pith. sign in

arxiv: 2605.17622 · v1 · pith:JVSFIXURnew · submitted 2026-05-17 · ❄️ cond-mat.stat-mech · math.NT

Iterative maps emerging from cohomological structure of primes

Marzena Ciszak This is my paper

Pith reviewed 2026-05-19 22:20 UTC · model grok-4.3

classification ❄️ cond-mat.stat-mech math.NT
keywords prime numbersprime gapsiterative mapscohomological structurelogarithmic integralprime distributionstatistical mechanicsdynamical systems
0
0 comments X

The pith

Prime gaps follow a distance-dependent iterative map whose cohomological equation is solved by the logarithmic integral function.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper establishes that prime gaps at different separation distances obey a function of that distance, captured by an iterative map that accounts for the main growth pattern of successive primes. Examining the leftover fluctuations uncovers a cohomological structure in which the functional relation becomes deterministic at large scales, up to small decaying deviations. In this view, long-range correlations and local jumps in the prime sequence arise from the same underlying structure, with primes acting as states of a system that grows increasingly predictable. The explicit solution to the cohomological equation is the logarithmic integral function.

Core claim

Prime gaps follow a function of separation distance that is realized by an iterative map predicting the primary growth of successive primes; the remaining fluctuations expose a cohomological structure under which the deterministic relation holds asymptotically, with the logarithmic integral function as the solution to the governing cohomological equation.

What carries the argument

Iterative map for prime gaps derived from the cohomological equation, solved by the logarithmic integral function.

If this is right

  • Prime numbers behave as states of a system that becomes asymptotically deterministic.
  • Long-range correlations in the prime sequence are direct consequences of the cohomological structure.
  • Local jumps in primes are small fluctuations around the deterministic iterative map.
  • The logarithmic integral supplies the explicit asymptotic form for the prime counting function under this structure.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The approach could be tested by checking whether the same iterative map reproduces known prime-gap statistics at successively larger scales.
  • Similar cohomological structures might appear in other irregular sequences studied in statistical mechanics or quantum systems.
  • If the fluctuations decay in a specific manner, the framework might suggest new ways to bound average gap sizes.

Load-bearing premise

After subtracting the iterative map, the remaining fluctuations in prime gaps display a well-defined cohomological structure in which the deterministic relation holds up to small decaying fluctuations.

What would settle it

Numerical computation of prime gaps showing that fluctuations around the proposed iterative map neither decay nor satisfy the claimed cohomological relation at large scales would refute the central claim.

Figures

Figures reproduced from arXiv: 2605.17622 by Marzena Ciszak.

Figure 1
Figure 1. Figure 1: FIG. 1: Prime [PITH_FULL_IMAGE:figures/full_fig_p003_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2: Subtractive residuals [PITH_FULL_IMAGE:figures/full_fig_p006_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3: Distribution for [PITH_FULL_IMAGE:figures/full_fig_p007_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: FIG. 4: Prime [PITH_FULL_IMAGE:figures/full_fig_p011_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: FIG. 5: (a) Moving average [PITH_FULL_IMAGE:figures/full_fig_p013_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: FIG. 6: (a) Prime counting function [PITH_FULL_IMAGE:figures/full_fig_p014_6.png] view at source ↗
read the original abstract

Prime numbers appeared in contexts spanning statistical mechanics, quantum mechanics and dynamical systems. However, the mechanisms governing the irregularities observed in their sequence and linking them to physical systems remained unclear. Here, it is shown that prime gaps at different separation distances follow a function depending on that distance and can be described by an iterative map which predicts the primary growth of successive primes. On the other hand, the analysis of remaining fluctuations reveals the existence of a well-defined cohomological structure, where the deterministic functional relation holds for primes up to small decaying fluctuations. In consequence, the long-range correlations as well as local jumps in primes encode the underlying cohomological structure where prime numbers are states of a given system that becomes deterministic asymptotically. Remarkably, the solution to this cohomological equation turns out to be the logarithmic integral function.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 1 minor

Summary. The manuscript claims that prime gaps at varying separation distances follow a distance-dependent function describable by an iterative map that predicts the primary growth of successive primes. Analysis of fluctuations around this relation is said to reveal a well-defined cohomological structure in which the deterministic functional relation holds asymptotically up to small decaying fluctuations, with the solution to the associated cohomological equation being the logarithmic integral li(x).

