Modeling Information Through Optimal Representation: An Introduction to Phase Theory of Representations (PTR)

Classical information theories typically assume fixed representational modalities (e.g., symbols for entropy, strings for Kolmogorov complexity), limiting their applicability to complex real-world systems where information manifests dynamically across discrete, continuous, or hybrid forms. Addressing this gap, we introduce the Phase Theory of Representations (PTR), a novel operational framework centered on the principle that the optimal representation R for an object O is the one that minimizes its computable descriptive complexity K*(R(O)). PTR formalizes this concept through the phase operator Φ* = arg min K*(R(O)), which selects the most efficient encoding from a candidate set R. We postulate and demonstrate that as the internal structure of an object O(α) changes (controlled by a parameter α), the optimal representation can undergo phase transitions, shifting dominance between, for instance, continuous and discrete modalities. This paper establishes the theoretical foundations of PTR, defines practical, computable proxies K* for representational complexity (e.g., DCT-based entropy for continuous forms, LZMA compression for discrete forms), and introduces the phase index Φ* = K*_disc / K*_cont (or its logarithmic variant Ψ*) to detect these transitions. We empirically validate PTR using controlled synthetic experiments on 1D signals (sine wave mixed with noise) and 2D images (gradient mixed with noise). By calculating K*_cont and K*_disc across varying noise levels (α), we clearly observe a representational phase transition point αc where the ratio Φ* crosses unity (or equivalently, ΔK = K*_disc - K*_cont crosses zero), signifying a shift in the most efficient representational form. This confirms the core prediction of PTR. PTR provides a universal, computable framework for analyzing data, signals, and potentially cognition, by evaluating the efficiency of expressing structure under different representational regimes. It reframes information processing as an adaptive, phase-driven process where data effectively "chooses" its optimal form based on complexity minimization. We conclude with a discussion of theoretical implications, practical applications in adaptive coding and AI, and future research directions.