PFUSRC-119 PFUSRC-119: The Optimal Coverage Problem of Discrete Anchors——A Paradigm Shift from Continuous Area Extremum to Topological Combinatorial Counting

PFUSRC-118 established the Discrete Line Segment as an independent topological concept. PFUSRC-118B articulated the fourfold systemic transition triggered by this concept—among which the third, “Practical Upgrade,” pointed to the paradigmatic shift from continuous area extremum to discrete anchor-point counting, but did not establish a quantifiable computational framework. As the quantitative implementation paper for PFUSRC-118B, this paper transforms the Discrete Line Segment concept from cognitive framework to computational framework. The core research question is: under the constraint of the 45° biconical topological breathing cycle, what is the minimum number of discrete anchors required to achieve full directional coverage in the plane? Relying on the PFUSRC 720° complete topological breathing cycle, this paper reformulates the optimal discrete anchor coverage problem as a combinatorial counting problem of biconical interlayer phase coverage. It establishes that the theoretical lower bound of the minimum anchor count is jointly determined by the biconical interlayer phase difference and the topological critical phase gap ΔA ≈0.0364. The precise numerical solution is beyond the scope of this paper and is reserved for PFUSRC-120. This paper completes the full mathematical framework, constraint definition, formula construction, and task partitioning. This paper marks the formal entry of the PFUSRC system from the qualitative interpretive phase of topological analysis into the quantifiable computational phase of Discrete Topological Dynamics, completing the critical paradigm transition.