Sharp Endpoint-Deletion Inequalities for Binary Words with Prescribed Directed-Pair Multisets

This preprint establishes computer-assisted sharp endpoint-deletion inequalities for binary words under interior symbol swaps preserving the multiset of directed adjacent pairs.

For a binary word s of length n, let d(s) denote the number of swaps of two unequal interior symbols that leave the directed adjacent-pair multiset unchanged. Define the endpoint-deletion losses by L_L(s) = d(s) − d(s[4:]) and L_R(s) = d(s) − d(s[:-4]).

We establish the sharp bounds

L_L(s) + L_R(s) ≤ 4n − 20 for n ≥ 9,

and

min{L_L(s), L_R(s)} ≤ 2n − 10 for n ≥ 8.

Sharpness is demonstrated by prefixes of the periodic binary word (0011)∞.

The proof decomposes the swap count into a quadratic expression in the frequencies of binary triples and a local correction for overlapping swap neighborhoods. Endpoint-deletion losses are then expressed as affine functionals of triple counts. Using the binary order-two de Bruijn graph, the unbounded-length problem reduces to a finite integer certificate involving 65,536 pairs of eight-bit endpoint windows, 22 vertex-simple directed paths, and six elementary directed cycles.

The accompanying reproducibility package includes the LaTeX source, compiled PDF, a pure-Python verification program, recorded console output, SHA-256 checksums, and technical review notes. The verification uses exact integer arithmetic and requires no external Python dependencies.

This manuscript is a revised research preprint supported by computer-assisted verification. It has not undergone external peer review or formal proof-assistant verification. Claims of originality remain subject to further literature assessment.

OpenAI ChatGPT was used substantially in mathematical exploration, programming, certificate design, and manuscript drafting. The named author is responsible for reviewing the mathematics, checking the released files, verifying citations and originality claims, and deciding whether to submit

Licensing Information

The mathematical manuscript and its accompanying documentation are made available under the Creative Commons Attribution 4.0 International License (CC BY 4.0).

All original Python source code included in the RT-IDENTITY 7.1 reproducibility package is released under the MIT License.

Copyright © 2026 Ivo Vajc.

The MIT License permits the use, copying, modification, distribution, and redistribution of the Python software, including for commercial purposes, provided that the original copyright notice and license text are retained.

The software is provided "as is", without warranty of any kind. The MIT License applies to the Python source code, while CC BY 4.0 applies to the manuscript and accompanying textual documentation.