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A Reconstruction Approach to the Schrödinger Equation via the principle of consistency

Rami M Habchi

Published Jun 28, 2026• 0 citations
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Abstract

Version note: This record is Version 5 of the work. Earlier versions were uploaded as separate Zenodo records. The preceding version is available at: [10.5281/zenodo.20509744.]. Earlier drafts are preserved for timestamp and version-history purposes.
DOI
10.5281/zenodo.20976072

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A Reconstruction Approach to the Schrödinger Equation via the principle of consistency part II

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Version note: This record is Version 3 of the work. Earlier versions were uploaded as separate Zenodo records. The preceding version is available at: [10.5281/zenodo.17846855]. Earlier drafts are preserved for timestamp and version-history purposes.

•Jun 2, 2026•0 citations

A Reconstruction Approach to the Schrödinger Equation via the principle of consistency

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Version 42

Andrea Oddone

====================================================================================================================================OFFICIAL PRIMACY RECORD & POST-QUANTUM CRYPTOGRAPHIC PROTECTION (V42 - Version 42)==================================================================================================================================== Author: Andrea OddoneDeposit Date (Zenodo): 28 February 2026 Note: The Post-Quantum Cryptographic Protection refers specifically to the PQC layer of the PDF asset named "Version42.pdf", corresponding to Version 42 (V42); the outer GPG layer provides classical encryption and access control. ------------------------------------------------------------------------------------------------------------------------------------1. DISCOVERY & EVIDENCE STATEMENT - PRIMACY ANCHOR------------------------------------------------------------------------------------------------------------------------------------ * FUNDAMENTAL GENESIS: The UNIFIED CONCEPTUAL SUBSTANCE and all pioneering discoveries of this research were officially established, then deposited and timestamped publicly on 10 January 2026 on Zenodo. * PRIMARY INCEPTION: The initial deposit served as a TIMESTAMP to secure the priority of the project. The overarching inception conveys and embodies the intellectual property, the manifestation, and the author’s intellectual mind, which establishes the ABSOLUTE CONCEPTUAL SYNTHESIS, thereby establishing as well the ranges of theoretical scopes and the INTEGRATIVE RESOLUTION of the entire METHOD, including all primary solutions, achieved IN ITS ENTIRETY and retained as the INTELLECTUAL BODY OF KNOWLEDGE inherent to that inception. * METHODICAL DOCUMENTATION: All the subsequent versions serve exclusively to formalize, progressively to unfold and extensively transcribe, so to provide the exhaustive articulation of the ORIGINAL INTEGRAL DISCOVERY. These records constitute the formal documentation of the discovery, which was already established and held by the author in its entirety on 10 January 2026. Every documentary record is an INHERENT MANIFESTATION of the complete intellectual substance formally established on 10 January 2026 (see Primary Record). These records provide the systematic documentation of the intellectual property ALREADY POSSESSED and HELD BY THE AUTHOR FROM THE ONSET. This includes and will include any detailed images, audio records, annotations, and analytical considerations that inherently pertain to the original discovery and are systematically rendered through subsequent versions solely for the purpose of facilitating the extensive transcription and formal articulation. * ELUCIDATION ON INTELLECTUAL COHESION: This synthesis pertains to the INTEGRAL INTELLECTUAL CORPUS inherently present at inception, remains valid across determinate enduring ranges of theoretical scopes, validity being retained irrespective of interpretative orientations and formulations, and likewise holds such validity with respect to the discoveries — including any ideas and findings originating from, traceable to, and stemming from the ranges — ensuring that the METHODOLOGY remains preserved regardless of interpretative orientation. 