Comment on "Unifying Aspects of Generalized Calculus"

Krzysztof Sienicki; Miko{\l}aj Sienicki

REVIEW 6 major objections 5 minor 10 references

Comment on "Unifying Aspects of Generalized Calculus"

T0 review · 6 major / 5 minor · reviewed 2026-08-05 · deepseek-v4-flash

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Pith's one-line read The paper argues that Czachor's non-Newtonian calculus is physically inert because it offers no principle for choosing the bijection that defines its arithmetic.

desk verdict The paper's broad worry about arbitrary bijections is real, but its new counterexamples violate the framework's own definitions, so the case collapses. read the letter →

arxiv 2508.06596 v1 pith:4FH4D5LE submitted 2025-08-08 quant-ph physics.hist-ph

Miko{l}aj Sienicki , Krzysztof Sienicki This is my paper · ORCID

classification quant-phphysics.hist-ph

keywords non-NewtoniancalculusgeneralizedarithmeticarbitrarybijectionsphysicalunderdeterminationrelativisticvelocityadditionBell'stheoremmeasurementindependenceCantorfunction

topics Dark Energy

open problems Dark Energy

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

This comment targets Marek Czachor's non-Newtonian calculus, which redefines addition and multiplication by pulling them back through a bijection between a physical quantity set and the real numbers. The authors' central claim is that the formalism supplies no physical principle for choosing among the uncountably many admissible bijections, so it cannot be tested and is not physically consistent on its own. They support this with concrete failure modes: a Cantor-function bijection makes the non-Newtonian derivative undefined, the cube-function bijection turns relativistic velocity addition into a formula that gives superluminal results, the exponential bijection makes entropy undefined, the cosmology is a reparametrization of ΛCDM, and Bell-type violations trace to measurement-dependent distributions. The point of the comment is that symbolic flexibility is not physical generality: until a bijection is fixed by symmetry, invariance, or observation, the framework is a relabeling rather than a theory.

What carries the argument

The central mechanism is the arbitrary bijection $f_X: X \to \mathbb{R}$ and the pullback arithmetic $x \oplus_X y = f_X^{-1}(f_X(x)+f_X(y))$, $x \odot_X y = f_X^{-1}(f_X(x)f_X(y))$, together with the non-Newtonian derivative $\frac{DA(x)}{Dx}=f_Y^{-1}(\frac{d(A\circ f_X)}{d f_X(x)})$ and its assumption that $f_X$ is differentiable. The mechanism carries the argument because every physical prediction is conditional on $f_X$; with no rule selecting $f_X$, the same formal structure yields correct and absurd results, so the calculus cannot constrain physics on its own.

What would settle it

Read Czachor's original articles for any stated admissibility condition on $f_X$ beyond bijectivity and differentiability. If the text contains a selection rule grounded in symmetry or measurement that excludes $f_X(\beta)=\beta^3$ and the Cantor function, the comment's examples miss the intended domain; if it does not, the underdetermination claim stands. A numerical check is also direct: with $f_X(\beta)=\beta^3$, the framework's own composition rule gives $\beta\approx 1.133$ for $\beta_1=\beta_2=0.9$, which any acceptable physical theory must reject.

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Extended reading notes

Core claim

The paper claims that Czachor's non-Newtonian calculus, defined by pulling back operations through an arbitrary bijection $f_X$, is internally consistent but physically inert because the choice of $f_X$ is unconstrained. Its demonstrations are that the same framework produces Einstein's velocity composition for $f_X(\beta)=\arctanh(\beta)$ and a superluminal composition $\beta_1\oplus_X\beta_2=(\beta_1^3+\beta_2^3)^{1/3}$ for $f_X(\beta)=\beta^3$; that the entropy formula becomes undefined for $f_X(x)=e^x$; that the dark-energy-free cosmology is a reparametrization of $\Lambda$CDM; and that the Bell-type violation uses a measurement-dependent distribution, abandoning measurement independence

Load-bearing premise

The argument rests on the premise that Czachor's framework supplies no physical principle to rule out bijections like the cube function or the Cantor function; the authors assert this in the introduction and conclusion but do not search the original text for such constraints, and if constraints exist the counterexamples lose their force.

