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arxiv: 2606.21571 · v1 · pith:5MOQLZ76new · submitted 2026-06-19 · 🧮 math.GT

Remarks on Fixed Point Assertions in Digital Topology, 12

Laurence Boxer This is my paper

Pith reviewed 2026-06-26 12:34 UTC · model grok-4.3

classification 🧮 math.GT
keywords fixed point theorydigital topologydigital imagesmetric spacesself-mapsliterature critiquemathematical errorstopology
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The pith

Many papers on fixed points in digital metric spaces contain incorrect, trivial or unclear assertions.

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

This review examines publications on fixed points of self-functions on digital images and identifies assertions that are incorrect, have flawed proofs, are trivial, or are stated incoherently. The discussion continues the author's earlier critiques of similar issues in the literature. A sympathetic reader would care because unreliable claims undermine the ability to build reliable results in digital topology. If the identifications of errors hold, then results in this area cannot be taken at face value without independent checks.

Core claim

Publications on fixed points in digital metric spaces continue to contain assertions that are incorrect, incorrectly proven, trivial, or incoherently stated, with the paper providing discussions of bad assertions concerning fixed points of self-functions on digital images.

What carries the argument

Specific identification and critique of erroneous fixed-point claims for self-maps on digital images.

If this is right

  • Results on fixed points for digital images should be re-checked before further use.
  • New papers in the area must avoid the types of errors catalogued here.
  • The existing body of work on digital fixed points may contain multiple unreliable claims.
  • Authors should state claims more clearly and prove them rigorously to avoid incoherence or triviality.

Where Pith is reading between the lines

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

  • Similar patterns of flawed claims could appear in other subtopics within digital topology.
  • The issues may reflect broader challenges in maintaining quality for specialized or rapidly growing publication areas.
  • Independent replication studies of the critiqued results would provide a direct test of the review's accuracy.

Load-bearing premise

The errors identified by the author in each cited publication are accurately described and real.

What would settle it

A re-examination showing that any one of the critiqued assertions is actually correct, non-trivial, and properly proved would undermine the paper's central point.

Figures

Figures reproduced from arXiv: 2606.21571 by Laurence Boxer.

Figure 1
Figure 1. Figure 1: The ”theorem” of [19] 2.4 The digital Banach contraction principle We have the following. Definition 2.9. Let (X, d, κ) be a digital metric space. Let f : X → X satisfy, for all x, y ∈ X and some c satisfying 0 < c < 1, d(fx, fy) < c · d(x, y). Then f is a digital contraction. The following is sometimes called the digital Banach contraction principle. Theorem 2.10. [16] A digital contraction has a unique f… view at source ↗
Figure 2
Figure 2. Figure 2: ‘Corollaries” 1, 2, 3 of [20] 8 [PITH_FULL_IMAGE:figures/full_fig_p008_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: “Theorem” 4.1 of [22] [PITH_FULL_IMAGE:figures/full_fig_p010_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: “Corollaries” 4.1 and 4.2 of [22] 10 [PITH_FULL_IMAGE:figures/full_fig_p010_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: “Theorem” 4.2 of [22] 6 Further remarks We paraphrase [9]: We have discussed several papers that seek to advance fixed point assertions for digital metric spaces. Most of their assertions are incorrect or incorrectly proven. The authors, the referees who ap￾proved their publication, and, perhaps, “predatory journals” that accept payment for publication regardless of quality - share respon￾sibility for defi… view at source ↗
read the original abstract

The topic of fixed points in digital metric spaces has drawn yet more publications with assertions that are incorrect, incorrectly proven, trivial, or incoherently stated. We discuss publications with bad assertions concerning fixed points of self-functions on digital images, as in some of our previous papers

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

0 major / 2 minor

Summary. The manuscript continues the author's series of papers critiquing the literature on fixed points in digital metric spaces. It identifies specific publications containing assertions about fixed points of self-maps on digital images that the author claims are incorrect, incorrectly proven, trivial, or incoherently stated, providing case-by-case analysis of these issues.

Significance. If the specific readings and identifications of errors in the cited works are accurate, the paper performs a useful corrective role by documenting flaws in recent publications on this specialized topic. This could help reduce the propagation of problematic claims in digital topology. The work does not introduce new theorems, parameter-free derivations, or machine-checked results, so its significance is primarily as a literature-cleaning contribution rather than an advance in geometric topology.

minor comments (2)
  1. The abstract and introduction refer to 'publications with bad assertions' without an explicit enumerated list or table of the specific papers critiqued in this installment; adding such a summary would improve readability.
  2. Since this is installment 12 in a series, a brief statement of how the current critiques differ from or extend the previous ones would help contextualize the contribution.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the review and the recommendation of minor revision. The referee's summary accurately describes the purpose of the manuscript as a continuation of our series identifying problematic assertions in the fixed-point literature on digital images. We address the referee's observations on significance below.

read point-by-point responses

  1. Referee: If the specific readings and identifications of errors in the cited works are accurate, the paper performs a useful corrective role by documenting flaws in recent publications on this specialized topic. This could help reduce the propagation of problematic claims in digital topology.