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CHRONOLOGY : CONNECTION TO VERSIONS AND RELATIVE DATA------------------------------------------------------------------------------------------------------------------------------------ This Primacy Record of Version 42 (V42) formally recognizes and records the prior existence and priority of the discovery. The corresponding PDF document of Version 42, named "Version42.pdf", constitutes the formalized manifestation of the discovery, being an inherent manifestation which unfolds and extensionally transcribes the findings of the inception and the discovery itself, which is formally established in the materials of the primary record including the deposited files dated 10 January 2026.The PDF document "Version42.pdf" is represented through its cryptographic encapsulation ("Version42.pdf.pqc.gpg") and through the offline holdings ("Version42.pdf", "Version42.pdf.pqc"), each referring to the identical underlying content. All assets deposited on Zenodo are also preserved in offline holdings. Below are listed the connections to the preceding versions of V42. * IMMEDIATE PRECEDENT VERSION (V41): - Version DOI (30 January 2026): https://doi.org/10.5281/zenodo.18428135 - Original Record URL: https://zenodo.org/records/18428135 - Original File Identifier: "VersionA6.zip" - Deposition Date: 30 January 2026 - Concept DOI: https://doi.org/10.5281/zenodo.18428134 - Note: This online URL provides access to the record of Version 41 (V41) of the original intellectual content deposited on Zenodo on 30 January 2026, with the Version DOI uniquely identifying that record and its associated timestamp. On 30 January 2026, a ZIP archive ("VersionA6.zip") containing the "VersionA6.pdf.asc" file was deposited of the corresponding version V41. Private Custody: The ZIP archive and the associated files ("VersionA6.zip", "VersionA6.pdf.asc", "VersionA6.pdf") are preserved in the author's private offline custody, each referring to the identical underlying intellectual content. ------------------------------------------------------------------------------------------------------------------------------------HISTORICAL CLARIFICATION OF V1 DEPOSITION------------------------------------------------------------------------------------------------------------------------------------ Regarding the corresponding material of Version 1, the archive (“draftandreaoddone.zip”) containing the “draftandreaoddone.pdf.asc”file reflects the same content as the PDF file “draftandreaoddone.pdf” that had been written prior, and which was prepared for deposition as of 10 January 2026 before its public deposition on Zenodo on the same day. 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Considering the deposition of 10 January 2026 the intellectual substance was not conceived arbitrarily, as discoveries of the methodology and corresponding ranges of theoretical scopes, and other discoveries and concepts, had been explored and reflected upon through prior investigation, and they were the focus of active intellectual involvement over preceding years. ------------------------------------------------------------------------------------------------------------------------------------3. CURRENT GPG & POST-QUANTUM PROTECTED ASSETS (V42)------------------------------------------------------------------------------------------------------------------------------------ This Primacy Record of Version 42 (V42) formally recognizes and records the prior existence and priority of the discovery. The corresponding PDF document of Version 42, named "Version42.pdf", constitutes the formalized manifestation of the discovery, being an inherent manifestation which unfolds and extensionally transcribes the findings of the inception and the discovery itself, which is formally established in the materials of the primary record including the deposited files dated 10 January 2026.The PDF document "Version42.pdf" is represented through its cryptographic encapsulation ("Version42.pdf.pqc.gpg") and through the offline holdings ("Version42.pdf", "Version42.pdf.pqc"), each referring to the identical underlying content. All assets deposited on Zenodo are also preserved in offline holdings. The following entries identify the current digital assets regarding this version (V42) and their respective cryptographic fingerprints. 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•Feb 28, 2026•0 citations

On a Variation of a Derivation of the Schrödinger Equation

Paul Mazur, Robert H. Barron

American Journal of Physics•Jul 1, 1974•2 citations

The Schrödinger Equation from Capacity Geometry: General forced-action theorem on a fixed capacity background