Editorial extensions

If this is right

If no principle selects the bijection, the framework is unfalsifiable: any observed law can be reproduced by choosing a suitable $f_X$.
Smooth bijections that meet the algebraic criteria can produce physically absurd results, such as superluminal velocity composition, so the formalism alone does not constrain dynamics.
Bell-type violations achieved by redefining expectation values depend on a measurement-dependent distribution, so the inequality is evaded rather than refuted.
Cosmological acceleration presented as needing no dark energy reduces to a reparametrization of $\Lambda$CDM with the constant hidden in a chosen bijection.
Non-Newtonian entropy can be undefined for legitimate probability distributions, showing that every application needs externally imposed physical input.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

A reader testing this critique should first search Czachor's own papers for implicit restrictions on the bijection, since the comment asserts an absence of constraints without systematically checking for them; this is the point where the argument stands or falls.
The same underdetermination pattern would afflict any physical formalism that redefines operations through an unrestricted invertible relabeling: unless observable statements are invariant across allowed relabelings, every law is relative to a hidden conventional choice.
A natural sharper standard suggested by the examples is that an admissible bijection must be derived from the target theory's symmetries, as $\arctanh$ arises from Lorentz geometry; under that standard the cube function is excluded on physical, not algebraic, grounds.
The appendix's Cauchy-additivity and Ohm/Kirchhoff arguments cite the authors' unpublished companion manuscript [8], so that portion of the critique cannot be independently checked from this comment alone.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

Quick read: this comment critiques Czachor's non-Newtonian calculus. It makes a legitimate broad point — without a physical principle to select a bijection, the framework is underdetermined — but that point was already made by Lambare and Pilat, both cited. What the paper adds are examples, and the examples are broken.

The velocity-addition example is the clearest problem. The paper defines the calculus via bijections f_X: X → R, but then uses f_X(β) = β^3 on [-1,1], which maps [-1,1] onto [-1,1], not onto R. When β1=β2=0.9, f(β1)+f(β2)=1.458, outside the image. f^{-1} of that value does not exist, so Eq. (4) is not a prediction of superluminal velocity; it is an undefined expression. Composing with a second bijection to fix the range changes the operation, and the advertised failure mode disappears. The Cantor function is equally inadmissible: it is not a bijection, not strictly increasing, and constant on intervals. The entropy example uses f_X=exp, which is not surjective onto R either. These flaws are not minor or fixable by editing; they are the demonstrations that are supposed to show the framework produces nonsense.

The paper does some things honestly. It cites the earlier critiques, it is clearly written, and the general worry about a missing selection principle is worth taking seriously. But the new material is not just weak; it is mathematically wrong on its own terms. The cosmology claim — that the model reduces to a reparametrization of ΛCDM — is asserted without derivation, and the paper leans on an unpublished manuscript [8] for key points, which is inaccessible and unverified. The central premise about the absence of a selection principle is not demonstrated; the paper assumes it rather than engaging Czachor's actual constraints.

Who should read this? If you want a compact summary of objections to Czachor, read Lambare and Pilat instead. This comment adds noise, not signal. It does not deserve referee time as a substantive contribution; a serious referee would catch the domain error within minutes.

Referee Report

6 major / 5 minor

Summary. This comment criticizes Czachor's non-Newtonian calculus, claiming that because it is built on arbitrary bijections f_X:X→R, it lacks a physical principle for selecting f_X and is therefore unfalsifiable, physically incoherent, and scientifically inert. The authors support this with several examples: the Cantor function supposedly breaks differentiability; the bijection f_X(β)=β³ supposedly yields superluminal velocity addition; an exponential bijection supposedly makes entropy undefined; the cosmological application is dismissed as a reparametrization of ΛCDM; and the treatment of Bell's theorem is described as semantically evasive. The comment concludes that the formalism is a 'symbolic relabeling' rather than a physical theory.

