Structural Failure Mode Analysis of the Binary Goldbach Conjecture

Ioannis Papadakis

doi:doi:10.11648/j.mcs.20261102.11

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Structural Failure Mode Analysis of the Binary Goldbach Conjecture

Ioannis Papadakis^*

Published in Mathematics and Computer Science (Volume 11, Issue 2)

Received: 5 February 2026 Accepted: 3 March 2026 Published: 18 March 2026

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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.

Published in	Mathematics and Computer Science (Volume 11, Issue 2)
DOI	10.11648/j.mcs.20261102.11
Page(s)	17-32
Creative Commons	This is an Open Access article, distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution and reproduction in any medium or format, provided the original work is properly cited.
Copyright	Copyright © The Author(s), 2026. Published by Science Publishing Group

Keywords

Arithmetic Constraints, Binary Goldbach Conjecture, Failure Mode Analysis, Factorial Spectrum, Inadmissible State, Partition Table, Prime Symmetry, Structural Identities

References

[1]	T. Oliveira e Silva, S. Herzog, and S. Pardi, “Empirical verification of the even Goldbach conjecture and computation of prime gaps up to 4 · 1018,” Math. Comp. 83 (2014), 2033–2060.
[2]	Helfgott, H. A. The Ternary Goldbach Conjecture is true. arXiv preprint arXiv: 1312.6029, 2014.
[3]	Maynard, J. Small gaps between primes. Annals of Mathematics, 181(1), 383–413, 2015. https://doi.org/10.4007/annals.2015.181.1.7
[4]	Nathanson, M. B. Additive Number Theory: The Classical Bases; Graduate Texts in Mathematics; Springer-Verlag: New York, NY, USA, 1996; Volume 164.
[5]	Selberg, A. Elementary Methods in the Theory of Primes. Norske Vid. Selsk. Forh., Trondheim, 19, 64–67, 1947.
[6]	Hardy, G. H., and Littlewood, J. E. Some problems of ’Partitio numerorum’; III: On the expression of a number as a sum of primes. Acta Mathematica, 44, 1–70, 1923.
[7]	Papadakis, I. On the Binary Goldbach Conjecture: Analysis and Alternate Formulations Using Projection, Optimization, Hybrid Factorization, Prime Symmetry and Analytic Approximation. Math. Comput. Sci. 2024, 9, 96–113.
[8]	Papadakis, I. Representation and Generation of Prime and Coprime Numbers by Using Structured Algebraic Sums. Math. Comput. Sci. 2024, 9, 57–63.
[9]	Papadakis, I. N. M. Algebraic Representation of Primes by Hybrid Factorization. Math. Comput. Sci. 2024, 9, 12–25.
[10]	Papadakis, I. N. M. On the Universal Encoding Optimality of Primes. Mathematics 2021, 9, 3155.
[11]	Hardy, G. H.; Ramanujan, S. The normal number of prime factors of a number n. Quart. J. Math. 1917, 48, 76–92.
[12]	Erdős, P.; Kac, M. The Gaussian law of errors in the theory of additive number theoretic functions. Am. J. Math. 1940, 62, 738–742.
[13]	Wolfram, S. A New Kind of Science; Wolfram Media: Champaign, IL, USA, 2002; pp. 737–750.

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APA Style

Papadakis, I. (2026). Structural Failure Mode Analysis of the Binary Goldbach Conjecture. Mathematics and Computer Science, 11(2), 17-32. https://doi.org/10.11648/j.mcs.20261102.11

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ACS Style

Papadakis, I. Structural Failure Mode Analysis of the Binary Goldbach Conjecture. Math. Comput. Sci. 2026, 11(2), 17-32. doi: 10.11648/j.mcs.20261102.11

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AMA Style

Papadakis I. Structural Failure Mode Analysis of the Binary Goldbach Conjecture. Math Comput Sci. 2026;11(2):17-32. doi: 10.11648/j.mcs.20261102.11

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@article{10.11648/j.mcs.20261102.11,
  author = {Ioannis Papadakis},
  title = {Structural Failure Mode Analysis of the Binary Goldbach Conjecture
},
  journal = {Mathematics and Computer Science},
  volume = {11},
  number = {2},
  pages = {17-32},
  doi = {10.11648/j.mcs.20261102.11},
  url = {https://doi.org/10.11648/j.mcs.20261102.11},
  eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.mcs.20261102.11},
  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.},
 year = {2026}
}

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TY  - JOUR
T1  - Structural Failure Mode Analysis of the Binary Goldbach Conjecture

AU  - Ioannis Papadakis
Y1  - 2026/03/18
PY  - 2026
N1  - https://doi.org/10.11648/j.mcs.20261102.11
DO  - 10.11648/j.mcs.20261102.11
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JF  - Mathematics and Computer Science
JO  - Mathematics and Computer Science
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PB  - Science Publishing Group
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AB  - 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.
VL  - 11
IS  - 2
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Author Information

Ioannis Papadakis

Independent Researcher, Charlotte, NC, USA

Contact Email

http://orcid.org/0009-0005-4830-6644

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APA Style

Papadakis, I. (2026). Structural Failure Mode Analysis of the Binary Goldbach Conjecture. Mathematics and Computer Science, 11(2), 17-32. https://doi.org/10.11648/j.mcs.20261102.11

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ACS Style

Papadakis, I. Structural Failure Mode Analysis of the Binary Goldbach Conjecture. Math. Comput. Sci. 2026, 11(2), 17-32. doi: 10.11648/j.mcs.20261102.11

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AMA Style

Papadakis I. Structural Failure Mode Analysis of the Binary Goldbach Conjecture. Math Comput Sci. 2026;11(2):17-32. doi: 10.11648/j.mcs.20261102.11

Copy | Download

@article{10.11648/j.mcs.20261102.11,
  author = {Ioannis Papadakis},
  title = {Structural Failure Mode Analysis of the Binary Goldbach Conjecture
},
  journal = {Mathematics and Computer Science},
  volume = {11},
  number = {2},
  pages = {17-32},
  doi = {10.11648/j.mcs.20261102.11},
  url = {https://doi.org/10.11648/j.mcs.20261102.11},
  eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.mcs.20261102.11},
  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.},
 year = {2026}
}

Copy | Download

TY  - JOUR
T1  - Structural Failure Mode Analysis of the Binary Goldbach Conjecture

AU  - Ioannis Papadakis
Y1  - 2026/03/18
PY  - 2026
N1  - https://doi.org/10.11648/j.mcs.20261102.11
DO  - 10.11648/j.mcs.20261102.11
T2  - Mathematics and Computer Science
JF  - Mathematics and Computer Science
JO  - Mathematics and Computer Science
SP  - 17
EP  - 32
PB  - Science Publishing Group
SN  - 2575-6028
UR  - https://doi.org/10.11648/j.mcs.20261102.11
AB  - 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.
VL  - 11
IS  - 2
ER  -

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