TAD Theory and Its Applications
A Mathematical Framework for Visualizing Histories
##article.authors##
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Yasuo Matsui
Independent Researcher
https://researchmap.jp/yasuo-matsui
DOI:
https://doi.org/10.51094/jxiv.1593Keywords:
History-Dependent Dynamics, Relaxation Function and Forgetting Curve, Logistic Growth Model, Kovacs Effect (Education and Memory Model), Inventory Management and Advertising Effect, Epidemiological Model (SIR), Economic Growth Model, Large Language Models (Transformer)Abstract
This study proposes TAD (Traced Allocation Dynamics), a general framework for representing dynamic phenomena through a two-time structure—the time of occurrence T and the time of observation t—and systematically presents its mathematical structure, universality, and applicability.
TAD is a unified framework composed of the input G(t,T), the historical structure g(t,T), the allocation rate μ(t,T), the natural decay γ(t,T), and the value conversion factor σ(t,T). We show that this structure consistently describes a wide range of phenomena—inventory management, population dynamics, learning and memory, marketing, economic growth, physical relaxation processes, and large language models—that have conventionally been treated as unrelated.
A notable feature of this work is that, while originating in the practical idea of inventory freshness management, it naturally leads to more abstract and universal mathematical structures such as history-based entropy and universal identities. Furthermore, we reveal for the first time the hidden cohort-time T dynamics underlying logistic growth, which has traditionally been formulated only in the observation-time axis ttt. We also generalize the Kovacs effect, known in physics, within the TAD framework and predict that an analogous transient overshoot phenomenon can arise in learning and memory, suggesting the existence of cross-domain historical effects.
In addition, we introduce TAD-DB, a two-dimensional historical database for applying TAD to real-world data, and use it to conduct a large-scale empirical analysis of Japan’s population dynamics from 1920 to 2023. We show that visualization of g(t,T), mortality and survival freshness, cohort contributions, entropy S(t), and future population projection can all be performed consistently within the TAD framework alone.
TAD requires no advanced mathematical prerequisites and enables unified understanding, visualization, estimation, and prediction of dynamic systems using history as the minimal source of information. This paper provides the first comprehensive and empirically operational presentation of this new meta-theory.
Conflicts of Interest Disclosure
This study was conducted independently by the author as an off-duty personal research activity. No funding, data, or facilities were provided by the author’s employer or any other organization. The paper aims to present a general theoretical framework and does not represent the work, products, or views of any specific company. The author declares no competing interests.Downloads *Displays the aggregated results up to the previous day.
References
John R. Anderson and Lael J. Schooler. The Adaptive Nature of Memory. In: Psychological Review 107.4 (2000), pp. 603–634. doi: 10.1037/0033-295X.107.4.603.
Iz Beltagy, Matthew E. Peters, and Arman Cohan. “Longformer: The Long-Document Transformer”. In: arXiv preprint arXiv:2004.05150. 2020.
Jean-Philippe Bouchaud. Weak Ergodicity Breaking and Aging in Disordered Systems. In: Journal de Physique I 2.9 (1992), pp. 1705–1713. doi: 10.1051/jp1:1992238.
Simon Broadbent. One way TV advertisements work. In: Journal of the Market Research Society 21.3 (1979), pp. 139–166.
Piers Coleman. “Fluctuation–dissipation theorem and linear response theory”. In: Introduction to Many-Body Physics. Cambridge University Press, 2015. Chap. 9. doi: 10.1017/CBO9781139020916.011.
Zihang Dai, Zhilin Yang, Yiming Yang, Jaime Carbonell, Quoc V. Le, and Ruslan Salakhutdinov. “Transformer-XL: Attentive Language Models Beyond a Fixed-Length Context”. In: International Conference on Learning Representations (ICLR). 2019.
Peter Debye. Polar Molecules. Chemical Catalog Company, 1929.
Hermann Ebbinghaus. Über das Gedächtnis: Untersuchungen zur experimentellen Psychologie. English translation: Memory: A Contribution to Experimental Psychology (1913). Leipzig: Duncker & Humblot, 1885.
Hermann Ebbinghaus. Memory: A Contribution to Experimental Psychology. English translation by Henry A. Ruger and Clara E. Bussenius (of the 1885 German original). New York: Teachers College, Columbia University, 1913.
Martin Egozcue, Albert Prat, and Enrique Vázquez. On some covariance inequalities for monotonic and non-monotonic functions. In: Journal of Inequalities in Pure and Applied Mathematics (JIPAM) 10.4 (2009), pp. 1–13.
Peter S. Fader and Bruce G. S. Hardie. Probability Models for Customer-Base Analysis. In: Journal of Interactive Marketing 23.1 (2009), pp. 61–69. doi: 10.1016/j.intmar.2008.11.003.
