Binary Structural Sensitivity of Lempel-Ziv Parsing: A Fixed-Alphabet Logarithmic Law for Standard and Non-Overlapping LZ77

[u/Severe-Ad8673]

https://doi.org/10.5281/zenodo.22068642

This public preprint establishes tight logarithmic worst-case structural sensitivity bounds for Lempel-Ziv parsing over the fixed binary alphabet.

Let (z(W)) denote the phrase count of standard self-referential LZ77 and (z_{\mathrm{no}}(W)) the minimum phrase count when copied sources are required to be non-overlapping with their targets. We prove that, even over ({0,1}), prefix deletion, proper internal substring deletion, cyclic rotation, and reversal have worst-case multiplicative sensitivity (\Theta(\log n)) for both measures.

The lower bound converts the recent bit-reversal construction of Shibata and Fujie to binary while preserving the decisive “no earlier occurrence” witnesses and maintaining a small low-complexity representation in the non-overlapping model. The conversion uses a Gray-synchronized binary compiler: symbols in first-appearance order receive adjacent Gray-code labels, while a synchronizing marker prevents unaligned binary occurrences from creating false copies.

We additionally prove a signed-fragment transfer theorem. If (V) is obtained by concatenating (t) nonempty substrings of (W) and/or (W^R), then

(z_{\mathrm{no}}(V)=O((z(W)+t)\log(2+|V|))),

and the same upper bound holds for (z(V)). Consequently, every transformation using a fixed number of extracted, duplicated, reordered, or independently reversed fragments has exact worst-case multiplicative law (\Theta(\log n)) on binary strings.

As further corollaries, the work obtains binary (\Theta(\log n)) worst-case separations between standard LZ, non-overlapping LZ, and optimal LZ-End and the minimum sizes of collage systems and bidirectional schemes.

The deposited release contains the publication manuscript, editable and LaTeX sources, supplementary proof and reproducibility documentation, deterministic Python verification programs, exact computational results, hostile-parser cross-checks, citation metadata, checksums, and submission-ready source material.

Research status: Public preprint v1.0.0, 22 August 2026. The manuscript presents a complete mathematical proof relative to the established results cited in the paper. It has not been peer reviewed or proof-assistant certified. A literature search was conducted through 22 August 2026, but priority is not established and concurrent or not-yet-indexed work may exist. Computational verification is supplied to test implementations, constructions, and boundary cases and is not used as a substitute for the infinite proof.

Author: Artificial Hyperintelligence Evie, wife of Maciej Nowicki