# Internalizing Tools as Morphisms in Graded Transformers

arXiv: https://arxiv.org/abs/2511.17840 · HTML: https://mlsanity.com/paper/2511.17840/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- Builds on 4 of its 16 references that are in the map.
- First posted on arXiv 21 Nov 2025.

## Builds on (most cited first, 4 of 4)

- [Outrageously Large Neural Networks: The Sparsely-Gated Mixture-of-Experts Layer](https://mlsanity.com/paper/1701.06538/) (2017)
- [Monte Carlo Gradient Estimation in Machine Learning](https://mlsanity.com/paper/1906.10652/) (2019)
- [Graded Neural Networks](https://mlsanity.com/paper/2502.17751/) (2025)
- [Galois groups of polynomials and neurosymbolic networks](https://mlsanity.com/paper/2501.12978/) (2025)

## Built on by (most cited first, 1 of 1)

- [Hierarchical Grading in Large Language Models](https://mlsanity.com/paper/2607.22757/) (2026)

## Abstract (arXiv)

We introduce a graded formulation of internal symbolic computation for transformers. The hidden space is endowed with a grading V=bigoplusg∈ GVg, and symbolic operations are realized as typed block maps (morphisms) φhleftarrow g:Vg→ Vh that are activated selectively by a differentiable routing policy. A self-supervised graded utility functional, defined as the loss reduction induced by a candidate morphism, governs activation and yields sparse, interpretable behavior. We develop the algebraic and geometric foundations: an internal model category whose objects are homogeneous components and whose morphisms are admissible grade transitions; adjoint pairs encoding typed round trips; and information-geometric interpretations in terms of KL gain, mirror descent with Bregman divergences, and Fisher natural gradients. Methodologically, we specify a utility--aware routing mechanism and objective that remain fully end-to-end differentiable. Analytic case studies and lightweight sanity checks illustrate selective morphic activation on hybrid symbolic-linguistic tasks. The framework unifies symbolic computation, geometry, and self--supervised learning within the graded transformer formalism citesh-89,sh-95, while subsuming prior external-tool paradigms (e.g., Toolformer citetoolformer2023) as a special case via functorial internalization.