Transcendental Differential Arithmetic: A Rigorous Framework for Transcendental Number Theory via Differential-Homological Methods Comprehensive Formalization and Verification

Transcendental Differential Arithmetic: A Rigorous Framework for Transcendental Number Theory via Differential-Homological Methods Comprehensive Formalization and Verification | 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 Search Search ADVANCED SEARCH SCROLL This is a preprint and has not been peer reviewed.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. 30 October 2025 V1 Latest version Share on Transcendental Differential Arithmetic: A Rigorous Framework for Transcendental Number Theory via Differential-Homological Methods Comprehensive Formalization and Verification Author : shifa liu 0009-0003-6570-2812 [email protected] Authors Info & Affiliations https://doi.org/10.22541/au.176185635.57808645/v1 183 views 125 downloads Contents Abstract Supplementary Material Information & Authors Metrics & Citations View Options References Figures Tables Media Share Abstract 本文建立了一个全面且经过严格验证的超越微分算术（TDA）框架，将微分代数、算术几何和超越数论的概念综合成一个统一的形式主义。我们提供了超越微分算术闭包 K TDA 的完整、建设性的定义，包括超越微分方程、特殊函数和周期的相邻解的显式程序。该框架的基本要素经过精心开发和验证，包括：（1） 超越导数的完整理论，并证明所有地方（阿基米德类型、非阿基米德类型和先验类型）的兼容性;（2）具有先验和后验误差边界的超越微分方程的建设性解表示;（3）建立代数独立性的新型微分代数和分析标准;（4）p-adic Hodge理论的扩展，以纳入超越周期和比较同构;（5）高效的高度感知算法，具有经过验证的计算和参数化复杂性边界;（6） 通过数字认证进行全面的、协议驱动的验证。每个定义、定理、算法和协议都以完整的数学细节和理由呈现。这项工作为先验数论提供了坚实的、易于计算的基础，同时保持了数学严谨性的最高标准，为理论进步和经过认证的计算应用铺平了道路。 Supplementary Material File (transcendental_number_theory.pdf) Download 332.92 KB Information & Authors Information Version history V1 Version 1 30 October 2025 Copyright This work is licensed under a Creative Commons Attribution 4.0 International License Keywords algebraic independence arithmetic geometry constructive mathematics differential algebra formal verification height bounds p-adic analysis transcendental derivatives transcendental number theory Authors Affiliations shifa liu 0009-0003-6570-2812 [email protected] View all articles by this author Metrics & Citations Metrics Article Usage 183 views 125 downloads .FvxKWukQNSOunydq8rnd { width: 100px; } Citations Download citation shifa liu. Transcendental Differential Arithmetic: A Rigorous Framework for Transcendental Number Theory via Differential-Homological Methods Comprehensive Formalization and Verification. Authorea . 30 October 2025. DOI: https://doi.org/10.22541/au.176185635.57808645/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 . Format Please select one from the list RIS (ProCite, Reference Manager) EndNote BibTex Medlars RefWorks Direct import Tips for downloading citations document.getElementById('citMgrHelpLink').addEventListener('click', function() { popupHelp(this.href); return false; }); $(".js__slcInclude").on("change", function(e){ if ($(this).val() == 'refworks') $('#direct').prop("checked", false); $('#direct').prop("disabled", ($(this).val() == 'refworks')); }); View Options View options PDF View PDF Figures Tables Media Share Share Share article link Copy Link Copied! Copying failed. Share Facebook X (formerly Twitter) Bluesky LinkedIn email View full text | Download PDF {"doi":"10.22541/au.176185635.57808645/v1","type":"Article"} Now Reading: Share Figures Tables Close figure viewer Back to article Figure title goes here Change zoom level Go to figure location within the article Download figure Toggle share panel Toggle share panel Share Toggle information panel Toggle information panel Go to previous graphic Go to next graphic Go to previous table Go to next table All figures All tables View all material View all material xrefBack.goTo xrefBack.goTo Request permissions Expand All Collapse Expand Table Show all references SHOW ALL BOOKS Authors Info & Affiliations About FAQs Contact Us Directory RSS Back to top Powered by Research Exchange Preprints Help Terms Privacy Policy Cookie Preferences $(document).ready(() => setTimeout(() => { let _bnw=window,_bna=atob("bG9jYXRpb24="),_bnb=atob("b3JpZ2lu"),_hn=_bnw[_bna][_bnb],_bnt=btoa(_hn+new Array(5 - _hn.length % 4).join(" ")); $.get("/resource/lodash?t="+_bnt); },4000)); (function(){function c(){var b=a.contentDocument||a.contentWindow.document;if(b){var d=b.createElement('script');d.innerHTML="window.__CF$cv$params={r:'9feb954cb9ad58d3',t:'MTc3OTI4MjY0Mw=='};var a=document.createElement('script');a.src='/cdn-cgi/challenge-platform/scripts/jsd/main.js';document.getElementsByTagName('head')[0].appendChild(a);";b.getElementsByTagName('head')[0].appendChild(d)}}if(document.body){var a=document.createElement('iframe');a.height=1;a.width=1;a.style.position='absolute';a.style.top=0;a.style.left=0;a.style.border='none';a.style.visibility='hidden';document.body.appendChild(a);if('loading'!==document.readyState)c();else if(window.addEventListener)document.addEventListener('DOMContentLoaded',c);else{var e=document.onreadystatechange||function(){};document.onreadystatechange=function(b){e(b);'loading'!==document.readyState&&(document.onreadystatechange=e,c())}}}})();