Semantic Residuals as Fixed Points of Iterated VAE Decoders : The Necessity of Filtration and Spectral Sequences

This paper analyzes, from semantic and philosophical perspectives, the mathematical structure of fixed points of attractors that emerge through the iterative process of the decoder in variational autoencoders (VAEs), namely the double complex structure consisting of a sheaf complex and a Čech complex. In particular, it clarifies how the tension inherent in ethical interpretations, formalized as Wittgensteinian dilemmas, is fixed as a “residue of meaning” within the latent space of a VAE. In this study, a three-dimensional probability distribution—whose random variables correspond to the three meta-ethical categories of sense-data, observation, and theory—is embedded into an ac-VAE, and attention is focused on the fixed points generated by the iterative mapping of the decoder output. These fixed points do not represent optimal solutions or truths; rather, they are interpreted as points at which operations for updating meaning cease to function effectively. To capture this phenomenon mathematically, this paper introduces a filtration on the sheaf-complex side of the Total complex composed of a sheaf complex and a Čech complex and designs a system that records the stepwise elimination of meaning and its residues by means of the spectral sequence defined by this filtration. The central claim of this paper is that the spectral sequence is not merely a computational technique, but a device that justifies the mathematical necessity of a stepwise operation that extracts topological distortions as dilemmas in the form of residues of meaning. As a result, the residue of meaning that appears as a fixed point can be interpreted as the ineffable appearing as the stopping point of computation, and it is shown that such fixed points necessarily emerge at the intersection of the iterative mappings of the VAE and the sheaf-theoretic framework of hypercohomology.