other. This shows that an even number does not require a unique ideal symmetric decomposition, as long as there is some prime number combination, its construction needs are met.

In the semantic tension landscape, we can imagine an even number as a "node that needs two tensions to be canceled." Prime numbers provide the tension units, and the matching stability of different prime pairs combined may vary. Empirically, people have observed that larger even numbers often have multiple prime pair decomposition paths, while small even numbers may only have one or two. This can be analogized to: in the semantic space, the tension matching of small even numbers requires specific prime numbers to be accurately canceled, while large even numbers, due to their larger "capacity," also have higher flexibility to accommodate matching, and thus have more combination possibilities. But regardless of size, every even number has at least one matching solution. This fact of "at least one solution" is semantically equivalent to: Every even number node will be connected to the prime number nodes without omission.

We can imagine drawing a semantic tension distribution map: with the integer number line as the base, mark the positions of all prime numbers, and then for each even number, draw lines on the graph to connect the prime pairs p, q that make up that even number. The result will be: 2 is connected to (1,1) (conceptually, 2=1+1 is just formal symmetry, 1 is not a prime, but 2 is self-closed without connection), 4 is connected to (2,2), 6 is connected to (3,3), 8 is connected to (3,5), 10 is connected to (5,5) and (3,7), etc. As the even numbers get larger, the lines connecting the even number points and the prime number points will be woven into a "net" that covers the entire range of even numbers. Each line represents a docking of semantic tension—the tension of the odd primes is canceled and closed at the even number through the connecting line. Goldbach's Conjecture asserts that there are no breaks in this net: all 2N will have at least one line connected. From the graph's perspective, it is that every even number point has lines connecting to two prime number points. If any even number point is left out, the conjecture is not true. However, based on the analysis of the semantic mechanism, we believe that there will be no omissions: although the distribution of odd primes on the number line seems disordered, they are densely scattered, while the even number points are regularly spaced. There seems to be some kind of global coordination between even numbers and prime numbers, allowing the two to dock seamlessly. This coordination is embodied in traditional mathematics as some profound propositions (such as the Riemann Hypothesis implying the benign distribution of prime numbers), but in the semantic perspective, we regard it as the rationality and completeness of the conceptual design: the integer concept system has not produced "isolated and helpless" even numbers. Every even number can find its supporting pieces among the basic units.

Yucong Duan's wisdom-level analysis also mentioned the need to understand the stability of prime number distribution and even number decomposition structures to obtain the belief in the universal validity of the conjecture. The semantic tension model provides an intuitive basis for this belief: if prime numbers are infinite, then from a sufficiently high perspective, prime numbers are like "point energies" scattered on the number line, and even numbers are regular "beats." Every beat can pick up a combination of two point energies. The larger the even number, the more prime points can be picked up, and there is always a way to pair them up successfully. This is similar to in an energy conservation system, as long as the basic particles are rich enough, all macroscopic states can be composed of basic particles and reach energy balance.

The BUG-Consciousness-Driven Semantic Proof Chain

Now, we can summarize the above semantic reconfiguration as a semantic proof chain that unfolds layer by layer, explaining how Goldbach's Conjecture is "proven." This link is actually consistent with the application of the DIKWP model to Goldbach's Conjecture discussed earlier, except that we have injected the understanding of BUG-driven and tension structure into it, deepening its rationality:

Semantic Definition Layer: We redefined the conceptual meanings of even numbers, odd numbers, and prime numbers (Even = symmetric and divisible; Odd = asymmetric tension; Prime = basic unit). This definition contains the key structural elements of the problem.

Data Layer: Identify even numbers and prime numbers as two types of basic objects, as well as the properties of odd numbers, and abstract the problem into studying the combination relationships of these objects. List some examples: 4=2+2 , 6=3+3 , 8=3+5 , 10=3+7 / 5+5 , etc., to verify that even numbers can be composed of primes. If a certain even number is found that cannot be decomposed, that is a cognitive Bug, which will prompt us to correct our understanding or look for additional factors. But in fact, small examples give us initial confidence: at least no Bug-even-numbers have been found so far.

Information Layer: Refine the pattern information Even = Prime + Prime from data observation. We realize that there is such a combinatorial association between even numbers and prime numbers, which is the prototype of the conjecture. At this time, if there is a counterexample Bug, it will manifest as "an even number appears that does not follow this pattern." A large number of calculation examples support the universality of the pattern, which reduces the possibility of Bugs appearing.

