Bansal Fiber Framework: A Stratified Organization of Schrödinger Electronic Structure

Bansal Fiber Framework: A Stratified Organization of Schrödinger Electronic Structure | 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 Search Search ADVANCED SEARCH SCROLL This is a preprint and has not been peer reviewed.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. 21 January 2026 V1 Latest version Share on Bansal Fiber Framework: A Stratified Organization of Schrödinger Electronic Structure Author : Abhishek Bansal 0000-0002-2572-9004 [email protected] Authors Info & Affiliations https://doi.org/10.22541/au.176901786.61472008/v1 88 views 52 downloads Contents Abstract Supplementary Material Information & Authors Metrics & Citations View Options References Figures Tables Media Share Abstract Nonrelativistic Schrödinger quantum mechanics successfully describes electronic structure across physics and chemistry, yet it admits multiple, apparently distinct representations of the electron, including localized particles, delocalized waves, continuous charge densities, fractional occupations, and transient carriers in open or nonequilibrium systems. These descriptions coexist within standard theory but lack a unified structural explanation. In this work, I introduce a stratified geometric framework that organizes Schrödinger-based electronic information without modifying the Schrödinger equation or its solutions. The central idea is to treat the electron as a derived, context-dependent projection of the Schrödinger state rather than as a primitive ontological object. Electronic information is decomposed into structurally distinct components corresponding to charge and occupation (R), phase and coherence (C), transient or virtual structure (D), and orientation-dependent internal degrees of freedom (H). This stratification is formalized using a graded algebra and a fibered geometric construction, referred to as the Bansal fiber. An algebra-valued Schrödinger equation is introduced as an organizational extension of the standard formalism. Its scalar projection recovers the ordinary Schrödinger equation exactly, while controlled nilpotent couplings encode short-lived electronic contributions without altering spectral properties or unitarity. A functional-analytic formulation based on self-adjoint operators and Kato-semigroup theory establishes well-posedness, projection consistency, and leading-order stability. Electron number, localization, and fractional occupation emerge naturally through energetic stabilization of charge-carrying projections. This leads to a nonlinear Poisson-projection equation that expresses electrostatic self-consistency in terms of projection admissibility, connecting the framework directly to quantum chemistry, density functional theory, and self-consistent field methods. The resulting theory provides a unified geometric and operator-theoretic language that clarifies the coexistence of multiple electron models within standard Schrödinger quantum mechanics. Supplementary Material File (abhishekbansal_schrödinger.pdf) Download 430.94 KB Information & Authors Information Version history V1 Version 1 21 January 2026 Copyright This work is licensed under a Creative Commons Attribution 4.0 International License Keywords algebravalued schrödinger equation bansal algebra charge stabilization density functional theory electron emergence functional analysis geometric phases kato-rellich theorem nilpotent perturbations projection operators schrödinger quantum mechanics self-consistent field methods stratified electronic structure Authors Affiliations Abhishek Bansal 0000-0002-2572-9004 [email protected] New Era Consultancy Services View all articles by this author Metrics & Citations Metrics Article Usage 88 views 52 downloads .FvxKWukQNSOunydq8rnd { width: 100px; } Citations Download citation Abhishek Bansal. Bansal Fiber Framework: A Stratified Organization of Schrödinger Electronic Structure. Authorea . 21 January 2026. DOI: https://doi.org/10.22541/au.176901786.61472008/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. 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