Published February 26, 2026 | Version v2
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Defining Information: Order from the Prime Number Structure of the Riemann Zeta Function

Authors/Creators

  • 1. Independent Researcher. No Affiliations.

Description

Abstract

Physics invokes “information” in contexts ranging from quantum measurement

to black hole evaporation, yet no consensus exists on what information physically is.

Shannon entropy quantifies surprise; von Neumann entropy quantifies mixedness;

Landauer’s principle assigns an energy cost; Englert’s distinguishability measures

which-path knowledge; Wheeler’s “it from bit” proposes a program. All provide

operational measures. None identify a physical referent. This paper proposes a

definition: information is order — the integer structure generated by the

prime numbers through multiplication, as encoded in the Euler product

of the Riemann zeta function. The quantitative measure is the Benford devi-

ation δB, which tracks how faithfully a physical system’s leading-digit distribution

reflects the integer-generated baseline given by Benford’s Law. Drawing on prior re-

sults connecting ζ(s) to the Schwarzschild metric, Bose–Einstein condensate phase

transitions, and causal set theory, we show that δB tracks quantum-to-classical

transitions continuously, that Benford’s Law emerges as the least-action distribu-

tion under scale invariance, and that the Second Law of Thermodynamics can be

reframed as the natural return to Benford conformance. Shannon entropy, Lan-

dauer’s principle, Englert’s duality relation, and Wheeler’s program all emerge as

consequences of the definition rather than axioms. The proposal is falsifiable: if

δB fails to track quantum-to-classical transitions in new experimental systems, the

framework fails. We present quantitative evidence from Bose–Einstein condensate

simulations, Kretschmann scalar profiles of black hole interiors, and a nine-model

quantum gravity comparison.

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Additional details

Related works

Is supplement to
Publication: 10.5281/zenodo.18553466 (DOI)
Publication: 10.5281/zenodo.18510250 (DOI)
Publication: 10.5281/zenodo.18731508 (DOI)
Publication: 10.5281/zenodo.18751909 (DOI)

Dates

Created
2026-02-26
We propose a physical definition of information: order — the integer structure generated by the prime numbers through multiplication, as encoded in the Euler product of the Riemann zeta function. The quantitative measure is the Benford deviation δ_B. We derive Benford's Law from scale invariance and maximum entropy, present evidence from Bose-Einstein condensate simulations and Kretschmann scalar profiles, and show that Shannon entropy, Landauer's principle, and Englert's duality relation emerge as consequences. The proposal is falsifiable.

