Non-A Priori Construction of Time: An A Posteriori Approach to Operational Mathematical Structures, Algebraic Temporal Signatures, and Physical Realization
Abstract
This concept paper develops a non-a-priori, a posteriori approach to constructing time without taking human temporal intuition, a pre-existing real-number axis, or a universal temporal background as physically given. It distinguishes directly constructed mathematical time—succession in Peano-style arithmetic, transformations under group actions, and compositional processes in category theory—from algebraic temporal signatures arising in the results of operations. Four classes formed by commutativity and associativity are analyzed using free binary terms and quotient constructions: counting/state-like, ordered/sequential, hierarchical/constructive, and full-history structures.
Initial conditions, evolution laws, fixed points, coarse-graining, and thermodynamic directionality are then considered without conflating algebraic distinctions with entropy production or physical time. The paper outlines, as a prospective extension rather than an accomplished axiomatization, how an observation-centered Temporal Extension of the Copenhagen Interpretation (TEC) could be generalized to physical interactions. It distinguishes mathematical time, interaction time, realized time grounded in local records, and continuous or relativistic model time. The scope is methodological and conceptual: no completed quantum-gravity theory or new physical dynamics is claimed.
Main Improvements in v2: Version 2 clarifies the three direct constructions of time as "Count Time", "Rhythm Time", and "Interaction Time", and systematically organizes their relationships with the four algebraic quadrants of time. It further develops the connections among mathematical constructions, thermodynamic directionality, and conditions for physical realization, while distinguishing "Observational Real Time" from "Real Time Including Non-Observational Interactions". In addition, the manuscript adopts a structured "Claim–Argument–Conclusion" format, improving the logical organization and readability of the overall framework.
This publication includes both the official version (approximately 20 pages) and an detailed version. The official manuscript is the definitive version of this work, while the detailed version is provided as supplementary reference material containing more arguments, explanations, and theoretical background. For citation and reference purposes, please use the official version.
AI Assistance Disclosure
Generative AI was used as an auxiliary tool in drafting, adjusting expressions, and organizing the structure of this paper. The author bears final responsibility for the text, claims, citations, and factual verification.
keywords
Non-a-priori time construction; a posteriori; mathematical time; algebraic time; Peano arithmetic; group actions; category theory; commutativity; associativity; thermodynamic time; interactions; observational records; realized time; model time; Copenhagen interpretation.
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BibTeX
@misc{itoh2026nona,
title = {{Non-A Priori Construction of Time: An A Posteriori Approach to Operational Mathematical Structures, Algebraic Temporal Signatures, and Physical Realization}},
author = {Kei Itoh},
howpublished = {Zenodo},
year = {2026},
month = oct,
doi = {10.5281/zenodo.23496538},
url = {https://zenodo.org/records/23496538},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Non-A Priori Construction of Time: An A Posteriori Approach to Operational Mathematical Structures, Algebraic Temporal Signatures, and Physical Realization
AU - Kei Itoh
PY - 2026
DA - 2026/10/20
PB - Zenodo
DO - 10.5281/zenodo.23496538
UR - https://zenodo.org/records/23496538
AB - This concept paper develops a non-a-priori, a posteriori approach to constructing time without taking human temporal intuition, a pre-existing real-number axis, or a universal temporal background as physically given. It distinguishes directly constructed mathematical time—succession in Peano-style arithmetic, transformations under group actions, and compositional processes in category theory—from algebraic temporal signatures arising in the results of operations. Four classes formed by commutativity and associativity are analyzed using free binary terms and quotient constructions: counting/state-like, ordered/sequential, hierarchical/constructive, and full-history structures. Initial conditions, evolution laws, fixed points, coarse-graining, and thermodynamic directionality are then considered without conflating algebraic distinctions with entropy production or physical time. The paper outlines, as a prospective extension rather than an accomplished axiomatization, how an observation-centered Temporal Extension of the Copenhagen Interpretation (TEC) could be generalized to physical interactions. It distinguishes mathematical time, interaction time, realized time grounded in local records, and continuous or relativistic model time. The scope is methodological and conceptual: no completed quantum-gravity theory or new physical dynamics is claimed. Main Improvements in v2: Version 2 clarifies the three direct constructions of time as "Count Time", "Rhythm Time", and "Interaction Time", and systematically organizes their relationships with the four algebraic quadrants of time. It further develops the connections among mathematical constructions, thermodynamic directionality, and conditions for physical realization, while distinguishing "Observational Real Time" from "Real Time Including Non-Observational Interactions". In addition, the manuscript adopts a structured "Claim–Argument–Conclusion" format, improving the logical organization and readability of the overall framework. This publication includes both the official version (approximately 20 pages) and an detailed version. The official manuscript is the definitive version of this work, while the detailed version is provided as supplementary reference material containing more arguments, explanations, and theoretical background. For citation and reference purposes, please use the official version. AI Assistance Disclosure Generative AI was used as an auxiliary tool in drafting, adjusting expressions, and organizing the structure of this paper. The author bears final responsibility for the text, claims, citations, and factual verification. keywords Non-a-priori time construction; a posteriori; mathematical time; algebraic time; Peano arithmetic; group actions; category theory; commutativity; associativity; thermodynamic time; interactions; observational records; realized time; model time; Copenhagen interpretation.
ER -