Nala Trees

This record contains Version 2.0 of The Schrödinger Equation from Capacity Geometry: General Forced-Action Theorem on a Fixed Capacity Background. Version 2.0 supersedes the earlier narrower local formulation (DOI: 10.5281/zenodo.19118007). The earlier record worked within a restricted quadratic-gradient density class. The present record proves the same result on the full local first-order homogeneous density class, with the weighted bulk-plus-flux bridge as the load-bearing structural principle rather than a minimality assumption on derivative order. The theorem lane is therefore broader, the claim boundaries are cleaner, and the companion-note dependence has been updated to reflect the current SGOC corpus. This record DOI. 10.5281/zenodo.19189901 should be treated as the source-of-record citation for the general fixed-background Schrödinger result. The paper proves a narrow but exact claim: within the full first-order homogeneous density class compatible with continuity-current structure, the Fisher-corrected Madelung action is the unique admissible action family up to the stated constant, and its Euler–Lagrange equations are exactly the Schrödinger equation. The decisive density correction is forced inside an upstream capacity-geometric theorem base rather than inserted by a diffusion or stochastic postulate. The logical scaffold is explicit and limited. Part 1A fixes the canonical logarithmic capacity scalar and the unique quadratic base tension. Parts 6 and 7A isolate the phase-blindness ceiling for the continuous quadratic sector. Part 6A sharpens the localisation-versus-smoothing and forced-balance framework. The present paper then shows that continuity-current compatibility fixes the phase sector, locality and first-order dependence force the full homogeneous density family, and a universal weighted bulk-plus-flux bridge with field-independent coefficients forces the Fisher form. The canonical member of that family is fixed by the bridge to the imported capacity scalar. In that precise sense the logical order at the Schrödinger layer is reversed: the geometric base comes first, and the Schrödinger action appears downstream of it. The claim is exact within its lane and no broader. This is a fixed-background, single-particle theorem stated away from nodal sets. It does not claim a many-body theory, a measurement theory, a relativistic completion, a full nodal-complement reconstruction theorem, or backreaction of the quantum sector on the background geometry. Appendix D records that on a nodal-complement component, once the phase-sensitive completion is an S¹-valued phase field, circulation quantisation and wavefunction reconstruction are automatic. Appendix G records a conditional H¹-closure lane inside the same fixed-background action. The discrete-curvature and branch-bookkeeping support used on nodal-complement triangulations belongs to the companion Closure Phase note, while the further complement-globalisation step belongs to the companion Relative Cohomology note. Keywords: Schrödinger equation; capacity geometry; Fisher information; Madelung action; phase-sensitive evolution; forced action; homogeneous density class; weighted Fisher bridge; REG; coherence scalar; fixed background; nodal complement; reproducibility Companion manuscripts cited in this paper's proof chain, from the SGOC programme: Part 1A: The Canonical Capacity Coordinate and the Unique Quadratic Tension in the Capacity Gauge. DOI: 10.5281/zenodo.19156316 Part 7A: The Second-Order Ceiling, Triadic Completion, and the Wilson-Triangle Lift. DOI: 10.5281/zenodo.19175902 Closure Phase as Discrete Curvature on a Triangulated 2-Complex. DOI: 10.5281/zenodo.19185885 Relative Cohomology and Cut-Graph Holonomy Trivialisation. DOI: 10.5281/zenodo.19185877 Additional provenance notes cited in the paper Part 6A: Localisation Rather than Saturation in Capacity Geometry: REG Dissipation, Forced Balance Laws, and a Restricted Ricci Bridge. DOI: 10.5281/zenodo.19174434 Landauer, Relative Entropy, and Fisher Length in Capacity Geometry. DOI: 10.5281/zenodo.19186309

