# What Iterated Self-Feeding Probes of Language Models Measure, and a test that separates the construction from the model

arXiv: https://arxiv.org/abs/2608.10986 · HTML: https://mlsanity.com/paper/2608.10986/

## Key facts

- Cited by 3 papers in the map (data through 14 Oct 2026).
- Placed in Test-time compute and efficient thinking, one of 17 directions in LLM reasoning.
- Builds on 10 of its 17 references that are in the map.
- First posted on arXiv 11 Aug 2026.

## Placed in

- LLM reasoning / [Test-time compute and efficient thinking](https://mlsanity.com/lr/test-time-compute-and-efficient-thinking/)
- LLM agents and tool use / [General LLM agent frameworks and benchmarks](https://mlsanity.com/agents/general-llm-agent-frameworks-and-benchmarks/)

## Builds on (most cited first, 10 of 10)

- [Self-Consistency Improves Chain of Thought Reasoning in Language Models](https://mlsanity.com/paper/2203.11171/) (2022)
- [The Hydra Effect: Emergent Self-repair in Language Model Computations](https://mlsanity.com/paper/2307.15771/) (2023)
- [Model Equality Testing: Which Model Is This API Serving?](https://mlsanity.com/paper/2410.20247/) (2024)
- [Explorations of Self-Repair in Language Models](https://mlsanity.com/paper/2402.15390/) (2024)
- [When LLMs Play the Telephone Game: Cultural Attractors as Conceptual Tools to Evaluate LLMs in Multi-turn Settings](https://mlsanity.com/paper/2407.04503/) (2024)
- [Unveiling Attractor Cycles in Large Language Models: A Dynamical Systems View of Successive Paraphrasing](https://mlsanity.com/paper/2502.15208/) (2025)
- [Phase Transitions in the Output Distribution of Large Language Models](https://mlsanity.com/paper/2405.17088/) (2024)
- [Which Model Is Actually Serving You? IRIS: Budgeted Black-Box Auditing of Model Substitution and Routing Dilution in LLM Gateways](https://mlsanity.com/paper/2607.20860/) (2026)
- [Mixing Times of Glauber Dynamics on Masked Language Models](https://mlsanity.com/paper/2605.16378/) (2026)
- [Auditing Question-Order Effects in Large Language Models with the QQ Equality: Mechanism Characterization and a Saturation Caveat](https://mlsanity.com/paper/2607.17219/) (2026)

## Built on by (most cited first, 3 of 3)

- [Prompt-Model Interaction Reaches the Fixed Points: A deterministic, task-free structural readout -- and the factorizations of it that failed](https://mlsanity.com/paper/2608.21315/) (2026)
- [What a Cross-Model Fixed-Point Census Can and Cannot Arbitrate About Repetition](https://mlsanity.com/paper/2609.29507/) (2026)
- [Where a Model Sends Its Own Repeated Token](https://mlsanity.com/paper/2609.31181/) (2026)

## Abstract (arXiv)

A growing class of methods probes a language model by feeding it its own output: self-consistency, iterated refinement, agentic loops. We ask what such a probe measures, in a construction chosen to make the question sharp: a ring of token cells resampled in place by the model's own windowed conditional p_r(x_i | x_{i+-r}). The substrate is Glauber dynamics on token sequences and is not new; what we change is the coupling. Advancing two rings that differ in one token under common random numbers makes undamaged copies diverge by exactly zero, so damage spreading becomes measurable where a maximal coupling gives mixing times instead. The answer is that it measures two different things at once, in readings that look alike. Some quantities are fixed by the construction: the damage light cone is kinematic, and the radius scaling of the token-space Lyapunov exponent lambda_ca(r) is model-invariant across 19 models and two scale ladders spanning 70x. Others genuinely track the model: lambda_ca crosses zero at a reproducible point in training, and the attractor share ranks models consistently however the lattice is built. Left undistinguished, the first kind is readily mistaken for the second -- we did so ourselves for four months, and report a phase transition we measured to three decimal places that belongs to the probe rather than to any language model. We give the test that separates them: hold the construction fixed and vary the model, or hold the model fixed and vary the construction, and see which readings move. We validate the instrument by reproduction first, recovering a Domany-Kinzel damage field bit-exactly against an independent prediction, and we report the estimator failures that this discipline caught -- four retracted verdicts, each on a quantity that looked like a measurement. The methodology ships as a package.