House of Mirrors: Monotone Nonlinear Transformations for Modeling and Quantifying Perceptual Distortion in Data-Driven and Psychometric Systems

This work presents a unified mathematical framework for understanding how monotone nonlinear transformations reshape data and generate structural forms of distortion, even when order is preserved. We model perception and algorithmic processing as the action of a monotone mapping h(x) applied to an underlying truth variable, showing that curvature alone can alter scale, emphasis, and information content. Using synthetic data drawn from uniform, normal, and bimodal distributions, we evaluate power, root, logarithmic, and logistic transformations and quantify their effects through four complementary measures: Truth Drift for positional change, Differential Entropy Difference for information content, Confidence Distortion Index for confidence shifts, and Kullback–Leibler Divergence for structural variation. Across all experiments, power functions with large exponents and steep logistic curves produced the strongest distortions, particularly for bimodal inputs. Even moderate transformations resulted in measurable changes in entropy, confidence, and positional truth, with strong correlations among the four metrics. The findings provide a geometric interpretation of bias, demonstrating that distortion arises naturally whenever a system curves the input space—whether in human perception or algorithmic pipelines. This framework offers a principled foundation for evaluating the hidden effects of scaling, compression, and saturation, and highlights how the appearance of neutrality can conceal systematic informational shifts. The proposed framework also provides a general mathematical basis for analyzing distortion in modern data-driven and artificial intelligence systems.