•Mar 23, 2026•0 citations

A new derivation of the schrödinger equation

H. N. Ioannidou

Lettere al nuovo cimento della societa italiana di fisica/Lettere al nuovo cimento•Aug 1, 1982•7 citations
====================================================================================================================================OFFICIAL PRIMACY RECORD & POST-QUANTUM CRYPTOGRAPHIC PROTECTION (V42 - Version 42)==================================================================================================================================== Author: Andrea OddoneDeposit Date (Zenodo): 28 February 2026 Note: The Post-Quantum Cryptographic Protection refers specifically to the PQC layer of the PDF asset named "Version42.pdf", corresponding to Version 42 (V42); the outer GPG layer provides classical encryption and access control. ------------------------------------------------------------------------------------------------------------------------------------1. DISCOVERY & EVIDENCE STATEMENT - PRIMACY ANCHOR------------------------------------------------------------------------------------------------------------------------------------ * FUNDAMENTAL GENESIS: The UNIFIED CONCEPTUAL SUBSTANCE and all pioneering discoveries of this research were officially established, then deposited and timestamped publicly on 10 January 2026 on Zenodo. * PRIMARY INCEPTION: The initial deposit served as a TIMESTAMP to secure the priority of the project. The overarching inception conveys and embodies the intellectual property, the manifestation, and the author’s intellectual mind, which establishes the ABSOLUTE CONCEPTUAL SYNTHESIS, thereby establishing as well the ranges of theoretical scopes and the INTEGRATIVE RESOLUTION of the entire METHOD, including all primary solutions, achieved IN ITS ENTIRETY and retained as the INTELLECTUAL BODY OF KNOWLEDGE inherent to that inception. * METHODICAL DOCUMENTATION: All the subsequent versions serve exclusively to formalize, progressively to unfold and extensively transcribe, so to provide the exhaustive articulation of the ORIGINAL INTEGRAL DISCOVERY. These records constitute the formal documentation of the discovery, which was already established and held by the author in its entirety on 10 January 2026. Every documentary record is an INHERENT MANIFESTATION of the complete intellectual substance formally established on 10 January 2026 (see Primary Record). These records provide the systematic documentation of the intellectual property ALREADY POSSESSED and HELD BY THE AUTHOR FROM THE ONSET. This includes and will include any detailed images, audio records, annotations, and analytical considerations that inherently pertain to the original discovery and are systematically rendered through subsequent versions solely for the purpose of facilitating the extensive transcription and formal articulation. * ELUCIDATION ON INTELLECTUAL COHESION: This synthesis pertains to the INTEGRAL INTELLECTUAL CORPUS inherently present at inception, remains valid across determinate enduring ranges of theoretical scopes, validity being retained irrespective of interpretative orientations and formulations, and likewise holds such validity with respect to the discoveries — including any ideas and findings originating from, traceable to, and stemming from the ranges — ensuring that the METHODOLOGY remains preserved regardless of interpretative orientation. This statement applies to all versions. ------------------------------------------------------------------------------------------------------------------------------------VERSION CLARIFICATION (V42)------------------------------------------------------------------------------------------------------------------------------------ This clause applies exclusively to Version 42. The specific PDF document corresponding to version V42 (named "Version42.pdf") was already written and prepared in the author's private offline custody as of 4 February 2026, prior to its public deposition on Zenodo dated 28 February 2026.This statement refers solely to this version (V42) as it existed on that date (4 February 2026). ------------------------------------------------------------------------------------------------------------------------------------2. CHRONOLOGY : CONNECTION TO VERSIONS AND RELATIVE DATA------------------------------------------------------------------------------------------------------------------------------------ This Primacy Record of Version 42 (V42) formally recognizes and records the prior existence and priority of the discovery. The corresponding PDF document of Version 42, named "Version42.pdf", constitutes the formalized manifestation of the discovery, being