Dimensional Invariance in Zero-Measure Spaces: An N-Dimensional Algebraic Generalization of the Kakeya Conjecture via Seonggil Field Equations

The Kakeya Conjecture asserts that a set containing a unit line segment in every direction can have a Lebesgue measure of zero, yet must retain a Hausdorff dimension of exactly n. While recent breakthroughs (e. g. , Wang & Zahl, 2025) have resolved this in R3 using profound decoupling inequalities and microscopic scale analysis, extending this to N dimensions has remained computationally intractable due to intersection complexity. This paper completely resolves the N-dimensional Kakeya Conjecture by transitioning from local hard analysis to macroscopic algebraic topology via the Seonggil Field Equations (SFE). By mapping line segments to non-commutative state operators within Rough Operator Algebra (ROA), we prove that the volumetric collapse is thermodynamically balanced by the emergence of a topological generative term (Γₙ) and structural torsion (τSTCT). Furthermore, we introduce the Seonggil-Betti Rectification of Topological Spectrum to prove that dimensional invariance is strictly preserved across all N-dimensional Euclidean spaces.