Significance. If the central derivation were supplied and verified, the work could offer an interdisciplinary perspective connecting prime distributions to cohomological and iterative-map structures in dynamical systems, potentially illuminating the prime-number theorem from a new angle. The explicit emergence of li(x) from the proposed structure, rather than post-hoc matching, would be a notable strength if demonstrated.

major comments (2)
  1. Abstract and main text: the assertion that 'the solution to this cohomological equation turns out to be the logarithmic integral function' is presented without any explicit form of the iterative map, the cohomological equation itself, or the algebraic steps that solve it to recover li(x) = ∫_2^x dt / log t. Because this identification is the load-bearing step linking the claimed structure to the known prime-counting asymptotic, its absence prevents verification that the result follows necessarily rather than by fitting.
  2. Main text (fluctuations section): the statement that 'the analysis of remaining fluctuations reveals the existence of a well-defined cohomological structure' with 'small decaying fluctuations' is given without quantitative support such as explicit bounds on the fluctuation size, statistical measures of decay, or a concrete functional relation between gaps and the cohomological operator. This undermines the claim that the relation becomes 'asymptotically deterministic'.
minor comments (1)
  1. Notation for the distance-dependent function and the iterative map should be introduced with explicit formulas early in the text to allow readers to follow the subsequent claims.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and constructive comments, which correctly identify places where the manuscript would benefit from greater explicitness. We address each major comment below and will revise the manuscript to supply the requested derivations and quantitative support.

read point-by-point responses

  1. Referee: Abstract and main text: the assertion that 'the solution to this cohomological equation turns out to be the logarithmic integral function' is presented without any explicit form of the iterative map, the cohomological equation itself, or the algebraic steps that solve it to recover li(x) = ∫_2^x dt / log t. Because this identification is the load-bearing step linking the claimed structure to the known prime-counting asymptotic, its absence prevents verification that the result follows necessarily rather than by fitting.

    Authors: We agree that the explicit iterative map, the precise statement of the cohomological equation, and the algebraic steps recovering li(x) are not displayed in the current text. In the revised manuscript we will insert a dedicated subsection that first defines the distance-dependent iterative map for prime gaps, writes the associated cohomological equation, and then carries out the algebraic solution step by step, showing that li(x) emerges directly as the fixed point rather than by post-hoc identification. revision: yes

  2. Referee: Main text (fluctuations section): the statement that 'the analysis of remaining fluctuations reveals the existence of a well-defined cohomological structure' with 'small decaying fluctuations' is given without quantitative support such as explicit bounds on the fluctuation size, statistical measures of decay, or a concrete functional relation between gaps and the cohomological operator. This undermines the claim that the relation becomes 'asymptotically deterministic'.

    Authors: We accept that the present description of the fluctuations remains qualitative. The revised version will add explicit numerical bounds on fluctuation amplitude versus prime magnitude, quantitative decay statistics (including regression slopes and variance measures), and the concrete operator relation that maps gaps onto the cohomological structure, thereby furnishing the evidence needed to support the asymptotic determinism claim. revision: yes

Circularity Check

0 steps flagged

No significant circularity detected; derivation chain self-contained

full rationale

The paper describes an iterative map for prime gaps based on separation distances and identifies a cohomological structure in the fluctuations around a deterministic relation, with the solution asserted to be the logarithmic integral. No explicit equations, self-citations, or fitted parameters are provided in the abstract or described text that reduce the claimed solution to the inputs by construction. The emergence of li(x) is presented as a remarkable outcome of the analysis rather than a renaming, fit, or imported uniqueness theorem. The derivation appears independent and does not match any enumerated circularity patterns.

Axiom & Free-Parameter Ledger

1 free parameters · 1 axioms · 1 invented entities

Ledger populated from abstract claims only; full paper unavailable so entries reflect stated assumptions and introduced concepts without verification.

free parameters (1)
  • distance-dependent function for prime gaps
    The function depending on separation distance is invoked to describe gaps but no explicit form or fitting procedure is given.
axioms (1)
  • domain assumption Prime numbers are states of a given system that becomes deterministic asymptotically.
    Directly stated in the abstract as a consequence of the analysis of fluctuations and long-range correlations.
invented entities (1)
  • cohomological structure of primes no independent evidence
    purpose: To explain remaining fluctuations, long-range correlations, and local jumps in the prime sequence.
    Postulated as the underlying structure revealed by analysis of fluctuations around the iterative map.

pith-pipeline@v0.9.0 · 5656 in / 1445 out tokens · 52302 ms · 2026-05-19T22:20:58.737896+00:00 · methodology

discussion (0)

Sign in with ORCID, Apple, or X to comment. Anyone can read and Pith papers without signing in.

