This paper establishes a direct analogue of the classical Coding Theorem in the setting of symmetry groups. We consider computable bijections on the set of binary strings, called symmetries and define the symmetry prior of a string as the probability that a randomly chosen symmetry from a given group has the string as its unique fixed point. We show that for any fix-retractable symmetry group, a group admitting a computable section that selects an isolating symmetry for every string, the symmetry prior is a universal lower semi-computable semi-measure. In this case, the Geometric Coding Theorem holds. We also develop a Galois connection between subgroups of G and subsets of binary strings, characterizing closed points and maximal closed subgroups, and explore the join-semilattice of dense subgroups. Our results unify algorithmic information theory with group theory and provide a framework for studying symmetry-induced complexity measures. This paper is the first in a series on Computational Algorithmic Statistics (CAS).
Wireless physical neural networks (WPNNs) embed neural computation directly into analog hardware, offering lower energy consumption and latency than conventional digital implementations. In this paper, we propose a deep WPNN in which nonlinear activations are realized by a multi-…
Linear attention promises constant-time recurrent inference but degrades sharply on associative recall. We formulate attention recall as a spherical-packing problem and introduce Kernelized Linear Attention Activations (KATA), a framework whose feature maps are derived from first…
Large language model (LLM)-driven evolutionary search is an emerging algorithm-discovery paradigm that has already produced novel results in several scientific fields. Yet its application to wireless communications remains largely unexplored. To bridge this gap, we introduce The…
Topos causal models recast causal inference inside a topos: a causal world is a presheaf, an intervention is a characteristic map into the subobject classifier, and reasoning is carried out in the intuitionistic internal language. We give the first machine-checked account of this…
Motivated by confidence-guided parallel unmasking in masked discrete diffusion, we study a single selection step in a stylized Gaussian random-field model. A locally dependent nonnegative score field represents position wise uncertainty, and the scheduler selects the K positions…
In multi-task learning (MTL) negative transfer is often considered as an optimization artifact, but it can also be viewed as a consequence of limited shared capacity and weak task redundancy. We investigate this effect through a Capacity--Redundancy (CR) identity that decomposes…