Information Topology

TL;DR

Information Topology reframes inference, learning, and memory as the formation, closure, and persistence of informational cycles. By replacing pointwise dot statistics with cycle-based invariants and using cycle closure $ abla^2=0$ to enforce consistency, the framework derives the SbS principle and the CCUP tradeoff to explain how stable structure enables generalization. It introduces extit{homological capacity} as a global, topological analog of Shannon capacity and shows how entropy concentrates on invariant cycles to produce robust prediction, memory, and conscious integration. The theory is illustrated across vision, working memory, and access consciousness, arguing that more is different as higher-order homology yields emergent cognitive capabilities. Together, these results offer a topological mechanism for how stable informational cores underpin scalable inference and cognition, with broad implications for learning, perception, and consciousness research.

Abstract

We introduce \emph{Information Topology}: a framework that unifies information theory and algebraic topology by treating \emph{cycle closure} as the primitive operation of inference. The starting point is the \emph{dot-cycle dichotomy}, which separates pointwise, order-sensitive fluctuations (dots) from order-invariant, predictive structure (cycles). Algebraically, closure is the cancellation of boundaries ($\partial^2=0$), which converts transient histories into stable invariants. Building on this, we derive the \emph{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 \emph{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 \emph{order invariance}: when mass collapses to a narrow tube around a closed cycle, reparameterizations of micro-steps leave predictive functionals unchanged. We then define \emph{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 \emph{more is different}: \textbf{visual binding}, \textbf{working memory}, and \textbf{access consciousness}. Together, these results recast inference, learning, and communication as \emph{topological stabilization}: the formation, closure, and persistence of informational cycles that make prediction robust and scalable.

Figures (5)

• Figure 1: Information Topology: an end-to-end view aligned with the paper’s sections. Broken symmetry leads to residual invariance; prediction requires order invariance for generalization. Dots (local statistics) and cycles (global invariants) are linked by cycle closure ($\partial^2\!=\!0$), which stabilizes information. SbS provides persistent structure, CCUP governs alignment between context and content, and homological capacity quantifies how many independent informational cycles a system can stably support, yielding robust prediction and planning.
• Figure 2: Dot-Cycle Dichotomy.Left: A cycle that is the boundary of a filled region ($\gamma=\partial S$) is null-homologous and therefore trivial in $H_1$: it can be “canceled” as a boundary when becoming a dot. Right: A cycle encircling a hole is not the boundary of any 2-chain in the space, so it represents a nontrivial class in $H_1$. In our framework, trivial cycles correspond to high-entropy, short-lived scaffolds ($\Psi$) that collapse under boundary cancellation ($\partial^2=0$), whereas nontrivial cycles correspond to low-entropy content invariants ($\Phi$) that persist as memory.
• Figure 3: Cycle formation as the mechanism behind concentration-based invariance: (A) symmetry breaking funnels probability to a low-dimensional set; (B) mass concentrates inside a tube around a closed cycle $\gamma$; (C) closure yields order invariance for any $F$ constant on the cycle class $[\gamma]$.
• Figure 4: SbS and CCUP, and their synthesis for cycle-based memory. (A) SbS: synchronization (context) and recurrence (content) drive closure into $H_1$ invariants. (B) CCUP: product constraint between contextual spread and content precision regulates balance. (C) Synthesis: ordered closure under balanced uncertainty bootstraps higher-order invariants ($H_2,H_3,\dots$), yielding homological capacity $\mathcal{C}_H$.
• Figure 5: Homological capacity $C_H(\delta)$ in a 6-neuron ring (Vietoris-Rips filtration). Small $\delta$: fragmented graph (high $\beta_0$, no loops). Medium $\delta$: a single robust loop $\Rightarrow$$\beta_1=1$ (capacity peak). Large $\delta$: triangle filling kills the loop $\Rightarrow$$\beta_1\to 0$. The peak at $\delta^\star$ visualizes the memory bandwidth scale.

Theorems & Definitions (43)

• Definition 1: Concentration on cycles
• Lemma 1: Broken symmetry $\Rightarrow$ entropy reduction $\Rightarrow$ concentration
• Theorem 1: Maximal predictive sufficiency of cycle class under concentration
• Remark 1: Topological compression meets predictive generalization
• Theorem 2: Dimension-robust concentration on a cycle
• Proposition 1: Cycle formation as the mechanism behind concentration-based invariance
• Corollary 1: Order invariance as a consequence of concentration
• Theorem 3: Cycle Closure Implies Order Invariance
• Remark 2: Topological realization of data processing inequality
• Proposition 2: GRR-style order invariance