Representation Precedes Solvability: An Epistemology of Knowledge, Computation, and Natural Problems

Abstract

This paper develops an operational account of representation choice in problem solving and model-relative complexity. A well-formed mathematical problem may already have determinate semantics and solvability questions in its native formal model. The additional representation index becomes essential when one changes encodings, proof systems, computational models, or physical realizations and asks which semantic or resource conclusions transfer. We define an admissible represented problem by specifying instance semantics, uniform preparation, an Exact Semantic Carrier, dynamics, readout, and a resource model. A transfer theorem makes explicit the elementary composition principle underlying representation changes: upper bounds transfer through semantics-preserving simulations with the required resource bounds, while lower bounds require simulations in the logically appropriate reverse direction. This is an audit principle, not a new representation-invariance theorem. The framework is situated relative to structural, inferential, perspectival, and pragmatic accounts of scientific representation. A Tseitin parity example supplies the mathematical anchor: for a residual graph cut, the component-syndrome vector is exactly the full-response carrier for an explicit component-charge query family, while a single extension query induces a coarser quotient. A global scalar-energy-shift argument shows that normalized state data cannot determine an absolute spectral offset without an operational reference. This physical case illustrates a broader distinction between static semantic representation, compatible dynamics, efficient convergence, and robust referenced readout: each constitutes a separate representation obligation.