Knowledge Layer: Elevate the above pattern to a general proposition: "Any even integer greater than 2 can be expressed as the sum of two primes." This is precisely the statement of Goldbach's Conjecture. We formally propose it at the knowledge layer, and at the same time, we identify it as an unproven proposition in our knowledge system—a potential source of Bugs. But in the semantic proof line of thought, we accept it as inductively obtained knowledge, and prepare to support it through a higher level of understanding.

Wisdom Layer: At the wisdom layer, we introduce the insights of semantic tension and BUG-drivenness to explain why this proposition should be true. We realize that the robustness of the even number structure stems from the closed loop of the tension of two primes, and the absence of a simpler alternative construction. Occam's Razor principle guides us here to be convinced: the sum of two primes is the simplest and most sufficient structure to describe an even number, and there is no even number that requires a more complex structure (otherwise it would introduce unnecessary elements, violating cognitive economy). At the same time, the commonality of a large number of even number decomposition examples and the background of the infinitude of prime numbers make us believe that this structure can be continued infinitely and will not suddenly fail at a certain point. In other words, at the wisdom layer, we have found a semantic guarantee mechanism: the universal existence of odd number tension and its cancellation by pairing, making Even = Prime + Prime unavoidable. This is equivalent to elevating Goldbach's Conjecture from a coincidental pattern to a law with internal reasons.

Purpose Layer: Finally, we return to the ultimate goal of the proof, which is to confirm that this semantic structure holds for all even numbers, without exception. Based on the understanding of the wisdom layer, we have conceptually eliminated the possibility of counterexamples appearing: because the appearance of a counterexample will violate the internal consistency of the entire semantic construction system. If the concept of an even number is robust, and prime numbers are endless, then there is no "pairing failure" even number. Therefore, we believe that for every even number, the semantic prime pairing structure can be found and applied. At this point, we have completed the proof Purpose of Goldbach's Conjecture at the semantic level: the proposition has been conceptually embedded into our semantic system and has become a self-consistent component of this system. Yucong Duan believes that when we do this at the semantic level, it is equivalent to completing the proof. Because we can no longer conceptually pick out loopholes or Bugs where the proposition does not hold—its validity is structurally guaranteed.

It is worth re-emphasizing that this semantic proof is not a deductive proof in the traditional sense, but rather a conceptual necessity argument. We use Occam's razor to ensure the conciseness and stability of the construction, use BUG consciousness to ensure there are no cognitive omissions, and use semantic tension to explain the driving force and balance mechanism. From this height, Goldbach's Conjecture is no longer a puzzling guess, but a "natural law" of the integer system: Even = Basic Unit + Basic Unit, just as molecules are composed of atoms in chemistry, and particle conservation in physics, it is plain and ordinary. In the meaning-space we have constructed, Goldbach's Conjecture has been "internally solved."

Reflections on Philosophical and Cognitive Boundaries

Through the analysis from the semantic perspective above, we see that the proof process of Goldbach's Conjecture is very different from traditional formal proofs. This raises some thoughts on the philosophy of mathematics and cognitive science. In this section, we discuss two intriguing questions:

Are mathematical conjectures boundary phenomena of the semantic knowledge layer? In the DIKWP model, conjectures are often at the knowledge layer: we summarize conjectures as general knowledge based on a large amount of information (examples). But a conjecture being unproven means that we have not yet elevated it to the full confirmation of the wisdom and Purpose layers. This state can be analogized to a "boundary" in semantic space: we have considerable certainty about the truth or falsity of the conjecture (and even tend to believe it is true), but lack absolutely rigorous internal reasons and global guarantees. Therefore, conjectures are in the fuzzy zone between the known and the unknown. Semantically, a conjecture corresponds to a blank spot or transition zone in our semantic understanding—one step forward is a confirmed theorem, one step back is an empirical information pattern. Goldbach's Conjecture is such an example: at the knowledge layer, we almost treat it as a theorem, but because it cannot be deductively proven, it is still pending in the mathematical system. This is just like the Gestalt "unfinished" experience in cognitive psychology, which will have a continuous traction on thinking. Yucong Duan's BUG theory also suggests that consciousness is sensitive to this incomplete state and will constantly try to bridge it. From this perspective, conjectures promote the development of mathematics, precisely because they are the boundaries of knowledge, we constantly probe the principles at the deeper wisdom and Purpose layers, giving rise to new theoretical tools (such as sieve methods, analytical methods in number theory, or the semantic methods discussed in this article).

