1. • The Law of Spatial Boundaries of the Proper Radical: A Deterministic Proof of the abc Conjecture, Fermat's Last Theorem, and the Beal Conjecture
Abstract
The paper establishes a novel spatial-geometric foundation for diophantine analysis by introducing three distinct classes of numerical radicals: individual, proper, and embedded. Based on the laws of individual radical peaks and quadratic symmetry within the first quadrant of the coordinate plane, the author derives strict, deterministic geometric boundaries for the proper radical of any coprime triplet (a, b, c). It is demonstrated that the upper bound is explicitly constrained by the geometric capacity of the system, providing an elementary and complete proof of the abc conjecture without probabilistic assumptions. Furthermore, Fermat's Last Theorem is resolved as a direct corollary of the derived boundary law, where higher powers inevitably breach the spatial limit of the proper radical, making integer solutions impossible for coprime bases. The Beal Conjecture is resolved through the same boundary law, proving that an embedded radical is the only possible spatial stabilizer capable of maintaining such a higher-power system in integers.
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BibTeX
@misc{gasanov2026spatial,
title = {{1. • The Law of Spatial Boundaries of the Proper Radical: A Deterministic Proof of the abc Conjecture, Fermat's Last Theorem, and the Beal Conjecture}},
author = {ramzes gasanov},
howpublished = {Zenodo},
year = {2026},
month = oct,
doi = {10.5281/zenodo.23559616},
url = {https://zenodo.org/records/23559616},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - 1. • The Law of Spatial Boundaries of the Proper Radical: A Deterministic Proof of the abc Conjecture, Fermat's Last Theorem, and the Beal Conjecture
AU - ramzes gasanov
PY - 2026
DA - 2026/10/22
PB - Zenodo
DO - 10.5281/zenodo.23559616
UR - https://zenodo.org/records/23559616
AB - The paper establishes a novel spatial-geometric foundation for diophantine analysis by introducing three distinct classes of numerical radicals: individual, proper, and embedded. Based on the laws of individual radical peaks and quadratic symmetry within the first quadrant of the coordinate plane, the author derives strict, deterministic geometric boundaries for the proper radical of any coprime triplet (a, b, c). It is demonstrated that the upper bound is explicitly constrained by the geometric capacity of the system, providing an elementary and complete proof of the abc conjecture without probabilistic assumptions. Furthermore, Fermat's Last Theorem is resolved as a direct corollary of the derived boundary law, where higher powers inevitably breach the spatial limit of the proper radical, making integer solutions impossible for coprime bases. The Beal Conjecture is resolved through the same boundary law, proving that an embedded radical is the only possible spatial stabilizer capable of maintaining such a higher-power system in integers.
ER -