    Authors: We confirm that the case-by-case analyses in the manuscript are accurate. Each referenced publication is examined for concrete issues (incorrect claims, flawed proofs, triviality, or incoherent statements) with direct quotations and counterexamples provided where appropriate. revision: no

  2. Referee: The work does not introduce new theorems, parameter-free derivations, or machine-checked results, so its significance is primarily as a literature-cleaning contribution rather than an advance in geometric topology.

    Authors: This characterization is correct. The manuscript is deliberately positioned as a literature critique within our ongoing series rather than a source of new theorems; its value lies in preventing the further dissemination of flawed results in digital topology. revision: no

Circularity Check

0 steps flagged

No significant circularity

full rationale

The paper is a literature critique that identifies errors, trivialities, or incoherences in prior publications on fixed points in digital metric spaces. It contains no derivations, predictions, fitted parameters, or mathematical claims that could reduce to inputs by construction. The sole self-reference (to the author's previous papers) is stylistic and non-load-bearing; each critique stands or falls on the accuracy of its reading of the cited external sources rather than any internal self-referential loop.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

No new mathematical theory, derivations, or fitted quantities are introduced; the paper rests on the domain assumption that the critiqued assertions can be evaluated against standard definitions in digital topology.

axioms (1)
  • domain assumption Assertions in the cited papers can be classified as incorrect, trivial, or incoherent using established definitions of digital metric spaces and fixed-point theorems.
    Invoked implicitly in the abstract's description of the topic.

pith-pipeline@v0.9.1-grok · 5548 in / 1055 out tokens · 17999 ms · 2026-06-26T12:34:29.416848+00:00 · methodology

discussion (0)

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Reference graph

Works this paper leans on

22 extracted references · 2 canonical work pages

  1. [1]

    M. S. Abdullahi, P. Kumam, J. Abubakar, and I. A. Garba, Coincidence and self-coincidence of maps between digital images,Topological Methods in Nonlinear Analysis56 (2) (2020), 607—628. https://www.tmna.ncu.pl/static/published/2020/v56n2-14.pdf

    2020

  2. [2]

    Boxer, A classical construction for the digital fundamental group,Jour- nal of Mathematical Imaging and Vision10 (1999), 51–62

    L. Boxer, A classical construction for the digital fundamental group,Jour- nal of Mathematical Imaging and Vision10 (1999), 51–62. https://link.springer.com/article/10.1023/A%3A1008370600456

    work page doi:10.1023/a 1999

  3. [3]

    Boxer, Remarks on Fixed Point Assertions in Digital Topology, 2,Ap- plied General Topology20, (1) (2019), 155–175

    L. Boxer, Remarks on Fixed Point Assertions in Digital Topology, 2,Ap- plied General Topology20, (1) (2019), 155–175. https://polipapers.upv.es/index.php/AGT/article/view/10667/11202

    2019

  4. [4]

    Boxer, Remarks on Fixed Point Assertions in Digital Topology, 3,Ap- plied General Topology20 (2) (2019), 349–361

    L. Boxer, Remarks on Fixed Point Assertions in Digital Topology, 3,Ap- plied General Topology20 (2) (2019), 349–361. https://polipapers.upv.es/index.php/AGT/article/view/11117 12

    2019

  5. [5]

    Boxer, Remarks on Fixed Point Assertions in Digital Topology, 4,Ap- plied General Topology21 (2) (2020), 265–284 https://polipapers.upv.es/index.php/AGT/article/view/13075

    L. Boxer, Remarks on Fixed Point Assertions in Digital Topology, 4,Ap- plied General Topology21 (2) (2020), 265–284 https://polipapers.upv.es/index.php/AGT/article/view/13075

    2020

  6. [6]

    Boxer, Remarks on Fixed Point Assertions in Digital Topology, 5,Ap- plied General Topology23 (2) (2022) 437–451 https://polipapers.upv.es/index.php/AGT/article/view/16655/14995

    L. Boxer, Remarks on Fixed Point Assertions in Digital Topology, 5,Ap- plied General Topology23 (2) (2022) 437–451 https://polipapers.upv.es/index.php/AGT/article/view/16655/14995

    2022

  7. [7]

    Boxer, Remarks on Fixed Point Assertions in Digital Topology, 6,Ap- plied General Topology24 (2) (2023), 281–305 https://polipapers.upv.es/index.php/AGT/article/view/18996/16097

    L. Boxer, Remarks on Fixed Point Assertions in Digital Topology, 6,Ap- plied General Topology24 (2) (2023), 281–305 https://polipapers.upv.es/index.php/AGT/article/view/18996/16097

    2023

  8. [8]

    Boxer, Remarks on Fixed Point Assertions in Digital Topology, 7,Ap- plied General Topology25 (1) (2024), 97 - 115 https://polipapers.upv.es/index.php/AGT/article/view/20026

    L. Boxer, Remarks on Fixed Point Assertions in Digital Topology, 7,Ap- plied General Topology25 (1) (2024), 97 - 115 https://polipapers.upv.es/index.php/AGT/article/view/20026