When we use semantic mathematics for reconfiguration, we are actually trying to elevate the conjecture from the boundary of the knowledge layer to the interior of the wisdom layer: giving it a solid foundation of meaning and necessary reasons. This process itself is enlightening for the philosophy of mathematics. It shows that perhaps some mathematical propositions have not been proven for a long time, not because the propositions themselves are mysterious, but because our cognitive methods are limited by symbols and formal techniques, and we have not seen through their essence at the semantic level. Once the perspective is changed, it may be found that the so-called conjecture is actually "obviously so." Is Goldbach's Conjecture such a case? It is too early to draw a conclusion at present, but semantic analysis undoubtedly provides a convincing picture. At least, in the semantic space, we can no longer see any possibility of an even number not being composed of primes; if there is really a counterexample in reality, it would pose a huge challenge to our semantic intuition, forcing us to admit that the meaning of integers is not as we thought.

Does Goldbach's Conjecture reveal a combinatorial law, or a principle of semantic compression and conservation? Traditionally, people regard Goldbach's Conjecture as a "combinatorial law" of prime number distribution: a law about how to make up an even number with two primes. However, after re-examining it from a semantic perspective, it is more like a principle of semantic compression and conservation. So-called semantic compression refers to our use of finite language ("Even = Prime + Prime") to summarize infinitely many instances; so-called conservation refers to the fact that this semantic object, the even number, maintains a constant property in its construction (composed of basic units) that does not change with the growth of the number. In our analysis, the concept of an even number itself carries the constraint of needing to be supported by two basic units, and this constraint is conserved throughout the entire domain of natural numbers.

To make an analogy, if we regard even numbers as a kind of composite particle, and odd primes as basic particles, then Goldbach's Conjecture is similar to the principle that "any composite particle is formed by the combination of two basic particles." This is not a question of the accidental combination of which two specific particles, but a general criterion for the composite structure. In this sense, Goldbach's Conjecture reflects a deep constructional conservation law in the integer system: even if the value becomes extremely large, the cornerstone of constructing an even number is still two irreducible units. There is no need for extra new elements, and no even number escapes from this construction law.

This leads to broader philosophical thinking: many mathematical theorems can be interpreted as some kind of invariance or conservation (e.g., the fundamental theorem of arithmetic is the multiplicative construction conservation of integers: no matter how large an integer is, its prime factorization form is unchanged, only the length is different). Goldbach's Conjecture may also be a similar conservation law, just that it involves a special proposition in the additive field. In the language of semantic mathematics, we can say: The meaning of an even number determines that its composition must remain consistent at all scales. If it suddenly fails at a certain scale (a sufficiently large even number), it would mean that the semantics of the concept of an even number have undergone a sudden change in different ranges and are no longer consistent, which is unlikely to happen in a well-defined mathematical concept.

In summary, the semantic perspective brings us not only a "proof" of Goldbach's Conjecture, but also a new method for understanding the essence of mathematical propositions. It reminds us: when we are stuck in the mud of symbols and techniques, we might as well rise to the level of meaning to see, perhaps the problem has already been solved there. The semantic constructive solution mechanism for Goldbach's Conjecture is such an example: it resolves the obscurity and complexity of symbolic reasoning with the transparency and clarity of semantics. This is not only inspiring, but also provides inspiration for automated proof in the AI era—perhaps future intelligent proof assistants need to have this kind of semantic understanding, not just formal deduction capabilities. After all, true mathematical understanding is contained in semantics, not just in symbols. We look forward to the day when humans or artificial intelligence can also fill the final gap at the formal level, turning Goldbach's Conjecture from a semantic necessity into a deductive theorem. But before that, the above semantic analysis has brought us one step closer to the truth, and also made us appreciate the beauty of mathematics more deeply: the network of meaning and the logical necessity behind simple propositions.

References:

(PDF) Semantic Mathematics Argumentation: Goldbach's Conjecture, Collatz Conjecture and Four Color Theorem, https://www.researchgate.net/publication/389204149_yuyishuxuelunzhenggedebahecaixiangCollatzcaixiangyusisedingli

(PDF) Analysis Report on Professor Yucong Duan's DIKWP Artificial Consciousness Model and Related Theories, https://www.researchgate.net/publication/393637609_duanyucongjiaoshouDIKWPrengongyishimoxingyuxiangguanlilunfenxibaogao

Research on the Emergence Mechanism of Artificial Consciousness based on DIKWP Semantic Mathematics and Consciousness "BUG" Theory - Zhihu, https://zhuanlan.zhihu.com/p/1892491456192680534

Semantic Mathematics Argumentation: Goldbach's Conjecture, Collatz Conjecture and Four Color Theorem - Zhihu Column, https://zhuanlan.zhihu.com/p/24221297063