From Dimensional Ambiguity to Controlled Rationality: Multi-Dimensional Projection and the SQAF Architecture for Auditable Inference

Across disciplines such as history, forensics, archaeology, intelligence studies, and legal inquiry, reasoning frequently proceeds amid incomplete, heterogeneous, contradictory, opposing, and structurally fragmented evidence. In such environments, expert interpretation often functions as a black-box process, in which inferential transitions remain only partially articulated and conclusions derive authority from the interpreter rather than from the transparent structure of reasoning itself. This study introduces the Structured Quantitative Assessment Framework (SQAF) as a methodological response to this limitation. Conceived not primarily as a collection of mathematical tools but as a form of cognitive infrastructure, SQAF transforms implicit interpretive reasoning into auditable logical trajectories. The framework emerged from multiple concrete historical and archaeological challenges, including the authentication of the Dinggong inscription, inquiries into the origin of the calabash in the Americas, and interpretive disputes over oracle bone inscriptions, all of which exposed the absence of an explicit structural pathway connecting evidence to conclusion.At the philosophical level, SQAF advances a central thesis: probability functions as a structural bridge between inductive uncertainty and deductive continuation. Rather than eliminating uncertainty, probabilistic reasoning converts inductive observations into operational premises of constrained credibility, enabling deductive inference under incomplete knowledge. Mathematics, in this context, serves as an instrument of historical philosophy, projecting interpretive processes into formal space while preserving both rigor and flexibility. Methodologically, SQAF organizes inference across three interconnected logical dimensions: ontological consistency, sequential feasibility, and strategic rationality. These dimensions project heterogeneous evidence into structured components prior to synthesis, preventing dimensional conflation and enabling the reconstruction of transparent reasoning trajectories. Through multi-dimensional probabilistic projection, dimensional interpretation, modeling of natural observational sequences, short-term optimal behavior matching, and robustness evaluation via Global Sensitivity Analysis (GSA), the framework establishes an operational structure of controlled rationality. Beyond its domain-specific applications, the broader implication of this work is conceptual. It suggests that many classical difficulties in probabilistic reasoning arise not from mathematical inconsistency, but from the compression of logically distinct inferential objects into a single scalar representation. By making inferential dimensions explicit, SQAF extends the applicability of structured probabilistic reasoning beyond specialized scientific domains into everyday interpretive contexts, where stability depends on the completeness of dimensional representation rather than numerical precision alone. The central contribution of this study is therefore methodological: the credibility of complex reasoning should rest not on interpretive authority alone, but on the structural integrity, transparency, and reproducibility of reasoning trajectories. SQAF thus represents an initial step toward a unified theory of controlled rationality applicable across historical, forensic, archaeological, intelligence, legal, policy, and other high-stakes interpretive disciplines.