# When Can Prefixes Compile LoRA? Exact Resource-Capped Tests for Frozen Attention

arXiv: https://arxiv.org/abs/2609.36766 · HTML: https://mlsanity.com/paper/2609.36766/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 43 of its 50 references that are in the map.
- First posted on arXiv 29 Sep 2026.

## Builds on (most cited first, 30 of 43)

- [Language Models are Few-Shot Learners](https://mlsanity.com/paper/2005.14165/) (2020)
- [Retrieval-Augmented Generation for Knowledge-Intensive NLP Tasks](https://mlsanity.com/paper/2005.11401/) (2020)
- [Parameter-Efficient Transfer Learning for NLP](https://mlsanity.com/paper/1902.00751/) (2019)
- [Prefix-Tuning: Optimizing Continuous Prompts for Generation](https://mlsanity.com/paper/2101.00190/) (2021)
- [The Power of Scale for Parameter-Efficient Prompt Tuning](https://mlsanity.com/paper/2104.08691/) (2021)
- [Convex optimization](https://mlsanity.com/paper/2106.01946/) (2021)
- [Rethinking the Role of Demonstrations: What Makes In-Context Learning Work?](https://mlsanity.com/paper/2202.12837/) (2022)
- [What Makes Good In-Context Examples for GPT-3?](https://mlsanity.com/paper/2101.06804/) (2021)
- [In-context Learning and Induction Heads](https://mlsanity.com/paper/2209.11895/) (2022)
- [An Explanation of In-context Learning as Implicit Bayesian Inference](https://mlsanity.com/paper/2111.02080/) (2021)
- [Few-Shot Parameter-Efficient Fine-Tuning is Better and Cheaper than In-Context Learning](https://mlsanity.com/paper/2205.05638/) (2022)
- [DoRA: Weight-Decomposed Low-Rank Adaptation](https://mlsanity.com/paper/2402.09353/) (2024)
- [Transformers learn in-context by gradient descent](https://mlsanity.com/paper/2212.07677/) (2022)
- [What Can Transformers Learn In-Context? A Case Study of Simple Function Classes](https://mlsanity.com/paper/2208.01066/) (2022)
- [What learning algorithm is in-context learning? Investigations with linear models](https://mlsanity.com/paper/2211.15661/) (2022)
- [Parameter-Efficient Fine-Tuning for Large Models: A Comprehensive Survey](https://mlsanity.com/paper/2403.14608/) (2024)
- [In-Context Learning Creates Task Vectors](https://mlsanity.com/paper/2310.15916/) (2023)
- [Function Vectors in Large Language Models](https://mlsanity.com/paper/2310.15213/) (2023)
- [Transformers as Statisticians: Provable In-Context Learning with In-Context Algorithm Selection](https://mlsanity.com/paper/2306.04637/) (2023)
- [Transformers learn to implement preconditioned gradient descent for in-context learning](https://mlsanity.com/paper/2306.00297/) (2023)
- [Transformers as Algorithms: Generalization and Stability in In-context Learning](https://mlsanity.com/paper/2301.07067/) (2023)
- [The Lipschitz Constant of Self-Attention](https://mlsanity.com/paper/2006.04710/) (2020)
- [Scaling Down to Scale Up: A Guide to Parameter-Efficient Fine-Tuning](https://mlsanity.com/paper/2303.15647/) (2023)
- [One Step of Gradient Descent is Provably the Optimal In-Context Learner with One Layer of Linear Self-Attention](https://mlsanity.com/paper/2307.03576/) (2023)
- [What In-Context Learning "Learns" In-Context: Disentangling Task Recognition and Task Learning](https://mlsanity.com/paper/2305.09731/) (2023)
- [The Learnability of In-Context Learning](https://mlsanity.com/paper/2303.07895/) (2023)
- [The Expressive Power of Low-Rank Adaptation](https://mlsanity.com/paper/2310.17513/) (2023)
- [When Do Prompting and Prefix-Tuning Work? A Theory of Capabilities and Limitations](https://mlsanity.com/paper/2310.19698/) (2023)
- [Learning without training: The implicit dynamics of in-context learning](https://mlsanity.com/paper/2507.16003/) (2025)
- [Universality and Limitations of Prompt Tuning](https://mlsanity.com/paper/2305.18787/) (2023)

## Abstract (arXiv)

Can a fixed continuous prefix replace a given low-rank adapter while the attention head stays frozen? In this research, we show that the answer depends on the adapter's target through three conditions. First, observability: at one causal readout, every independent key--value prefix sees the content only through the query, attention partition, and value numerator, so a target that differs on two inputs with equal summaries incurs an error floor at every prefix length; norm caps extend this floor to nearly equal summaries. Second, realizability: at a common query, any prefix reduces exactly to two aggregate variables, and the norm-capped optimum is an attained second-order-cone program, also after a fixed output projection; it places two equal-norm rank-one value updates on opposite sides of compilability. Third, implementation: under affine query exposure, 2r signed slots approximate a rank-r value update, but their values grow as O(ε-3/2), and the construction passes all 400 tolerance checks in float64 yet only 38 in bfloat16. A first-layer GPT-2 readout with fixed token and position meets the common-query condition without clamping activations; at three such heads, the capped optimum leaves 18.4% to 74.2% of the projected adapter effect uncompiled, with a head-dependent value--query ordering. All claims concern local approximation at one head, not whole-network equivalence.