A Positional Realizability Theory of Length Generalization in Single-Head Attention

The Problem

Why does length generalization (LG) depend so sharply on the positional encoding (PE)? Relative schemes extrapolate fixed offsets, absolute schemes fix positions, and neither handles growing offsets. We give a realizability theory for a deliberately controlled abstraction, a single positional-only attention head, that derives this well-documented empirical pattern and says precisely where it holds and breaks. The device is a simulation transfer operator

$T_L$

that re-evaluates a selected length-

$L_0$

-realizing positional kernel on the longer grid, with a scalar

$α_L \in [0,1]$

read off before any training from

$(Φ, t_n)$

alone. Here

$α_L$

serves as the computable witness of that taxonomy's derivation, not as a stronger empirical predictor.

Results

Our results are a relative-head deep-row characterization for kernels with a strict unique global maximizer (Theorem 2), an absolute/relative duality that derives the pattern (Proposition 4), a learned-absolute underdetermination result, and a static expressivity bound; a sufficiency theorem (Theorem 1) bridges

<