Observer-Limited Emergent Dimensionality via Minimum Description Length: No-Go Theorems, Universal Scaling, and Pathological Spectra

We develop a complete and self-contained theory of observer-limited emergent dimensionality based on the Minimum Description Length (MDL) principle. We prove a strengthened No-Go Theorem showing that no observer-independent notion of intrinsic dimension can be compatible with MDL optimality over any nontrivial spectral class. For broad families of spectra, including power-law and logarithmically corrected decay, we derive universal scaling laws for the MDL dimension and identify a critical exponent governing its divergence with observer resolution. We establish convergence properties in the infinite-system limit, classify logarithmically pathological spectra exhibiting arbitrarily slow dimensional growth, and provide numerical validation of all scaling results on both synthetic and real-world datasets. We further compare MDL dimensionality with classical estimators and discuss connections to information geometry, renormalization group ideas, and quantum systems. Our results demonstrate that observer-limited MDL dimensionality provides a principled and practically relevant notion of effective dimension. Note: numerical results (Section 6) are illustrative computations on standard benchmark datasets. Theoretical results (Theorems 3.1, 4.1, 5.2) are self-contained and independent of numerical validation.