Testability: Factor Hierarchy Theory and Scientific Methodology for Multi-Factor Complex Systems

What is the relationship among factors in any complex system driven by multiple quantifiable factors? This paper advances two core propositions. The law proposition: Factors naturally fall into two tiers—rule factors determine which set of rules currently applies, and execution factors carry out pricing under the given rules. The function of a rule factor is not reflected in its direct explanatory power but in the interaction effect through which it adjusts the exposure coefficients of execution factors. When a rule factor crosses a critical threshold, the factor weights of the entire system are systematically reset. The principle proposition: Any claim about the importance of factors in a multi-factor system must satisfy testability—only after exhaustively enumerating candidate factors, testing interaction effects, and confirming structural breaks can the claim be elevated from a hypothesis to reliable knowledge. A claim that "this factor is important" without specifying "under which regime" is an incomplete proposition. Popper's principle of falsifiability stipulates the entry criterion for scientific knowledge: a proposition must be capable of being overturned by facts. Yet the classic criticism of Popper's theory within the philosophy of science has always centered on it having "only an attitudinal principle, no operational procedure": Popper required scientists to possess a spirit of falsification but did not specify, within multi-factor empirical research, "what counts as a rigorous test." This paper proposes that science requires a second threshold—testability: a procedural standard that specifies the concrete steps of testing. Exhaustive enumeration replaces prior beliefs; interaction effects and Chow tests replace the assumption of stable coefficients; multiple verification replaces single significance. Falsifiability guards the entrance to science; testability guards the exit of scientific discovery. Together they constitute the complete chain of scientific methodology. The universality of this methodology rests not on the physical material of the systems—financial assets or microscopic particles—but solely on their epistemological property: whether the values of factors are independent of the subjective judgment of the observer. Any data satisfying this condition—generated by natural forces or automated equipment, uncontaminated by human scoring—should be subjected to the test of factor hierarchy theory and must satisfy the requirement of testability. This paper takes the Four-Test Procedure as the current optimal operationalization of the testability principle, completes a full verification across five major financial markets, and provides an exhaustive classification of all possible configurations of factor hierarchies. A companion paper (Tang, 2026h) has independently replicated the identical factor hierarchy structure in astronomy, identifying for the first time the new structural form of "proxy rule factor." The critical phenomena of phase transitions in physics, switchable catalysis in chemistry, and master regulators in biology—these core natural-science phenomena are likewise concrete manifestations of the rule–execution hierarchy structure, and the specialized tools already existing in each discipline are functionally equivalent to the Four-Test Procedure. The ultimate contribution of this paper lies in establishing a second threshold for science after Popper's falsifiability—testability—and offering a systematic methodological response to the reproducibility crisis in science.