# Prompt-Model Interaction Reaches the Fixed Points: A deterministic, task-free structural readout -- and the factorizations of it that failed

arXiv: https://arxiv.org/abs/2608.21315 · HTML: https://mlsanity.com/paper/2608.21315/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- Builds on 13 of its 17 references that are in the map.
- First posted on arXiv 21 Aug 2026.

## Builds on (most cited first, 13 of 13)

- [Efficient Streaming Language Models with Attention Sinks](https://mlsanity.com/paper/2309.17453/) (2023)
- [Fantastically Ordered Prompts and Where to Find Them: Overcoming Few-Shot Prompt Order Sensitivity](https://mlsanity.com/paper/2104.08786/) (2021)
- [Quantifying Language Models' Sensitivity to Spurious Features in Prompt Design or: How I learned to start worrying about prompt formatting](https://mlsanity.com/paper/2310.11324/) (2023)
- [When Attention Sink Emerges in Language Models: An Empirical View](https://mlsanity.com/paper/2410.10781/) (2024)
- [When Benchmarks are Targets: Revealing the Sensitivity of Large Language Model Leaderboards](https://mlsanity.com/paper/2402.01781/) (2024)
- [A Theoretical Analysis of the Repetition Problem in Text Generation](https://mlsanity.com/paper/2012.14660/) (2020)
- [Mind Your Format: Towards Consistent Evaluation of In-Context Learning Improvements](https://mlsanity.com/paper/2401.06766/) (2024)
- [On the Worst Prompt Performance of Large Language Models](https://mlsanity.com/paper/2406.10248/) (2024)
- [MAPO: Boosting Large Language Model Performance with Model-Adaptive Prompt Optimization](https://mlsanity.com/paper/2407.04118/) (2024)
- [Unveiling Attractor Cycles in Large Language Models: A Dynamical Systems View of Successive Paraphrasing](https://mlsanity.com/paper/2502.15208/) (2025)
- [Repetitions are not all alike: distinct mechanisms sustain repetition in language models](https://mlsanity.com/paper/2504.01100/) (2025)
- [One prompt is not enough: Instruction Sensitivity Undermines Embedding Model Evaluation](https://mlsanity.com/paper/2605.22544/) (2026)
- [What Iterated Self-Feeding Probes of Language Models Measure, and a test that separates the construction from the model](https://mlsanity.com/paper/2608.10986/) (2026)

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

- [What a Cross-Model Fixed-Point Census Can and Cannot Arbitrate About Repetition](https://mlsanity.com/paper/2609.29507/) (2026)

## Abstract (arXiv)

That a prompt's effect is not a property of the prompt is established: prompts optimised for one model degrade on another, and rankings reorder under neutral reformatting. That evidence is about task accuracy, which cannot say whether the interaction is a fact about task machinery or about the conditional distribution itself. We ask on a readout with no task in it: the fixed-point structure of the short-window argmax map x_{t+1} = argmax_x p(x | x_{t-1}, x_t), censused from 96 starts. It is deterministic, so nothing can be helped or hurt, and it exists only at short windows -- four of six models lose it entirely by window 16 -- so everything here concerns how a model reads a fragment. Two results. First, the interaction reaches this readout at full magnitude: nine tokens of conditioning move the fixed-point fraction across most of its range, change a four-way structural class, and reorder models, while instruction tuning worth 60.5 IFEval points moves the class by zero. Second, nothing we proposed carries it. Prefix length fails: the effect is not monotone. Four phenomenological factors -- prose-versus-markup, a universal direction, bidirectionality, instruct-resistance -- were each withdrawn within one run of being proposed, dissolved by widening the sample. And the nearest mechanistic account, attention-sink dominance of early tokens, predicts the sign of the shift on 2 of 5 models -- chance -- while a length-by-content cross shows it holds on real text and fails on our probe's uniformly random input, so we are outside its regime, not against it. One fixed nine-token prefix drives four models toward 0 and two toward 1; the bidirectionality survives in-distribution starts. On this readout the unit of explanation is the prompt-model pair. The recurring error it caught in us has a name: a criterion with a shape applied to a quantity with no room to vary.