Information loss and quotient geometry of local sequence encoders: representation lattices, fibre entropy and diameter bounds

A sequence encoder is local of order k if it depends on a sequence only through its multiset of k-mers and its first (k−1)-mer. Amino acid and k-mer composition, pseudo amino acid composition with small lag, and mean-pooled convolutional networks are local. This theoretical paper describes exactly what such encoders lose. The kernels of local encoders of order k are coarsenings of one equivalence relation, which we show is a congruence of the word semigroup. Its classes are the spectral fibres F_k(S). We define the fibre entropy B_k(S) = log2 |F_k(S)| and prove that, for every law of a random sequence X and every encoder φ, H(X | φ(X)) ≤ E B_φ(X), with equality exactly when the law is uniform on fibres. The same quantity bounds the information-bottleneck loss I(X;Y) − I(φ(X);Y) for every target Y. That loss is fixed by the encoder before any training. For the Hamming diameter D_k(S) of a fibre we prove B_k(S)/log2(qL) ≤ D_k(S) ≤ j0 − i0 − k + 1, where q is the alphabet size, L the length, and i0, j0 are the first branching and last merging positions of the de Bruijn trail of S. The lower bound comes from the BEST determinant, the upper bound from forced arcs. A lower bound for the edit diameter is also given. We show that no upper bound on the diameter in terms of fibre size alone exists. Ordered by refinement, encoders form a complete lattice dual to the partition lattice. It is lower semimodular and not modular. The local encoders of order k form a principal ideal closed under joins, so no combination of order-k features resolves more than order k. Asymptotically, B_k(X_1^L)/L converges almost surely to the conditional entropy H(X_k | X_1^{k−1}) for stationary ergodic sources, with an explicit error of order q^{k−1} log L / L. For uniform random sequences, the expected blindness per symbol tends to log2 q when k ≤ (1−ε) log_q L, and the fibre is a singleton with high probability when k ≥ (2+ε) log_q L. The main statements were checked by exhaustive enumeration of all spectral fibres for alphabets of size 2 to 4 (158,211 cases, no failure). This record contains the paper (PDF and LaTeX source), the TikZ source and PDF of Figure 1, the verification script and its transcript. The paper uses no external data. Related papers: doi:10.5281/zenodo.22950095 and doi:10.5281/zenodo.22959211, which compute spectral fibres and spectral resolution for the human proteome. Code: https://github.com/Ruqing1963/quotient-geometry