PFUSRC‑175D: The Illusion of Static Geometric Objects and the Mathematical Zeroing Assumption Subtitle: Dynamic Ontological Localization of the Lead‑Ball–Watermelon Homologous Mechanism, the Banach–Tarski Paradox, and the Kakeya Conjecture from the C

Classical mathematics and classical physics presuppose static three‑dimensional geometric objects as their bottom layer: spheres and lead balls are treated as stationary entities; cutting merely delineates geometric boundaries; reassembly can restore the original. This paper points out that this presupposition is built upon a silently omitted action — mathematical cutting forcibly zeroes out all flow, loss, and state change. The cutting line can be defined as a purely conceptual boundary that has no effect on the gap. This is the premise that enables mathematical abstraction to carry out cut‑and‑reassemble deductions, but this premise holds only at the symbolic level. The PFUSRC framework takes the four‑dimensional flow‑variable architecture (three‑dimensional dynamic variables + flow dimension) as its bottom‑layer ontology, in which no truly static three‑dimensional object exists. Cutting is not a three‑dimensional spatial demarcation but a four‑dimensional event: the flow dimension intervenes immediately, the cut generates a topological fissure, and phase undergoes irreversible migration. The cut‑surface changes of a watermelon, as a living body, are directly visible; a lead ball is not exempt — its cut surface undergoes local lattice tearing and topological continuity disruption, sharing the same set of flow mechanisms as the watermelon, differing only in evolution rate. Therefore, simple reassembly can only align coordinate boundaries at the projection level and cannot rebuild the already‑broken flow continuity chain. Two famous mathematical facts are thereby re‑localized: first, the “one ball becomes two” of the Banach–Tarski paradox is not three‑dimensional material replication but a fissure effect produced by the re‑projection of flow phase; three‑dimensional volume conservation is only a projection‑layer law. Second, the modern Kakeya conjecture (the Kakeya set conjecture) has completed a presuppositional switch from the dynamic needle‑rotation problem to a static set problem. The three‑dimensional proof by Wang–Zahl is strictly valid within the static Euclidean axiomatic framework; its premises (δ‑tubes, convex Wolff axioms, sticky estimates) contain no time or flow variables throughout — the so‑called 720° phase‑minus‑one snapshot belongs to PFUSRC‑internal ontological extrapolation and is not part of the mathematical proof text. Since the ontological layer contains no instantaneous cross‑section capable of isolating flow, the snapshot is only a projection abstraction; viewed from the dynamic ontological perspective, the “flow‑free perturbation motion” presupposed by the Kakeya conjecture cannot be realized at the physical‑reality level, and the conjecture’s design carries an inherent limitation. This paper concludes: the bottom‑layer ontology of the world is dynamic; static three‑dimensionality is only an observational illusion of the projection layer; using static axioms to directly deduce the ontology necessarily produces arithmetic illusions beneath projection fissures.