Research Article
Structural Failure Mode Analysis of the Binary Goldbach Conjecture
Ioannis Papadakis*
Issue:
Volume 11, Issue 2, April 2026
Pages:
17-32
Received:
5 February 2026
Accepted:
3 March 2026
Published:
18 March 2026
DOI:
10.11648/j.mcs.20261102.11
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Abstract: This paper analyzes the Binary Goldbach Conjecture (bGC) through a deterministic structural lens, employing a Failure Mode Analysis (FMA) framework to map prime and composite inventories onto the Left-Right Partition Table (LRPT). The FMA framework identifies the specific structural conditions—categorized into three distinct Tiers—that render the existence of a counterexample (a “Failure State”) structurally inadmissible under standard density constraints. We establish structural identities governing the conservation of partition elements, demonstrating that the count of Prime-Prime (P P) pairs functions as a necessary deterministic residual. The analysis identifies tiered inadmissible failure states where, in each Tier, the exhaustion of composite inventories mathematically forces prime-prime partitions into existence to preserve information conservation. Numerical analysis for N up to 106 shows how the tiered FMA framework quantifies the structural mechanisms through which the bGC remains valid mathematically. Furthermore, as explained in Appendix I, by leveraging the midpoint symmetry of Goldbach primes, the FMA approach yields a ”Mirror Search” mechanism for distal primes that demonstrates superior discovery efficiency compared to sequential scanning methods guided by the Prime Number Theorem. The analysis also reveals, as detailed in Appendix II, that the failure state (P P (N) = 0) implies a deterministic dependency between partition components, allowing the primality characteristic function on the interval [3, 2N − 3] to be determined by testing π(N) fewer odd integers.
Abstract: This paper analyzes the Binary Goldbach Conjecture (bGC) through a deterministic structural lens, employing a Failure Mode Analysis (FMA) framework to map prime and composite inventories onto the Left-Right Partition Table (LRPT). The FMA framework identifies the specific structural conditions—categorized into three distinct Tiers—that render the e...
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Research Article
Graph-based Representation of Arbitrary Binary Linear Codes: A Novel Algorithmic Approach
Issue:
Volume 11, Issue 2, April 2026
Pages:
33-44
Received:
21 April 2026
Accepted:
30 April 2026
Published:
12 May 2026
DOI:
10.11648/j.mcs.20261102.12
Downloads:
Views:
Abstract: This study presents a novel algorithm for transforming binary linear codes with parameters (n,k,d) into a bipartite graph representation. The proposed method explicitly represents each codeword as a node, enabling a complete structural visualization of the code. The algorithm is implemented and its computational performance is analyzed, demonstrating linear complexity with respect to code length (n) and exponential complexity with respect to dimension (k). The correctness and interpretability of the approach are illustrated using representative examples of (4,2) and (6,3) linear codes, where the resulting graphical structures reveal clear and meaningful patterns. In addition, the proposed representation is shown to be information-complete and to preserve code equivalence through graph isomorphism. The representation is particularly well-suited for integration with modern graph-based machine learning techniques, such as Graph Neural Networks, where structural information plays a central role in learning. Furthermore, the scalability characteristics of the algorithm make it applicable to a wide range of code parameters, while maintaining consistency in representation. To further assess the effectiveness of the proposed algorithm, it is compared with existing methodologies, including Trellis and Tanner graph representations, demonstrating advantages in structural analysis, effectiveness for graph-based learning, and its unique representation of the zero codeword. This framework therefore serves as a foundation for structural analysis of linear codes, facilitates equivalence testing, and is naturally suited for integration with graph-based machine learning models.
Abstract: This study presents a novel algorithm for transforming binary linear codes with parameters (n,k,d) into a bipartite graph representation. The proposed method explicitly represents each codeword as a node, enabling a complete structural visualization of the code. The algorithm is implemented and its computational performance is analyzed, demonstrati...
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