Symbolic Representation of Mathematical Structures: A Categorical Framework with Application to TSP

Symbolic Representation of Mathematical Structures: A Categorical Framework with Application to TSP | Authorea try { document.documentElement.classList.add('js'); } catch (e) { } var _gaq = _gaq || []; _gaq.push(['_setAccount', 'G-8VDV14Y67G']); _gaq.push(['_trackPageview']); (function() { var ga = document.createElement('script'); ga.type = 'text/javascript'; ga.async = true; ga.src = ('https:' == document.location.protocol ? 'https://ssl' : 'http://www') + '.google-analytics.com/ga.js'; var s = document.getElementsByTagName('script')[0]; s.parentNode.insertBefore(ga, s); })(); Skip to main content Preprints Collections Wiley Open Research IET Open Research Ecological Society of Japan All Collections About About Authorea FAQs Contact Us Quick Search anywhere Search for preprint articles, keywords, etc. Search Search ADVANCED SEARCH SCROLL This is a preprint and has not been peer reviewed. Data may be preliminary. 14 October 2025 V1 Latest version Share on Symbolic Representation of Mathematical Structures: A Categorical Framework with Application to TSP Author : Sergey Kotikov 0009-0009-4367-7859 [email protected] Authors Info & Affiliations https://doi.org/10.22541/au.176046400.01707319/v1 149 views 138 downloads Contents Abstract Supplementary Material Information & Authors Metrics & Citations View Options References Figures Tables Media Share Abstract We present a symbolic mathematical framework that reinterprets classical mathematics through structural representations, where numbers, graphs, logical statements, and computational processes are unified as objects in a category of symbolic sequences. Rather than inventing new mathematics, we view familiar constructions from a different perspective: mathematical objects are not abstract values but concrete symbolic structures whose semantics emerge through transformations. The framework is built on finite strings over alphabet Σ = {S, P, I, Z, Ω, Λ}. We show how: • Integers are represented as balanced sequences of S (successor) and P (predecessor) symbols, with arithmetic operations performed via string concatenation and involution in O(1) time. Experimental validation on 176 graphs (random, chaotic, pathological) achieves 100% optimality. The framework demonstrates that many mathematical problems become computationally accessible when formulated as structural morphisms rather than discrete combinatorics. We discuss physical implementation through memristor crossbar networks and other analog computing substrates. Supplementary Material File (symstructures_v3.pdf) Download 502.71 KB Information & Authors Information Version history V1 Version 1 14 October 2025 Copyright This work is licensed under a Non Exclusive No Reuse License. Keywords bitopological spaces categorical framework expansive operator gödel's theorems hypercomputing memristor networks self-applicable functors structural numbers symbolic arithmetic tsp Authors Affiliations Sergey Kotikov 0009-0009-4367-7859 [email protected] View all articles by this author Metrics & Citations Metrics Article Usage 149 views 138 downloads .FvxKWukQNSOunydq8rnd { width: 100px; } Citations Download citation Sergey Kotikov. Symbolic Representation of Mathematical Structures: A Categorical Framework with Application to TSP. Authorea . 14 October 2025. DOI: https://doi.org/10.22541/au.176046400.01707319/v1 If you have the appropriate software installed, you can download article citation data to the citation manager of your choice. Simply select your manager software from the list below and click Download. For more information or tips please see 'Downloading to a citation manager' in the Help menu . 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