Three-Dimensional Entropic Domains for TSP Complexity via 3-SAT: Foundations of the GCT-STL Research Program on P versus NP

This paper establishes the mathematical and methodological foundations of the GCT-STL research program on the P versus NP problem, using the decision Traveling Salesman Problem (TSP_dec) as a canonical NP-complete representative and its polynomial reduction to 3-SAT with an exact materialized 3-CNF artifact. The framework organizes nine provenance-controlled descriptors (D1–D9) across three computational-information domains: semantic/geometric (S), temporal/computational (T), and logical/syntactic (L). These descriptors form the canonical raw Valiant–González tensorial signature (R0), supported by explicit SourceMap and Provenance Bundle contracts and three foundational entropy results. Paper I-A situates R0 within a branched four-representation architecture: R1 is a componentwise logarithmic scale complement, R2 is the canonical entropic/information representation, and R3 is the exact-envelope normalized representation. The Full STL Tensorial Coupling Principle remains an open and falsifiable hypothesis rather than an established obstruction theorem. The logical-complexity architecture distinguishes the internal classes P_STL and NP_STL and the conditional F1–F4 route toward the classical P versus NP problem. F1 and F2 remain open mathematical obligations. Empirical robustness, tensorial diagnostics, and candidate obstructions do not constitute universal computational lower bounds. Accordingly, Paper I-A does not prove P ≠ NP. It establishes the representational, logical, methodological, and formal-certification foundations for subsequent research within the GCT-STL program.