PFUSRC-155 The Topological Essence of Fractional Dimensions — Information Compression Traces of Discrete Anchor Points at the Continuous Projection Layer, and the Hierarchical Reassignment of L4-L5 Morphing Projections

Abstract

Hausdorff dimension and box-counting fractional dimension are widely applied in geometric measure theory to depict scale characteristics of fractal structures such as coastlines, snowflakes and Kakeya sets, yet academia has long failed to reach a consensus on whether fractional dimensions are intrinsic spatial attributes or merely abstract mathematical constructs. Integrating three core prior PFUSRC theoretical frameworks—rotational non-integrability theory (PFUSRC-13), critiques of time-latching cyclic illusion (PFUSRC-13B), and living calculus with reconstructed Kakeya paradigm (PFUSRC-117)—this paper establishes a complete topological ontology of fractional dimensions. Fractional dimension is neither an innate non-integer dimension of cosmic space nor a hollow mathematical fiction. It denotes the dynamic projection residue generated when discrete topological anchor points under L4 ontological morphing propagate through PNRMTS space to the continuous L5 projection layer, and are statistically approximated by continuous-scale measuring instruments at the L3 observation layer. Its non-integer numerical value acts as a scale-density index, essentially an observational snapshot of morphing variables extracted via motion-variable measuring tools. This paper uncovers two time-concealment mechanisms that trigger misinterpretations of three-dimensional space. First, increments and decrements of morphing variables counteract each other at the L5 layer under dynamic equilibrium, freezing observational readings and diluting temporal evolution. Second, a 360° rotation only resets spatial coordinates without restoring topological phase, forming a half topological cycle that conceals the passage of time; a full 720° rotation is required to simultaneously reset space and phase and complete an entire bicone respiratory cycle. The integer dimension value 3 proven for the 3D Kakeya set is a solidified projection reading at the L5 layer, while the genuine ontological carrier of morphing processes is four-dimensional, combining three spatial dimensions and one temporal dimension. Four inherent topological constants of the PFUSRC system impose rigid quantitative constraints on all physically valid fractional dimension values: the minimum planar full-coverage anchor number N_{min}=11, the global upper limit of steady-state anchor points 55, the incompressible critical measurement gap \Delta A≈0.0364, and the self-similar bicone gauge ratio \kappa=12/11. These constants define the effective value range of fractional dimensions and ban the physically meaningless mathematical ideal of infinite subdivision where \varepsilon \to 0. The paper reinterprets the Kakeya conjecture from an ontological perspective: fractional dimensions merely serve as auxiliary exclusion tools for mathematical proofs, while the core judgment criterion relies on the minimum 11 directional anchor points required for full spatial coverage. Natural fractals and artificial Kakeya fractals share identical topological generation logic rooted in discrete anchor projection residues, differing only in anchor classification. Furthermore, this paper contrasts traditional slicing calculus and fractal geometry with PFUSRC’s living calculus, pointing out their shared paradigm defect: processing discrete underlying morphing structures with continuous-limit mathematical tools, and mistakenly equating projection-layer observational readings with intrinsic ontological structures of the universe. Together with living calculus, the topological ontology of fractional dimensions completes systematic paradigm revision for mathematical analysis at the PFUSRC projection layer.