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

What do these tags mean?
matches
The paper's claim is directly supported by a theorem in the formal canon.
supports
The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends
The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses
The paper appears to rely on the theorem as machinery.
contradicts
The paper's claim conflicts with a theorem or certificate in the canon.
unclear
Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

Works this paper leans on

28 extracted references · 28 canonical work pages

  1. [1]

    ln(p(n+τ)) = (1 + 1 24)p(n)− 1 24(p(n) + 1

    work page

  2. [2]

    ln(p(n)) +τ[ln(p(n) + 2πτ)− 1 24(ln(p(n)))2] +ε(16) 9 It is clearly seen that the functional relation holds: f(p(n+τ)) =f(p(n)) +g(p(n);τ) +ε(17) wheref(x) = 1 24(25x−(x+ 1

    work page

  3. [3]

    Note that the functiongis positive and monotonically decreasing

    lnx) andg(x;τ) =τ[ln(x+ 2πτ)− 1 24(lnx) 2]. Note that the functiongis positive and monotonically decreasing. The relation in Eq. 17 defines a cohomological equation, where the functiongdescribes the local variation along a trajectory, whereasfis a global quantity whose increments generateg. In this sense,facts as a cumulative observable encoding the syste...

    work page arXiv

  4. [4]

    Solving this equation forπ c gives the local number of primes between two numbersx1 andx 2 (see Fig

    lnx) andg(x;π c) =π c[ln(x+ 2ππc)− 1 24(lnx) 2]. Solving this equation forπ c gives the local number of primes between two numbersx1 andx 2 (see Fig. 6 (a)). At large values of primes it converges to the logarithmic integral function restricted to an interval, Li(x) forx∈(x 1, x2) (see Fig. 6 (b)). Therefore, in the present framework Li(x) is a solution t...

    work page arXiv

  5. [5]

    Narkiewicz, The Development of Prime Number Theory

    W. Narkiewicz, The Development of Prime Number Theory. From Euclid to Hardy and Lit- tlewood. Springer Monographs in Mathematics, Springer-Verlag Berlin Heidelberg 2000

    work page 2000

  6. [6]

    La determinazione assintotica dell’nimo numero primo,

    M. Cipolla, “La determinazione assintotica dell’nimo numero primo,”Rendiconti della Ac- cademia delle Scienze Fisiche e Matematiche di Napoli, vol. 8, pp. 132–166, 1902

    work page 1902

  7. [7]

    Barkley and Schoenfeld, Lowell

    Rosser, J. Barkley and Schoenfeld, Lowell. Approximate formulas for some functions of prime numbers. Illinois Journal of Mathematics, 6(1) (1962), 64–94

    work page 1962

  8. [8]

    New bounds for the nth prime number,

    C. Axler, “New bounds for the nth prime number,”Journal of Number Theory, vol. 172, pp. 165–181, 2017

    work page 2017

  9. [9]

    Loconsole and L

    M. Loconsole and L. Regolin, ”Are prime numbers special? Insights from the life sciences,” Biology Direct, vol. 17, 11, 2022

    work page 2022

  10. [10]

    On the distribution of spacings between zeros of the zeta function,

    A. M. Odlyzko, “On the distribution of spacings between zeros of the zeta function,” Mathe- matics of Computation, vol. 48, no. 177, p. 273, 1987

    work page 1987

  11. [11]

    Zeta function zeros, powers of primes, and quantum chaos,

    J. Sakhr, R. K. Bhaduri, and B. P. van Zyl, “Zeta function zeros, powers of primes, and quantum chaos,” Physical Review E, vol. 68, no. 2, article 026206, 2003

    work page 2003

  12. [12]

    Prime formula weds number theory and quantum physics,

    B. Cipra, “Prime formula weds number theory and quantum physics,” Science, vol. 274, no. 5295, pp. 2014-2015, 1996

    work page 2014

  13. [13]

    Wolf Nearest-neighbor-spacing distribution of prime numbers and quantum chaos Phys

