Position-colour-preserving isomorphisms of binary rewrite graphs with unbordered difference words
Authors/Creators
Description
For any two binary keywords of the same length L ≥ 2 whose adjacent-difference words are unbordered, the paper proves that their rewrite graphs admit a position-colour-preserving isomorphism for every word length n ≥ 0. Applying the result to 10000 and 01000 gives a negative answer to Open Problem 18 in the JIS version of Bates et al. (2025). Version 0.3.0 recasts the earlier Chinese working paper in formal English and AMS-style LaTeX while retaining the theorem and recorded computational ranges. The complete proof, appendix, references and reproducibility materials support scrutiny and further research. Finite computation is distinguished from the general proof; completed independent human line-by-line verification and journal peer review are not recorded.
對任意兩個等長且長度 L ≥ 2 的二元關鍵字,若其相鄰差分字均為 unbordered(沒有相同的非空真前綴與真後綴),本文證明其替換圖對每個字長 n ≥ 0 都存在保位置顏色同構。套用於10000與01000,得到Bates等人(2025)JIS版Open Problem 18的負答。v0.3.0將較早中文工作論文重整為正式英文與AMS風格LaTeX,保留同一定理與計算範圍,提供完整證明、附錄、參考文獻及重現材料,供論證檢查與延伸研究。有限計算與一般證明分開,尚未完成獨立人類逐行核驗或期刊同儕審查。
Notes
資料來源與可用性 / Sources and availability
較早中文工作論文 / Earlier Chinese working paper; 較早公開研究 / Earlier public investigation
本Zenodo版本提供英文PDF、完整LaTeX來源ZIP、重現材料ZIP、AI-USE.md、LICENSE.md、README、版本說明與校驗碼。問題來源為Bates等人《A New Combinatorial Interpretation of Partial Sums of m-Step Fibonacci Numbers》(JIS 28, Article 25.6.6, 2025);對應問題在arXiv:2503.11055v1為Open Problem 4.1。原創文字與資料採CC BY 4.0,程式採MIT。
This Zenodo version supplies the English PDF, complete LaTeX-source ZIP, reproducibility ZIP, AI-USE.md, LICENSE.md, README, version notes and checksums. The problem originates in Bates et al., A New Combinatorial Interpretation of Partial Sums of m-Step Fibonacci Numbers (JIS 28, Article 25.6.6, 2025), where it is Open Problem 18; it is Open Problem 4.1 in arXiv:2503.11055v1. Original text and data are CC BY 4.0; code is MIT.
Methods
方法與品質說明 / Methods and quality
本版明引AlyciaBHZ的較早公開研究及其留言,不將有限文獻查核當首次性認證。一般圖同構的雙向邊驗證涵蓋1≤n≤26,較小獨立實作另查位置顏色;未擴大計算範圍。程式與原驗證紀錄公開,大型映射檔可依原說明重建。生成式AI實質參與構想、證明、程式、運算、文獻、翻譯與寫作,Codex、Claude與Gemini分工見AI-USE.md;本紀錄未列更細實際型號。具名作者負責稿件與訂正,但其完整逐行驗證尚無紀錄。本Zenodo存放不表示arXiv投稿或期刊接受。
The paper explicitly cites AlyciaBHZ's earlier public investigation and comments; a bounded literature check does not certify firstness. Ordinary graph isomorphism was verified in both edge directions for 1≤n≤26, with smaller independent implementations checking position colours; no larger range is claimed. Programs and original verification records are public, and large mapping files are described as regenerable. Generative AI substantially contributed to ideas, proof development, code, computation, literature analysis, translation and writing. Roles of Codex, Claude and Gemini are documented in AI-USE.md; this record does not identify more specific actual models. The named author remains responsible for the manuscript and corrections, while the author's own complete line-by-line verification is not recorded. This deposit is neither an arXiv submission nor journal acceptance.
Files
paper-v0.3.0.pdf
Files
(181.5 kB)
| Name | Size | |
|---|---|---|
|
md5:7895895ef964e3afa407f1e8413baec2
|
2.6 kB | Preview Download |
|
md5:fa976f40c2a1b46f693d5cbacd244e5e
|
12.5 kB | Preview Download |
|
md5:9d486f2b717cc37deaa37273644d543c
|
479 Bytes | Preview Download |
|
md5:adf5ea4059b3ce0d59b48b5017e61c3e
|
116.6 kB | Preview Download |
|
md5:22d9b9ad6e3d7031db5278a2e4a72218
|
2.1 kB | Preview Download |
|
md5:af0f0f6f845ca7ecb8c7347f5e597dda
|
43.7 kB | Preview Download |
|
md5:f6908cb25be7660f1193d5323adbd090
|
578 Bytes | Preview Download |
|
md5:8bc5f7f9d38a2073d2cb164f2745ef96
|
2.9 kB | Preview Download |
Additional details
References
- AlyciaBHZ (2026-09-21). prereg: Bates et al. (JIS 28, 25.6.6) Open Problems 18/19 — an invariant of the keyword's difference word. GitHub, the-omega-institute/trureturing, issue #9241, including two subsequent comments. https://github.com/the-omega-institute/trureturing/issues/9241 . Accessed 2026-10-24; open computational research record, not an all-length proof.
- Erik Bates, Blan Morrison, Mason Rogers, Arianna Serafini and Anav Sood (2025). A New Combinatorial Interpretation of Partial Sums of m-Step Fibonacci Numbers. Journal of Integer Sequences 28, Article 25.6.6. https://cs.uwaterloo.ca/journals/JIS/VOL28/Bates/bates5.html . References in this manuscript use the JIS numbering, including Open Problem 18 and Proposition 15.
- Erik Bates, Blan Morrison, Mason Rogers, Arianna Serafini and Anav Sood (2025). A new combinatorial interpretation of partial sums of m-step Fibonacci numbers. arXiv:2503.11055v1, 14 March 2025. https://arxiv.org/abs/2503.11055v1 . Open Problem 4.1 corresponds to JIS Open Problem 18; Proposition 3.6 corresponds to JIS Proposition 15; Open Problem 4.2 corresponds to JIS Open Problem 19.