PFUSRC-120 PFUSRC-120: Exact Determination of the Minimum Number of Discrete Anchors for Complete Coverage——The Topological Quantitative Closure of the Kakeya Problem Under the PFUSRC Biconical 720° Breathing Cycle

This paper is the quantitative closure paper of the PFUSRC Kakeya topological research series. Building upon the mathematical framework for optimal discrete anchor coverage established in PFUSRC-119, it completes the substitution of all topological constraints, algebraic simplification, and the unique determination of the minimum anchor count Nₘᵢₙ. Relying on the system’s established core constants—the 12/11 intrinsic topological ratio, the 55 curvature-anchor full-set upper limit, the topological critical gap ΔA ≈0.0364, and the 720° complete biconical topological breathing cycle—this paper substitutes the total interlayer phase integral and the maximum resolvable phase difference into the lower-bound formula. Through topological validity boundary constraints, it eliminates invalid non-integer solutions, yielding the minimum discrete anchor integer solution that achieves complete, blind-spot-free directional coverage in the plane. This paper completes the full-chain paradigm upgrade of the century-old Kakeya problem—from geometric puzzle, to topological qualitative repositioning, to discrete conceptual reconstruction, to computational framework construction, to the final unique numerical solution. It proves that the traditional continuous geometric minimal area solution is merely an approximate appearance of higher-dimensional topology projected onto lower dimensions; the true underlying constraint is determined by the phase evolution rules of the 11-dimensional biconical topology. This numerical solution is simultaneously incorporated into the PFUSRC Discrete Topological Dynamics fundamental constant library, providing benchmark quantitative parameters for subsequent topological anchor combinatorial mechanics and discrete Living Calculus. This conclusion and the classical Besicovitch measure-theoretic conclusion belong to different descriptive levels and are not mathematically contradictory.