What finite-receptive-field encoders cannot distinguish: invariance theorems and architectural boundaries of convolutional sequence models

One-dimensional convolutional encoders with global pooling are a standard tool for DNA, RNA and protein sequences and for text. This paper characterizes exactly which inputs such an encoder cannot tell apart, and which architectural components change that. Consider an encoder built from stride-1 convolutions of total receptive field R, with any padding, position-wise operations, and a pooling that depends only on the multiset of position features, with no positional information. For every value of its weights it gives identical outputs on any two sequences with the same multiset of R-mers and the same first (R−1)-mer, i.e. on the spectral fibres of order R. No training can change this. The bound is tight: a single convolution layer with sum pooling separates every pair whose R-spectra differ. The invariant class is larger than it looks. Attention pooling, max pooling (which is even coarser), dilation, self-attention without positional encoding (which sees only the composition), and relative position biases clipped symmetrically all stay inside it; they only change R. What escapes is absolute position (sufficient even with R = 1), strides (which expose position modulo the stride), relative biases clipped in a sign-aware way (as in Shaw et al. and T5), unclipped relative encodings (RoPE, ALiBi), and state-space layers. For state-space layers with fading memory we prove an explicit approximate-invariance bound: a layer with decay ρ behaves, to precision ε, like a local encoder of order about log(C/ε)/log(1/ρ). The theory is verified in float64 PyTorch on human proteins paired with fibre-mates, i.e. different sequences with the same k-spectrum, differing at up to 1,347 positions. Fifteen randomly initialized encoders were tested on 180 pairs (2,700 model–pair combinations) with no violation. A random convolution of window R separates exactly the pairs whose R-spectra differ, for all 17 tested values of R. This record contains the paper (PDF and LaTeX source), the experiment code, the UniProt snapshot used, all results and the figures. Companion papers: doi:10.5281/zenodo.22950095, doi:10.5281/zenodo.22959211, doi:10.5281/zenodo.22960439. Code: https://github.com/Ruqing1963/finite-receptive-field