John D. Ferry. Viscoelastic Properties of Polymers. 3rd. New York: Wiley, 1980.
H. von Foerster. “Some Remarks on Changing Populations”. In: The Kinetics of Cellular Proliferation. 1959, pp. 382–407.
I. M. Gelfand and S. V. Fomin. Calculus of Variations. Prentice Hall, 1963.
Michael E. Gilpin and Francisco J. Ayala. Global Models of Growth and Competition. In: Proceedings of the National Academy of Sciences 70.12 (1973), pp. 3590–3593. doi: 10.1073/pnas.70.12.3590.
Herbert Goldstein, Charles Poole, and John Safko. Classical Mechanics. 3rd. Addison-Wesley, 2001. isbn: 978-0201657029.
Benjamin Gompertz. On the Nature of the Function Expressive of the Law of Human Mortality. In: Philosophical Transactions of the Royal Society of London 115 (1825), pp. 513–585.
A. Hahn and W. H. Fietz. On the magnetic viscosity of SmCo5. In: Physics Letters A 45.5 (1973), pp. 397–398. doi: 10.1016/0375-9601(73)90890-6.
Leif Johansen. Substitution versus Fixed Production Coefficients in the Theory of Economic Growth. North-Holland, 1959.
Nathan Keyfitz and Hal Caswell. Applied Mathematical Demography. 3rd. New York: Springer, 2005.
Thomas B. L. Kirkwood. Deciphering death: a commentary on Gompertz (1825) ‘On the nature of the function expressive of the law of human mortality…’. In: Philosophical Transactions of the Royal Society B 370.1666 (2015), p. 20140379. doi: 10.1098/rstb.2014.0379.
R. Kohlrausch. Theorie des elektrischen Rückstandes in der Leidener Flasche. In: Annalen der Physik und Chemie 91 (1854), pp. 179–214. doi: 10.1002/andp.18541670203.
Mark Kot. Elements of Mathematical Ecology. Cambridge University Press, 2001.
A. J. Kovacs. Transition vitreuse dans les polymères amorphes. Étude phénoménologique. In: Fortschritte der Hochpolymeren-Forschung 3 (1963), pp. 394–507. doi: 10.1007/BF01806805.
Ryogo Kubo. Statistical-Mechanical Theory of Irreversible Processes. I. In: Journal of the Physical Society of Japan 12 (1957), pp. 570–586. doi: 10.1143/JPSJ.12.570.
L. D. Landau and E. M. Lifshitz. Mechanics. 3rd. Course of Theoretical Physics, Vol. 1. Butterworth Heinemann, 1976. isbn: 978-0750628969.
B. P. Lathi. Linear Systems and Signals, 3rd ed. Oxford University Press, 2018.
P. Maltoni, G. Giangrandi, F. Spizzo, et al. Time and temperature dependent magnetic viscosity experiments on ferrite nanoparticles. In: Journal of Applied Physics 133.16 (2023), p. 163902. doi: 10.1063/5.0139615.
A. G. McKendrick. Applications of Mathematics to Medical Problems. In: Proceedings of the Edinburgh Mathematical Society 44 (1926), pp. 98–130. doi: 10.1017/S0013091500002568.
Ministry of Health, Labour and Welfare. Vital Statistics of Japan. https://www.mhlw.go.jp/english/database/db-hw/vs01.html
. Accessed 23 November 2025.
C. T. Moynihan, A. J. Easteal, J. Wilder, and J. Tucker. Dependence of the Fictive Temperature of Glass on Cooling Rate. In: Journal of Physical Chemistry 78.26 (1976), pp. 2673–2677. doi: 10.1021/j100562a006.
J. D. Murray. Mathematical Biology I: An Introduction. 3rd. Springer, 2002.
Jaap M. J. Murre and Joeri Dros. Replication and Analysis of Ebbinghaus’ Forgetting Curve. In: PLOS ONE 10.7 (2015), e0120644. doi: 10.1371/journal.pone.0120644.
Prasad A. Naik and Kalyan Raman. Understanding the Impact of Synergy in Multimedia Communications. In: Journal of Marketing Research 40.4 (2003), pp. 375–388. doi: 10.1509/jmkr.40.4.375.19386.
O. S. Narayanaswamy. A Model of Structural Relaxation in Glass. In: Journal of the American Ceramic Society 54.10 (1971), pp. 491–498. doi: 10.1111/j.1151-2916.1971.tb12186.x.
Alan V. Oppenheim, Alan S. Willsky, and S. Hamid Nawab. Signals and Systems, 2nd ed. Prentice Hall, 1997.
Raymond Pearl and Lowell J. Reed. On the Rate of Growth of the Population of the United States since 1790 and Its Mathematical Representation. In: PNAS 6.6 (1920), pp. 275–288. doi: 10.1073/pnas.6.6.275.