Significance. If the central critique were correct, it would be a significant challenge to Czachor's program, with implications for the interpretation of non-Newtonian calculus in physics. The paper does correctly identify a genuine question: what physical principles, if any, select the bijection in Czachor's framework? However, the manuscript's own demonstrations fail. The two main counterexamples—Cantor function and β³ velocity addition—are not admissible under the framework's stated definition of f_X as a bijection onto R. The cosmology claim is asserted without derivation, and key arguments rely on an unpublished manuscript. Thus the paper does not establish its central thesis. No new proofs, reproducible code, or falsifiable predictions are provided.

major comments (6)

[Velocity addition, Eqs. (3)–(4)] The central counterexample is invalid. The framework requires f_X to be a bijection from the physical domain X to R. The function f_X(β)=β³ maps (-1,1) to (-1,1), not onto R. For β₁=β₂=0.9, f_X(β₁)+f_X(β₂)=1.458 lies outside the image of f_X, so f_X⁻¹ is undefined and Eq. (4) has no value in the calculus. The 'superluminal' β≈1.133 is a domain error, not a prediction. A valid bijection such as f_X(β)=(artanh β)³ yields a subluminal result for the same input. The paper therefore does not construct the advertised physically absurd velocity addition.
[Cantor function, Eq. (1) and Figure 1] The Cantor function is not a bijection: it is not injective, is constant on the removed middle-thirds, and is not strictly increasing. It is nondecreasing and has derivative zero almost everywhere, but it does not map [0,1] onto R. Hence it cannot serve as an admissible f_X in Czachor's construction, and the claimed breakdown of differentiability is irrelevant to the framework. The repeated description 'strictly increasing' is factually wrong.
[Entropy, Eq. (6)] The entropy counterexample is also inadmissible. f_X(x)=eˣ is not a bijection from R to R; its range is (0,∞). Moreover, for any probability distribution, -Σ pᵢ² < 0, so f_X⁻¹=ln is undefined. The formula in Eq. (6) is not a well-defined non-Newtonian entropy for any legitimate probability distribution, so it cannot be used to show that the framework produces 'meaningless results'.
[Cosmology paragraph] The claim that Czachor's cosmological model 'turns out to be nothing more than a reparametrization of the standard ΛCDM solution' is unsupported. No equations from Ref. [1] are reproduced, and no derivation is given for the asserted relation f_X(t)∼sinh((3/2)√0.7 t). This is a load-bearing assertion for the paper's conclusion that the framework has no new predictive power, and it is neither demonstrated nor referenced to a specific section of Czachor's paper.
[Bell-type theorems and Appendix] The paper's treatment of Bell's theorem relies on Lambare's analysis [5] and on the authors' own unpublished manuscript [8]. The unpublished source is not accessible to readers and cannot provide independent verification. The argument that non-Newtonian expectation values 'sidestep' Bell's inequality is presented as established, but the manuscript does not derive the expectation-value formula or show why it constitutes evasion rather than a different model. This weakens the broader claim that the framework is scientifically inert.
[Introduction and conclusion, underdetermination] The central charge that the calculus is 'arbitrary by construction' assumes that Czachor's framework places no constraints on admissible bijections. The paper never examines [1] for such constraints—e.g., symmetry, invariance, or consistency requirements. Without this examination, the conclusion that the framework is unfalsifiable is not established. The underdetermination critique may be defensible in a modified form, but this manuscript does not supply the needed argument.

minor comments (5)

[Eq. (8), Appendix] The displayed equation f_X(x+y)=f_X(x)⊕f_X(y) mixes the definition of ⊕_X with a property. The standard pullback addition is x⊕_X y = f_X⁻¹(f_X(x)+f_X(y)). Please clarify whether Eq. (8) is meant as a definition or as a claimed identity.
[Footnote 1] The footnote notes that f_X(β)=βⁿ is bijective on [-1,1] only for odd n, but even for odd n the image is [-1,1], not R. This does not satisfy the framework's f_X:X→R condition; the footnote should acknowledge this.
[Eq. (7)] The text says 'nowhere ⊙ denotes a non-Newtonian product'; this should be 'where'.
[Figure 1 caption] The caption repeats the incorrect statement that the Cantor function is 'strictly increasing' on [0,1]. The function is nondecreasing and constant on intervals.
[Abstract] The abstract states the Cantor function breaks 'core assumptions' of the framework. Since the Cantor function is not a bijection, it is not an admissible f_X; this claim should be corrected or removed.

