The Shortest Binary 3-Collapsing Word Has Length 38

The Shortest Binary 3-Collapsing Word Has Length 38 This preprint presents a proposed proof of the exact value c(3,2) = 38, where c(3,2) denotes the minimum length of a binary 3-collapsing word. The manuscript combines two complementary approaches. Upper bound: A candidate binary word of length 38 is presented, together with a structural argument based on the classification of proper binary 3-compressible automata developed by Cherubini, Frigeri and Liu (2017). The manuscript examines the relevant classification cases and proposes explicit arguments for the remaining exceptional cases. Lower bound: A finite collection of 269 individually 3-compressible binary automata (260 with five states and nine with six states) is used to exclude common 3-compressing words of length 37. Exact meet-in-the-middle searches were performed with complementary splits of 18+19 and 19+18. An independently constructed SAT encoding returned UNSATISFIABLE using CaDiCaL 3.0.1, and the resulting DRAT certificate was successfully checked by drat-trim. Computational reproducibility The source code, automata witnesses, SAT instance, full DRAT certificate, execution logs and SHA-256 hashes are publicly available in Zenodo: https://doi.org/10.5281/zenodo.23664594 Scientific status This manuscript is an original research preprint that has not yet undergone independent peer review. In particular, the structural upper-bound argument remains subject to external mathematical scrutiny. Publication of this preprint does not constitute independent confirmation of the proposed equality. Author and research transparency Alberto Ambroj is an independent researcher. Mathematical exploration, software development, drafting and manuscript preparation were assisted by OpenAI ChatGPT. The author assumes responsibility for the submitted research and its scientific claims.