The Law of Orthogonal Pruning: Polynomial Computability of Two-Stage Extensive-Intensive Resolution and Dual-Direction Construction Principle

Across ultra-large-scale manufacturing networks, smart power grids, global logistics corridors, and national resource allocation systems—all prime examples of Open Complex Giant Systems (OCGS)—computer science and Operations Research (OR) have long confronted a seemingly insurmountable mathematical barrier: Combinatorial Explosion (the NP-Hard curse). When system scale expands to 500,000 demand orders, a 20-level Bill of Materials (20-level BOM), and 2,000,000 material items, conventional Mixed-Integer Linear Programming (MILP) solvers and commercial Advanced Planning and Scheduling (APS) packages routinely suffer multi-hour runtime timeouts or memory exhaustion. Simultaneously, in engineering implementation, industry practice overwhelmingly favors a purely "Top-Down" construction sequence—erecting monthly executive dashboards (S&OP/IBP) first while leaving daily shop-floor execution to manual spreadsheets—causing top-level plans to decouple from shop-floor reality. Grounded in Value Chain Physics and first principles of complex systems, this paper formulates and proves the physical computation and engineering construction law that pierces combinatorial explosion: The Law of Orthogonal Pruning. First, we prove that conventional mathematical programming encounters factorial O(N!)O(N!) combinatorial explosion because uniform algebraic symbols obliterate the boundary between two physically opposite quantities in a controlled network: Extensive Flows (such as multi-period material kitting flows, which obey quantity additivity and inter-temporal flow conservation) and Intensive/Discrete Capacities (such as bottleneck machine time slots, which possess strict spatiotemporal exclusivity and cannot be accumulated across periods), mixing both into a single monolithic matrix for coupled search. Second, by orthogonally separating these two physical classes—Stage 1 executes network-wide parallel water-level pulling on Extensive Flows along true multi-echelon kitting bottleneck dates in linear time O(N)O(N), pruning over 95% of spurious un-kitted branching combinations; Stage 2 executes local fine-grained sorting and swapping on Intensive Capacities exclusively within the narrowed kitted feasible window in log-linear time O(Nlog⁡N)O(NlogN)—combined with a finite causal propagation depth KK bounded by the physical instability window, factorial O(N!)O(N!) explosion collapses into deterministic polynomial-time closed-loop computability T(N)=O(K⋅Nlog⁡N)∈PT(N)=O(K⋅NlogN)∈P. Third, because the effective control dimension is bounded by the observable and writable feedback dimension (dim⁡C≤dim⁡OdimC≤dimO), engineering construction must follow the Dual-Direction Construction Principle: strategic guardrails Top-Down, closed-loop capability foundation Bottom-Up. Implemented in the general-purpose Intelligent Planning & Control (IPC) Engine, this law solves 500,000 demand orders, a 20-level BOM, and 2,000,000 material items in 296 seconds on a single commodity PC. Keywords: The Law of Orthogonal Pruning; Extensive Flow and Intensive Capacity; Combinatorial Explosion; Polynomial Closed-Loop Computability; Dual-Direction Construction Principle; Bottom-Up Foundation Building; 20-level BOM; IPC Engine