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

Abstract: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factors -- prose-versus-markup, a universal direction, bidirectionality, instruct-resistance -- were each withdrawn within one run of being proposed, dissolved by widening the sample.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.

11 pages, 4 tables. Companion to arXiv:2608.10986. Code, per-run results, and the findings ledger: this https URL (archived: this https URL)

Computation and Language (cs.CL)

arXiv:2608.21315 [cs.CL]

(or arXiv:2608.21315v1 [cs.CL] for this version)

https://doi.org/10.48550/arXiv.2608.21315 Focus to learn more arXiv-issued DOI via DataCite (pending registration)

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