    2024

  9. [9]

    Boxer, Remarks on Fixed Point Assertions in Digital Topology, 8,Ap- plied General Topology25 (2) (2024), 457-473 https://polipapers.upv.es/index.php/AGT/article/view/21074/16938

    L. Boxer, Remarks on Fixed Point Assertions in Digital Topology, 8,Ap- plied General Topology25 (2) (2024), 457-473 https://polipapers.upv.es/index.php/AGT/article/view/21074/16938

    2024

  10. [10]

    Boxer, Remarks on Fixed Point Assertions in Digital Topology, 9,Ap- plied General Topology26 (1) (2025), 501–527 https://polipapers.upv.es/index.php/AGT/article/view/22510/17360

    L. Boxer, Remarks on Fixed Point Assertions in Digital Topology, 9,Ap- plied General Topology26 (1) (2025), 501–527 https://polipapers.upv.es/index.php/AGT/article/view/22510/17360

    2025

  11. [11]

    Boxer, Remarks on Fixed Point Assertions in Digital Topology, 10, Applied General Topology26 (2) (2025), 853 - 869 https://polipapers.upv.es/index.php/AGT/article/view/23678/17739

    L. Boxer, Remarks on Fixed Point Assertions in Digital Topology, 10, Applied General Topology26 (2) (2025), 853 - 869 https://polipapers.upv.es/index.php/AGT/article/view/23678/17739

    2025

  12. [12]

    Boxer, Remarks on Fixed Point Assertions in Digital Topology, 11, Applied General Topology27 (1) (2026) https://polipapers.upv.es/index.php/AGT/article/view/25239/18382

    L. Boxer, Remarks on Fixed Point Assertions in Digital Topology, 11, Applied General Topology27 (1) (2026) https://polipapers.upv.es/index.php/AGT/article/view/25239/18382

    2026

  13. [13]

    Boxer, O

    L. Boxer, O. Ege, I. Karaca, J. Lopez, and J. Louwsma, Digital fixed points, approximate fixed points, and universal functions,Applied General Topology17(2), 2016, 159–172 https://polipapers.upv.es/index.php/AGT/article/view/4704/6675

    2016

  14. [14]

    Boxer and P.C

    L. Boxer and P.C. Staecker, Remarks on fixed point assertions in digital topology,Applied General Topology20 (1) (2019), 135–153. https://polipapers.upv.es/index.php/AGT/article/view/10474/11201

    2019

  15. [15]

    Chartrand and S

    G. Chartrand and S. Tian, Distance in digraphs.Computers&Mathe- matics with Applications34 (11) (1997), 15–23. https://www.sciencedirect.com/science/article/pii/S0898122197002162

    1997

  16. [16]

    Ege and I

    O. Ege and I. Karaca, Banach fixed point theorem for digital images, Journal of Nonlinear Science and Applications8 (3) (2015), 237–245 https://www.isr-publications.com/jnsa/articles-1797-banach-fixed-point- theorem-for-digital-images 13

    2015

  17. [17]

    S.E. Han, Banach fixed point theorem from the viewpoint of digital topology.Journal of Nonlinear Science and Applications9 (2016), 895–905 https://www.isr-publications.com/jnsa/articles-1915-banach-fixed-point- theorem-from-the-viewpoint-of-digital-topology

    2016

  18. [18]

    Khalimsky, Motion, deformation, and homotopy in finite spaces,Proc

    E. Khalimsky, Motion, deformation, and homotopy in finite spaces,Proc. IEEE Intl. Conf. Systems, Man, Cybernetics(1987), 227–234

    1987

  19. [19]

    Mishra, P.K

    A. Mishra, P.K. Tripathi, A.K. Agrawal, and R. Mehrotra, Application of contraction conditions,Solid State Technology63 (6) (2020), 201045 – 201052 https://www.solidstatetechnology.us/index.php/JSST/article/view/8631

    2020

  20. [20]

    ¨Ozkapu and ¨O

    A.S. ¨Ozkapu and ¨O. Acar, New common fixed point results in digital metric space,Demonstratio Mathematica2026; 59(1): 20250208 https://www.degruyterbrill.com/document/doi/10.1515/dema-2025- 0208/html

    work page doi:10.1515/dema-2025- 2025

  21. [21]

    Rosenfeld, ‘Continuous’ functions on digital pictures,Pattern Recogni- tion Letters4, 1986, 177–184

    A. Rosenfeld, ‘Continuous’ functions on digital pictures,Pattern Recogni- tion Letters4, 1986, 177–184. https://www.sciencedirect.com/science/article/pii/01678655

    arXiv 1986

  22. [22]

    Singh, S

    D.S. Singh, S. Parveen, C.K. Yadav, and B. K. Gupta, Fixed point theorems for four mappings in a complete digital metric space,Boletim da Sociedade Paranaense de Matematica44 (7) (2026), 1–10. https://periodicos.uem.br/ojs/index.php/BSocParanMat/article/view/80695/751375161968 14

    2026