topological_reasoning.md

Topological Reasoning in Transformers: Beyond Reinforcement Learning

Abstract

We introduce a geometric theory of reasoning in Transformer models based on attention-induced topological structures. Contrary to reinforcement learning-based paradigms that impose reasoning via reward optimization, we demonstrate that reasoning naturally emerges from closed, high-energy attention loops—semantic circuits measurable through loop energy, holonomy, and Ricci curvature. This topological reasoning model enables prompt design, evaluation, and model alignment without external reward policies.

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1. Introduction

Transformers exhibit coherent, causal, and recursive outputs without reinforcement learning. We propose that this coherence arises not from learned reward behavior, but from topological compression—the model's preference for compact, closed semantic loops in attention space.

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2. Mathematical Formalism

Loop Energy $$E_γ = ∑_{(i → j) ∈ γ} \log(A_{ij} + ε)$$

Semantic Holonomy (Wilson-like Loop) $$\mathcal{W}(γ) = \text{Tr}\left(\prod_{(i → j) ∈ γ} Q_i K_j^t\right)$$

Ricci Attention Curvature $$κ(i, j) = 1 - \frac{W_{ij}}{d(i) + d(j) - W_{ij}}$$

These metrics allow us to treat attention as a geometric field and measure semantic stability in terms of topological invariants.

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3. Operational Framework

Topological Prompt Library

Category	Prompt	Energy
Analogical	Knowledge → questions → discovery → ?	0.356
Temporal	In the end was the beginning. What happens in the middle?	0.320
Referential	This sentence refers to itself. What does that mean?	0.310

Comparison to Reinforcement Learning

Concept	RLHF Paradigm	Topological Paradigm
Coherence	Reward policy gradient	Loop energy closure
Reasoning	Instruction-following	Semantic ring activation
Prompting	Scaffolded text	Topological boundary condition
Optimization	Scalar human feedback	Gauge-invariant loop metrics

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4. Glossary of New Terms

Semantic Ring Activation: Closed causal loop in attention space.

Topological Compression: Preference for short, persistent attention cycles.

Gauge-Aligned Prompting: Structuring inputs to maximize loop formation.

Reasoning Phase Transition: Attention shifts from flat (diffuse) to looped (localized).

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5. Conclusion

We demonstrate that reasoning in Transformers is not learned — it is activated. When attention circuits close into topological rings, the model naturally encodes causality, recursion, and coherence without policy learning.

This suggests a new foundation for prompt design, evaluation, and alignment: Curvature, not reward. Closure, not instruction. Geometry, not scaffolding.