# DALM: A Domain-Algebraic Language Model via Three-Phase Structured Generation

arXiv: https://arxiv.org/abs/2604.15593 · HTML: https://mlsanity.com/paper/2604.15593/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 17 of its 24 references that are in the map.
- First posted on arXiv 17 Apr 2026.

## Builds on (most cited first, 17 of 17)

- [Retrieval-Augmented Generation for Knowledge-Intensive NLP Tasks](https://mlsanity.com/paper/2005.11401/) (2020)
- [REALM: Retrieval-Augmented Language Model Pre-Training](https://mlsanity.com/paper/2002.08909/) (2020)
- [Estimating or Propagating Gradients Through Stochastic Neurons for Conditional Computation](https://mlsanity.com/paper/1308.3432/) (2013)
- [Atlas: Few-shot Learning with Retrieval Augmented Language Models](https://mlsanity.com/paper/2208.03299/) (2022)
- [Large Language Diffusion Models](https://mlsanity.com/paper/2502.09992/) (2025)
- [Poincar'e Embeddings for Learning Hierarchical Representations](https://mlsanity.com/paper/1705.08039/) (2017)
- [Fast-dLLM: Training-free Acceleration of Diffusion LLM by Enabling KV Cache and Parallel Decoding](https://mlsanity.com/paper/2505.22618/) (2025)
- [Lexically Constrained Decoding for Sequence Generation Using Grid Beam Search](https://mlsanity.com/paper/1704.07138/) (2017)
- [Constrained Language Models Yield Few-Shot Semantic Parsers](https://mlsanity.com/paper/2104.08768/) (2021)
- [A Phase Transition in Diffusion Models Reveals the Hierarchical Nature of Data](https://mlsanity.com/paper/2402.16991/) (2024)
- [How Deep Neural Networks Learn Compositional Data: The Random Hierarchy Model](https://mlsanity.com/paper/2307.02129/) (2023)
- [Attention Residuals](https://mlsanity.com/paper/2603.15031/) (2026)
- [Probing the Latent Hierarchical Structure of Data via Diffusion Models](https://mlsanity.com/paper/2410.13770/) (2024)
- [dLLM: Simple Diffusion Language Modeling](https://mlsanity.com/paper/2602.22661/) (2026)
- [Domain-constrained knowledge representation: A modal framework](https://mlsanity.com/paper/2604.01770/) (2026)
- [Domain-Contextualized Inference: A Computable Graph Architecture for Explicit-Domain Reasoning](https://mlsanity.com/paper/2604.04344/) (2026)
- [Reasoning as Data: Representation-Computation Unity and Its Implementation in a Domain-Algebraic Inference Engine](https://mlsanity.com/paper/2604.10908/) (2026)

## Abstract (arXiv)

Large language models compress heterogeneous knowledge into a single parameter space, allowing facts from different domains to interfere during generation. We propose DALM, a Domain-Algebraic Language Model that replaces unconstrained token generation with structured denoising over a domain lattice. DALM follows a three-phase generation path: it first resolves domain uncertainty, then relation uncertainty, and finally concept uncertainty, so each stage operates under explicit algebraic constraints. The framework requires only three ingredients: a lattice of domains with computable meet, join, and implication; a typing function over relations that controls inheritance across domains; and a fiber partition that localizes knowledge to domain-specific subsets. Given these ingredients, DALM yields a three-phase encoder-decoder architecture in which generation is confined to a domain fiber, cross-domain contamination is structurally prevented in closed-vocabulary mode and auditably bounded in open-vocabulary mode, and a single query can produce a domain-indexed multi-perspective answer space. We instantiate the framework with the CDC knowledge representation system and outline training and evaluation on validated domain-annotated crystal libraries. DALM reframes language generation as algebraically constrained structured denoising rather than unconstrained decoding over a flat token space.