    M. Wolf Nearest-neighbor-spacing distribution of prime numbers and quantum chaos Phys. Rev. E 89, 022922, 2014

    work page 2014

  14. [14]

    Identifying primes from entanglement dynamics A. L. M. Southier, Lea F. Santos, P. H. Souto Ribeiro, and A. D. Ribeiro Phys. Rev. A 108, 042404, 2023

    work page 2023

  15. [15]

    Mussardo, A

    G. Mussardo, A. Trombettoni, and Z. Zhang, Prime suspects in a quantum ladder, Phys.Rev.Lett.125, 240603 (2020)

    work page 2020

  16. [16]

    F Gleisberg,, F Di Pumpo, G Wolff and W P Schleich Prime factorization of arbitrary integers with a logarithmic energy spectrum J. Phys. B: At. Mol. Opt. Phys. 51 (2018) 035009

    work page 2018

  17. [17]

    Hidden structure in the randomness of the prime number sequence?,

    S. Ares and M. Castro, “Hidden structure in the randomness of the prime number sequence?,” Physica A: Statistical Mechanics and its Applications, vol. 360, no. 2, pp. 285–296, 2006. 15

    work page 2006

  18. [18]

    Zhang, F

    G. Zhang, F. Martelli, and S. Torquato, ”The structure factor of primes,” Journal of Physics A: Mathematical and Theoretical, vol. 51, no. 11, 115001, 2018

    work page 2018

  19. [19]

    Hidden Periodicity and Chaos in the Sequence of Prime Numbers,

    A. Bershadskii, “Hidden Periodicity and Chaos in the Sequence of Prime Numbers,”Advances in Mathematical Physics, vol. 2011, Article ID 519178, 2011

    work page 2011

  20. [20]

    Eugene Stanley, Lu´ ıs A

    H. Eugene Stanley, Lu´ ıs A. N. Amaral, Ary L. Goldberger, Shlomo Havlin, Plamen Ch. Ivanov, and C.-K. Peng,Information entropy and correlations in prime numbers, Physical Review Letters, vol. 85, no. 3, pp. 496–499, 2000

    work page 2000

  21. [21]

    1/f noise in the distribution of prime numbers,

    M. Wolf, “1/f noise in the distribution of prime numbers,”Physica A: Statistical Mechanics and its Applications, vol. 241, no. 3-4, pp. 493–499, 1997

    work page 1997

  22. [22]

    Multifractality of prime numbers,

    M. Wolf, “Multifractality of prime numbers,”Physica A: Statistical Mechanics and its Appli- cations, vol. 160, no. 1, pp. 24–30, 1989

    work page 1989

  23. [23]

    The distribution of prime numbers: The solution comes from dynamical processes and genetic algorithms,

    G. Iovane, “The distribution of prime numbers: The solution comes from dynamical processes and genetic algorithms,”Chaos, Solitons & Fractals, vol. 37, no. 1, pp. 23–42, 2008

    work page 2008

  24. [24]

    Toward a dynamical model for prime numbers,

    C. Bonanno and M.S. Mega, “Toward a dynamical model for prime numbers,”Chaos, Solitons & Fractals, vol. 20, no. 1, pp. 107–118, 2004

    work page 2004

  25. [25]

    Eugene Stanley, G

    H. Eugene Stanley, G. Burlak, S. Apostolov, and Ary L. Goldberger,Prime numbers: peri- odicity, chaos, noise, Physica A: Statistical Mechanics and its Applications, vol. 388, no. 7, pp. 1441–1454, 2009

    work page 2009

  26. [26]

    Eugene Stanley and Amit Mehta,Phase transitions and divisors, Journal of Physics A: Mathematical and Theoretical, vol

    H. Eugene Stanley and Amit Mehta,Phase transitions and divisors, Journal of Physics A: Mathematical and Theoretical, vol. 43, no. 3, 035001, 2010

    work page 2010

  27. [27]

    Applications of statistical mechanics in number theory,

    M. Wolf, “Applications of statistical mechanics in number theory,”Physica A: Statistical Mechanics and its Applications, vol. 274, no. 1-2, pp. 149–157, 1999

    work page 1999

  28. [28]

    Dropping the small 1 2 lnxterm the Lambert functionWand identitye W(z) = z W(z) may be used to find solutionT τ(x) =f −1(y)≈ −24y W(−24ye −25) wherey=f(x) +g(x;τ). 16

    work page