From the Zero-Dimensional Illusion to the Geometric Quantum: Rethinking the Hodge Conjecture and Other Problems Through the Infinitum

This paper does not offer another proof of the Hodge Conjecture within the standard axiomatic framework. Instead, it analyzes the reasons why this and other fundamental problems — the Riemann Hypothesis, P vs NP, and Fermat's Last Theorem before Wiles — have remained unsolved for decades despite the investment of enormous intellectual and material resources. The thesis advanced and justified here is that the root of the difficulty lies not in the intrinsic complexity of these problems, but in a logical contradiction embedded in the foundations of mathematics: the use of a structureless, dimensionless point as the primitive object. As an alternative, Δ‑ontology is proposed, in which the primitive element is a structural quantum — the right isosceles triangle (RIT, the Infinitum). It is shown how this system naturally dissolves the identified contradictions, and why the old problems within it either do not arise or become transparent.