Abstract:
This paper introduces and formalizes Yıldırım’s Geometry, a novel axiomatic system founded on a ternary generative principle. This system gives rise to an infinite family of convex polytopes, {Yn}. We demonstrate that this family is a specific, previously uncharacterized class of zonotopes, and we derive exact recurrence relations for their f-vectors, revealing a rate of combinatorial growth significantly faster than that of hypercubes. The discrete polytopes are then elevated to a family of smooth, compact Riemannian manifolds, {M(Yn)}, through established smoothing techniques. We frame and analyze the central question of whether these manifolds admit a Ricci-flat metric. A detailed comparison with Calabi-Yau manifolds is conducted, highlighting that the {M(Yn)} are unlikely to possess a complex structure, a conclusion supported by topological obstruction theory. This positions them not as alternatives to Calabi-Yau manifolds within supersymmetric contexts, but as a potentially groundbreaking class of candidates for the geometry of compactified extra dimensions in non-supersymmetric string vacua. The existence of such Ricci-flat manifolds with generic holonomy is a long-standing open problem in mathematics, and the {M(Yn)} family presents a constructive path toward its resolution.
Yıldırım, E. (2025). A Formalization of Yıldırım’s Geometry: A Novel Family of Zonotopes and their Application to M-Theory Compactification. Zenodo. https://doi.org/10.5281/zenodo.17036121
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