an inherent manifestation which unfolds and extensionally transcribes the findings of the inception and the discovery itself, which is formally established in the materials of the primary record including the deposited files dated 10 January 2026.The PDF document "Version42.pdf" is represented through its cryptographic encapsulation ("Version42.pdf.pqc.gpg") and through the offline holdings ("Version42.pdf", "Version42.pdf.pqc"), each referring to the identical underlying content. All assets deposited on Zenodo are also preserved in offline holdings. Below are listed the connections to the preceding versions of V42. * IMMEDIATE PRECEDENT VERSION (V41): - Version DOI (30 January 2026): https://doi.org/10.5281/zenodo.18428135 - Original Record URL: https://zenodo.org/records/18428135 - Original File Identifier: "VersionA6.zip" - Deposition Date: 30 January 2026 - Concept DOI: https://doi.org/10.5281/zenodo.18428134 - Note: This online URL provides access to the record of Version 41 (V41) of the original intellectual content deposited on Zenodo on 30 January 2026, with the Version DOI uniquely identifying that record and its associated timestamp. On 30 January 2026, a ZIP archive ("VersionA6.zip") containing the "VersionA6.pdf.asc" file was deposited of the corresponding version V41. Private Custody: The ZIP archive and the associated files ("VersionA6.zip", "VersionA6.pdf.asc", "VersionA6.pdf") are preserved in the author's private offline custody, each referring to the identical underlying intellectual content. ------------------------------------------------------------------------------------------------------------------------------------HISTORICAL CLARIFICATION OF V1 DEPOSITION------------------------------------------------------------------------------------------------------------------------------------ Regarding the corresponding material of Version 1, the archive (“draftandreaoddone.zip”) containing the “draftandreaoddone.pdf.asc”file reflects the same content as the PDF file “draftandreaoddone.pdf” that had been written prior, and which was prepared for deposition as of 10 January 2026 before its public deposition on Zenodo on the same day. The deposition on Zenodo of Version 1 (V1) took place on 10 January 2026, uniquely anchoring the record through its DOI and timestamp. At deposition time, as a historical establishment, the record encompassed the archive file (“draftandreaoddone.zip”) containing the “draftandreaoddone.pdf.asc” file, which faithfully represents the original content corresponding to Version 1. The files remain preserved in the author’s private custody, ensuring the integrity and traceability of the intellectual content. The transition notice dated 08 February 2026 referred to a prospective organizational optimization of the archival container; such notice does not affect the historical fact that the ZIP archive was deposited and timestamped on 10 January 2026. Considering the deposition of 10 January 2026 the intellectual substance was not conceived arbitrarily, as discoveries of the methodology and corresponding ranges of theoretical scopes, and other discoveries and concepts, had been explored and reflected upon through prior investigation, and they were the focus of active intellectual involvement over preceding years. ------------------------------------------------------------------------------------------------------------------------------------3. CURRENT GPG & POST-QUANTUM PROTECTED ASSETS (V42)------------------------------------------------------------------------------------------------------------------------------------ This Primacy Record of Version 42 (V42) formally recognizes and records the prior existence and priority of the discovery. The corresponding PDF document of Version 42, named "Version42.pdf", constitutes the formalized manifestation of the discovery, being an inherent manifestation which unfolds and extensionally transcribes the findings of the inception and the discovery itself, which is formally established in the materials of the primary record including the deposited files dated 10 January 2026.The PDF document "Version42.pdf" is represented through its cryptographic encapsulation ("Version42.pdf.pqc.gpg") and through the offline holdings ("Version42.pdf", "Version42.pdf.pqc"), each referring to the identical underlying content. All assets deposited on Zenodo are also preserved in offline holdings. The following entries identify the current digital assets regarding this version (V42) and their respective cryptographic fingerprints. All assets are secured via GPG and post-quantum cryptographic shielding: - FILE (Deposited on Zenodo): "Version42.pdf.pqc.gpg" MD5: 5a70766373d19d229ee2d40af3da7e3f SHA-256: b6c5d6695d7123fcfea7d8edd85acae688db2f5475cfc7ec9505457f1dae4bd3 SHA3-256: 7aedcc1f6f6b32ec5b95f25339630303c8342090c4260599717bb8740428f240 BLAKE2: 68a7b6fb5ab87c69c69bc5845e2c1547973ff3ecf3e88857944560afaa120adfcc48e59088914760574317e59307f99eb7260311408a68b7b2b6f11f8043a7b2 BLAKE3: 8d647687e854070bb666919db8a6dd25ee067decabf4567185bf25d038582c6d - FILE (Held in Author's Private Custody): "wrapped42_key.bin.gpg" MD5: bb3f4dd38fb8d7c21b5df34f7b4d5c47 SHA-256: 71ee2317e05b0805eb609c2cec47748eba10fd67ed3562c823088d96bf78cbc6 SHA3-256: 633b078cabe85de47166b
