# Information Topology

arXiv: https://arxiv.org/abs/2210.03850 · HTML: https://mlsanity.com/paper/2210.03850/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 5 of its 52 references that are in the map.
- First posted on arXiv 7 Oct 2022.

## Builds on (most cited first, 5 of 5)

- [Geometric Deep Learning: Grids, Groups, Graphs, Geodesics, and Gauges](https://mlsanity.com/paper/2104.13478/) (2021)
- [A Dissection of Overfitting and Generalization in Continuous Reinforcement Learning](https://mlsanity.com/paper/1806.07937/) (2018)
- [The Sensory Neuron as a Transformer: Permutation-Invariant Neural Networks for Reinforcement Learning](https://mlsanity.com/paper/2109.02869/) (2021)
- [Regularizing Towards Permutation Invariance in Recurrent Models](https://mlsanity.com/paper/2010.13055/) (2020)
- [How to Compute Invariant Manifolds and their Reduced Dynamics in High-Dimensional Finite-Element Models](https://mlsanity.com/paper/2103.10264/) (2021)

## Abstract (arXiv)

We introduce Information Topology: a framework that unifies information theory and algebraic topology by treating cycle closure as the primitive operation of inference. The starting point is the dot-cycle dichotomy, which separates pointwise, order-sensitive fluctuations (dots) from order-invariant, predictive structure (cycles). Algebraically, closure is the cancellation of boundaries (partial2=0), which converts transient histories into stable invariants. Building on this, we derive the Structure-Before-Specificity (SbS) principle: stable information resides in nontrivial homology classes that persist under perturbations, while high-entropy contextual details act as scaffolds. The Context-Content Uncertainty Principle (CCUP) quantifies this balance by decomposing uncertainty into contextual spread and content precision, showing why prediction requires invariance for generalization. Measure concentration onto residual invariant manifolds explains order invariance: when mass collapses to a narrow tube around a closed cycle, reparameterizations of micro-steps leave predictive functionals unchanged. We then define homological capacity, the topological dual of Shannon capacity, as the sustainable number of independent informational cycles supported by a system. This capacity links dynamical (KS) entropy to structural (homological) capacity and refines Euler characteristics from a ``net'' summary to a ``gross'' count of persistent invariants. Finally, we illustrate the theory across three domains where more is different: visual binding, working memory, and access consciousness. Together, these results recast inference, learning, and communication as topological stabilization: the formation, closure, and persistence of informational cycles that make prediction robust and scalable.