Software

Repository URL
https://github.com/jackwayne234/research-hub

References

  • Riner, C. J. W. (2025). Modified Schwarzschild Metric via Benford's Law. Zenodo. DOI: 10.5281/zenodo.18553466. (Paper #1 in series.)
  • Riner, C. J. W. (2025). Bose–Einstein Condensates + Benford's Law. Zenodo. DOI: 10.5281/zenodo.18510250. (Paper #2 in series.)
  • Riner, C. J. W. (2025). Prime Numbers as Causal Set Theory. Zenodo. DOI: 10.5281/zenodo.18731508. (Paper #6 in series.)
  • Riner, C. J. W. (2026). Emergence of General Relativity from the Prime Number Structure of the Riemann Zeta Function. Zenodo. DOI: 10.5281/zenodo.18751909. (Paper #7 in series.)
  • Shannon, C. E. (1948). A Mathematical Theory of Communication. Bell System Technical Journal, 27(3), 379–423.
  • Von Neumann, J. (1932). Mathematische Grundlagen der Quantenmechanik. Springer, Berlin.
  • Landauer, R. (1961). Irreversibility and Heat Generation in the Computing Process. IBM Journal of Research and Development, 5(3), 183–191.
  • Wheeler, J. A. (1989). Information, physics, quantum: the search for links. Proceed- ings of the 3rd International Symposium on Foundations of Quantum Mechanics, 354–368.
  • Englert, B.-G. (1996). Fringe Visibility and Which-Way Information: An Inequality. Physical Review Letters, 77(11), 2154–2157.
  • Berut, A. et al. (2012). Experimental verification of Landauer's principle. Nature, 483, 187–189.
  • Yan, L. L. et al. (2018). Single-atom demonstration of quantum Landauer principle. Physical Review Letters, 120, 210601.
  • Newcomb, S. (1881). Note on the Frequency of Use of the Different Digits in Natural Numbers. American Journal of Mathematics, 4(1), 39–40.
  • Benford, F. (1938). The Law of Anomalous Numbers. Proceedings of the American Philosophical Society, 78(4), 551–572.
  • Burke, J. & Kincanon, E. (1991). Benford's law and physical constants. American Journal of Physics, 59(10), 952.
  • Sen(De), A. & Sen, U. (2011). Benford's law detects quantum phase transitions similarly as earthquakes. EPL, 95, 10006.
  • Rane, A., Mishra, A., Biswas, A., Sen(De), A. & Sen, U. (2014). Benford's law gives better scaling exponents in phase transitions. Physical Review E, 90, 022144.
  • Bera, S., Mishra, A., Roy, S. S., Biswas, A. & De, A. S. (2018). Benford analysis of quantum critical phenomena. Physics Letters A, 382, 1639–1644.
  • Cong, P., Li, M. & Ma, L. (2019). First digit law from Laplace transform. Physics Letters A, 383, 1836–1844.
  • Weyl, H. (1916). Annalen, 77, 313–352. Uber die Gleichverteilung von Zahlen mod. Eins. Mathematische
  • Einsiedler, M. & Ward, T. (2011). Ergodic Theory with a View Towards Number Theory. Graduate Texts in Mathematics 259, Springer.
  • aynes, E. T. (1957). Information Theory and Statistical Mechanics. Physical Review, 106(4), 620–630.
  • Scully, M. O., Englert, B.-G. & Walther, H. (1991). Quantum Optical Tests of Complementarity. Nature, 351, 111–116.
  • Kim, Y.-H., Yu, R., Kulik, S. P., Shih, Y. & Scully, M. O. (2000). Delayed 'Choice' Quantum Eraser. Physical Review Letters, 84, 1–5.
  • Walborn, S. P., Terra Cunha, M. O., Padua, S. & Monken, C. H. (2002). Double-Slit Quantum Eraser. Physical Review A, 65, 033818.
  • Coles, P. J., Kaniewski, J. & Wehner, S. (2014). Equivalence of wave–particle duality to entropic uncertainty. Nature Communications, 5, 5814.
  • Batelaan, H., Jones, E., Huang, W. C.-W. & Bach, R. (2020). Momentum exchange in the electron double-slit experiment. arXiv: 2012.02141.
  • Sinha, U., Couteau, C., Jennewein, T., Laflamme, R. & Weihs, G. (2010). Ruling Out Multi-Order Interference in Quantum Mechanics. Science, 329(5990), 418–421.
  • S¨ollner, I., Gsch¨osser, B., Mai, P., Preber, B., V¨or¨os, Z. & Weihs, G. (2012). Testing Born's rule in quantum mechanics for three mutually exclusive events. Foundations of Physics, 42, 742–751.
  • Elitzur, A. C. & Vaidman, L. (1993). Quantum Mechanical Interaction-Free Mea- surements. Foundations of Physics, 23, 987–997.
  • Kwiat, P. G. et al. (1999). High-Efficiency Quantum Interrogation via Quantum Zeno Effect. Physical Review Letters, 83, 4725.
  • Robens, C., Alt, W., Emary, C., Meschede, D. & Alberti, A. (2017). Atomic Bomb Testing: Elitzur–Vaidman Violates Leggett–Garg. Applied Physics B, 123, 12.
  • Elouard, C., Herrera-Marti, D., Clusel, M. & Auff`eves, A. (2016). The role of quan- tum measurement in stochastic thermodynamics. npj Quantum Information, 3, 9.
  • Jordan, A. N., Elouard, C. & Auff`eves, A. (2019). Quantum measurement engines and their relevance for quantum interpretations. arXiv: 1911.06838.
  • Kammerlander, P. & Anders, J. (2015). Coherence and measurement in quantum thermodynamics. Scientific Reports, 6, 22174.
  • Latune, C. & Elouard, C. (2024). A thermodynamically consistent approach to the energy costs of quantum measurements. Quantum, 8, 1526.
  • Berry, M. V. & Keating, J. P. (1999). The Riemann Zeros and Eigenvalue Asymp- totics. SIAM Review, 41(2), 236–266.
  • Sierra, G. & Townsend, P. K. (2008). Landau levels and Riemann zeros. Physical Review Letters, 101, 110201.
  • LeClair, A. & Mussardo, G. (2023). Riemann zeros as quantized energies of scattering with impurities. JHEP, 2023, 62.
  • Hartnoll, S. A. & Yang, E. (2025). The conformal primon gas at the end of time. JHEP, 2025, 34.
  • Godet, V. (2025). Mobius randomness in the Hartle–Hawking state. arXiv: 2505.03068.
  • Tamburini, F. (2025). Majorana particle spectrum in Rindler spacetime encoded by zeta. arXiv: 2503.09644.
  • Yakaboylu, E. (2024). Hamiltonian for the Hilbert–P´olya conjecture. Journal of Physics A, 57, 235203.
  • Kalauni, P. & Milton, K. A. (2023). Supersymmetric quantum mechanics and the Riemann hypothesis. arXiv: 2211.04382.
  • Sorkin, R. D. (2003). Causal sets: Discrete gravity. In Lectures on Quantum Gravity, Springer, 305–327.
  • Modesto, L. (2004). Disappearance of black hole singularity in quantum gravity. Physical Review D, 70, 124009.
  • Bonanno, A. & Reuter, M. (2000). Renormalization group improved black hole space- times. Physical Review D, 62, 043008.
  • Nicolini, P., Smailagic, A. & Spallucci, E. (2006). Noncommutative geometry inspired Schwarzschild black hole. Physics Letters B, 632, 547–551.
  • Bekenstein, J. D. (1973). Black holes and entropy. Physical Review D, 7(8), 2333– 2346.
  • Hawking, S. W. (1975). Particle creation by black holes. Communications in Math- ematical Physics, 43(3), 199–220.
  • 't Hooft, G. (1993). Dimensional reduction in quantum gravity. arXiv: gr- qc/9310026.
  • Susskind, L. (1995). The world as a hologram. Journal of Mathematical Physics, 36(11), 6377–6396.