Testability: The Factor Hierarchy Law and Scientific Methodology for Multi-Factor Complex Systems—Complete Empirical Evidence from Global Asset Pricing, Multi-Dimensional Astronomical Verification, and First-Principles Calibration in Physics

Abstract

What is the relationship among factors in any complex system driven by multiple quantifiable factors? This paper advances two core propositions. The law proposition—the Factor Hierarchy Law: Factors naturally fall into two tiers. Rule factors determine which set of rules currently applies; their function is not reflected in direct explanatory power but in the interaction effect through which they adjust 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—Testability: Any claim about the importance of factors in a multi-factor system must pass a complete regime-dependence test before it can 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 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 functional dimensions that testing must cover. Falsifiability guards the entrance to science; Testability guards the exit of scientific discovery. Together they constitute the complete chain of scientific methodology. This paper formally designates the operational implementation of the Testability principle as the Testability Norms—comprising six functional dimensions: four core dimensions of search completeness, modulation relationship, structural breakpoint, and causal direction, plus two completeness dimensions of question completeness and process closed-loop. The Testability Norms are not bound to any specific tools—the functional dimensions are eternal, while the implementation tools are replaceable. 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. The Factor Hierarchy Law has received cross-disciplinary evidential support from three entirely independent disciplines. Finance: The China–U.S. interest rate spread has been repeatedly verified as a rule factor across five major markets and dozens of assets; when the spread crosses the zero axis, pricing equations undergo structural breaks. Astronomy: Five independent dimensions—including transit depth measurement and stellar atmospheric model self-consistency—all undergo structural breaks at the regime boundary (Tang Break) constituted by the K/G transition zone (approximately 4400–5800 K) and surface gravity log g = 4.64, with 4762 K being the most representative critical point. Physics: Calibration proof on the Onsager exact solution of the 2D Ising model confirms the existence of an exact information-theoretic duality between the present norms and the Ehrenfest phase transition classification, with analytic mathematical proof confirming that the detection bias is strictly zero. The contribution of this paper lies in establishing a second threshold for science after Popper's falsifiability—Testability—and, under the Testability principle, establishing the Testability Norms as an operational standard. The Factor Hierarchy Law reveals the structure; Testability prescribes the test; the Testability Norms define completeness. Research Paradigm Statement: The core methodology, research direction, and final decisions were independently directed by the author. DeepSeek assisted with code implementation, data presentation, and text drafting. The author takes full academic responsibility for the final content.