L. S. Pontryagin, V. G. Boltyanskii, R. V. Gamkrelidze, and E. F. Mishchenko. The Mathematical Theory of Optimal Processes. Interscience, 1962.
Ofir Press, Noah A. Smith, and Mike Lewis. Train Short, Test Long: Attention with Linear Biases. arXiv:2108.12409. 2021.
Samuel H. Preston, Patrick Heuveline, and Michel Guillot. Demography: Measuring and Modeling Population Processes. Blackwell, 2001.
Jack W. Rae, Anna Potapenko, Siddhant M. Jayakumar, and Timothy P. Lillicrap. “Compressive Transformers for Long-Range Sequence Modelling”. In: ICLR. 2019.
Walter Rudin. Real and Complex Analysis. 3rd ed. McGraw-Hill, 1987.
Norman B. Ryder. The Cohort as a Concept in the Study of Social Change. In: American Sociological Review 30.6 (1965), pp. 843–861. doi: 10.2307/2090964.
Peter Shaw, Jakob Uszkoreit, and Ashish Vaswani. “Self-Attention with Relative Position Representations”. In: NAACL. 2018.
Julian L. Simon and Johan Arndt. The shape of the advertising response function. In: Journal of Advertising Research 20.4 (1980), pp. 11–28.
Robert M. Solow. A Contribution to the Theory of Economic Growth. In: Quarterly Journal of Economics 70.1 (1956), pp. 65–94. doi: 10.2307/1884513.
Robert M. Solow, James Tobin, C. C. von Weizsäcker, and Menahem E. Yaari. Neoclassical Growth with Fixed Factor Proportions. In: Review of Economic Studies 33.2 (1966), pp. 79–115. doi: 10.2307/2296445.
Elias M. Stein and Rami Shakarchi. Real Analysis: Measure Theory, Integration, and Hilbert Spaces. Princeton University Press, 2005.
Gerard J. Tellis and David L. Weiss. Does TV advertising really affect sales? The role of measures, models, and data aggregation. In: Journal of Advertising Research 35.2 (1995), pp. 1–13.
Albert Q. Tool. Relation between Inelastic Deformability and Thermal Expansion of Glass in its Annealing Range. In: Journal of the American Ceramic Society 29.9 (1946), pp. 240–253. doi: 10.1111/j.1151-2916.1946.tb11592.x.
Ashish Vaswani et al. “Attention is All You Need”. In: NeurIPS. 2017, pp. 5998–6008.
Pierre-François Verhulst. Notice sur la loi que la population poursuit dans son accroissement. In: Correspondance Mathématique et Physique 10 (1838), pp. 113–121.
Waloddi Weibull. A Statistical Distribution Function of Wide Applicability. In: Journal of Applied Mechanics 18.3 (1951), pp. 293–297.
G. Williams and D. C. Watts. Non-symmetrical dielectric relaxation behaviour arising from a simple empirical decay function. In: Transactions of the Faraday Society 66.565 (1970), pp. 80–85. doi: 10.1039/TF9706600080.
John T. Wixted. The Psychology and Neuroscience of Forgetting. In: Annual Review of Psychology 55 (2004), pp. 235–269. doi: 10.1146/annurev.psych.55.090902.141555.
Yang Yang and Kenneth C. Land. Age-Period-Cohort Analysis: New Models, Methods, and Empirical Applications. Chapman and Hall/CRC, 2013.
Manzil Zaheer et al. “Big Bird: Transformers for Longer Sequences”. In: NeurIPS. 2020.
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Submitted: 2025-10-04 12:15:52 UTC
Published: 2025-10-28 01:25:43 UTC — Updated on 2025-11-27 00:05:38 UTC
Versions
- 2025-11-27 00:05:38 UTC (3)
- 2025-11-06 07:08:37 UTC (2)
- 2025-10-28 01:25:43 UTC (1)
Reason(s) for revision
In this revised version (ver. 1.2), the chapter structure has been extensively reorganized and the content significantly expanded in order to clarify the foundational theory of TAD and its range of applications. In particular, the design of the historical database TAD-DB and a newly added empirical analysis using Japan’s population data (1920–2023) constitute major enhancements, demonstrating in a systematic manner how far the TAD framework can explain and predict real-world social data. Through the visualization of g(t,T), analyses of mortality and survival freshness, entropy S(t), future population projections, and cohort-wise contributions, this revision makes clear that TAD is not merely a conceptual model but a practical framework applicable across diverse domains. This update also includes unified notation and terminology, additional related work, and correction of minor inconsistencies. Overall, ver. 1.2 represents the most refined version of the TAD theory to date, offering a more robust foundational structure together with empirically supported applicability.License
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Yasuo Matsui
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