This record contains Version 2.0 of The Schrödinger Equation from Capacity Geometry: General Forced-Action Theorem on a Fixed Capacity Background. Version 2.0 supersedes the earlier narrower local formulation (DOI: 10.5281/zenodo.19118007). The earlier record worked within a restricted quadratic-gradient density class. The present record proves the same result on the full local first-order homogeneous density class, with the weighted bulk-plus-flux bridge as the load-bearing structural principle rather than a minimality assumption on derivative order. The theorem lane is therefore broader, the claim boundaries are cleaner, and the companion-note dependence has been updated to reflect the current SGOC corpus. This record DOI. 10.5281/zenodo.19189901 should be treated as the source-of-record citation for the general fixed-background Schrödinger result. The paper proves a narrow but exact claim: within the full first-order homogeneous density class compatible with continuity-current structure, the Fisher-corrected Madelung action is the unique admissible action family up to the stated constant, and its Euler–Lagrange equations are exactly the Schrödinger equation. The decisive density correction is forced inside an upstream capacity-geometric theorem base rather than inserted by a diffusion or stochastic postulate. The logical scaffold is explicit and limited. Part 1A fixes the canonical logarithmic capacity scalar and the unique quadratic base tension. Parts 6 and 7A isolate the phase-blindness ceiling for the continuous quadratic sector. Part 6A sharpens the localisation-versus-smoothing and forced-balance framework. The present paper then shows that continuity-current compatibility fixes the phase sector, locality and first-order dependence force the full homogeneous density family, and a universal weighted bulk-plus-flux bridge with field-independent coefficients forces the Fisher form. The canonical member of that family is fixed by the bridge to the imported capacity scalar. In that precise sense the logical order at the Schrödinger layer is reversed: the geometric base comes first, and the Schrödinger action appears downstream of it. The claim is exact within its lane and no broader. This is a fixed-background, single-particle theorem stated away from nodal sets. It does not claim a many-body theory, a measurement theory, a relativistic completion, a full nodal-complement reconstruction theorem, or backreaction of the quantum sector on the background geometry. Appendix D records that on a nodal-complement component, once the phase-sensitive completion is an S¹-valued phase field, circulation quantisation and wavefunction reconstruction are automatic. Appendix G records a conditional H¹-closure lane inside the same fixed-background action. The discrete-curvature and branch-bookkeeping support used on nodal-complement triangulations belongs to the companion Closure Phase note, while the further complement-globalisation step belongs to the companion Relative Cohomology note. Keywords: Schrödinger equation; capacity geometry; Fisher information; Madelung action; phase-sensitive evolution; forced action; homogeneous density class; weighted Fisher bridge; REG; coherence scalar; fixed background; nodal complement; reproducibility Companion manuscripts cited in this paper's proof chain, from the SGOC programme: Part 1A: The Canonical Capacity Coordinate and the Unique Quadratic Tension in the Capacity Gauge. DOI: 10.5281/zenodo.19156316 Part 7A: The Second-Order Ceiling, Triadic Completion, and the Wilson-Triangle Lift. DOI: 10.5281/zenodo.19175902 Closure Phase as Discrete Curvature on a Triangulated 2-Complex. DOI: 10.5281/zenodo.19185885 Relative Cohomology and Cut-Graph Holonomy Trivialisation. DOI: 10.5281/zenodo.19185877 Additional provenance notes cited in the paper Part 6A: Localisation Rather than Saturation in Capacity Geometry: REG Dissipation, Forced Balance Laws, and a Restricted Ricci Bridge. DOI: 10.5281/zenodo.19174434 Landauer, Relative Entropy, and Fisher Length in Capacity Geometry. DOI: 10.5281/zenodo.19186309