Circularity Check

2 steps flagged · score 6.0 of 10

Central 'arbitrariness' charge presupposes the absence of a selection principle, relies on invalid counterexamples, and invokes an unpublished self-citation.

self definitional [Comment, opening paragraph (p. 1) and closing paragraph (p. 4)]
"There are uncountably many such bijections between continuum-sized sets, and without a physical principle to narrow the field, the calculus turns arbitrary by construction. ... It offers no principle for choosing among the vast set of possible bijections, no way to test its predictions empirically, and no assurance of physical consistency."

The paper's central conclusion—that Czachor's framework offers no principle for choosing a bijection—is exactly the antecedent it inserts at the start ('without a physical principle...'). The absence of such a principle is never demonstrated from Czachor's text; it is assumed, and then reported as a finding. The two headline counterexamples (f_X=β^3 and the Cantor function) do not satisfy the framework's own stated requirements (f_X:X→R and differentiability), so they cannot provide the missing evidence. The claim of arbitrariness is therefore not derived from the framework but is presupposed and restated: the 'derivation' reduces to the assumption.
self citation load bearing [Appendix, sections 1–4 (pp. 6–7)]
"But as we've shown [8], this use of the harmonic mean naturally emerges from linear relationships, not from any departure from ordinary arithmetic. ... As discussed in [8], this breakdown undercuts the interpretation of ⊕X as a meaningful composition law in physics. ... That's not scientific explanation; that's symbolic relabeling [8]."

The appendix's key supports—that Czachor misreads harmonic mean, that the Cauchy additivity failure undermines ⊕X, and that the framework is 'symbolic relabeling'—are all attributed to the authors' own unpublished manuscript [8]. No argument from [8] is reproduced or independently verified. Since these claims underpin the paper's broader verdict that the framework is 'scientifically inert,' the critique leans on a self-referential, inaccessible prior work. The negative conclusion is thereby supported by the authors' own assertion rather than by verifiable analysis, making the citation load-bearing and circular.

full rationale

This is a comment paper rather than a novel derivation, so circularity must be assessed in its critique. The central charge—that the non-Newtonian calculus is arbitrary because no bijection-selection principle exists—is not established. The opening conditional ('without a physical principle... arbitrary by construction') presupposes the very absence that the later paragraphs present as a conclusion. The advertised counterexamples fail the framework's own definitions: f_X(β)=β^3 maps [-1,1] onto [-1,1], not onto R, so the 'superluminal' β≈1.133 is an undefined operation, not a valid prediction; the Cantor function is not a bijection and is not differentiable in the required sense. These are correctness defects that leave the underdetermination thesis unsupported, but they are not themselves circular. The genuine circularity is twofold: the thesis is assumed in its own premise, and the appendix leans on the authors' unpublished self-citation [8] for load-bearing points. Those factors warrant a score of 6 rather than a lower score. The paper does cite independent sources (Lambare [5], Pilat [6]) for the Bell-inequality portion, so the work is not wholly self-referential; this prevents a higher circularity score.

Assumptions & free parameters 0 free parameters · 3 assumptions · 0 invented entities

The comment's argument depends on the premises that physical theories need a unique bijection and that Czachor imposes no restrictions on bijections; both are asserted rather than proven. The Cantor function example incorrectly assumes a non-bijection is admissible, and the reliance on the authors' unpublished work further weakens the independent grounding.

assumptions (3)

domain assumption A physical theory must provide a principle for choosing a unique bijection among uncountably many to be predictive.
This premise underlies the entire critique. It is stated in the introduction and repeated in the conclusion, but it is not derived from any standard philosophy of science or from a direct analysis of Czachor's framework.
domain assumption Czachor's framework treats all bijections equally and imposes no constraints on their physical admissibility.
The comment assumes the framework allows any bijection without restriction. It does not examine Czachor's own conditions for admissible bijections or his discussion of physical principles such as symmetry and invariance.
ad hoc to paper The Cantor function is an admissible bijection in the framework.
The comment presents the Cantor function as a counterexample, but the Cantor function is not injective and therefore not a bijection. This is a factual error that invalidates that specific example.

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Cite this review

Pith. "Pith review of Comment on "Unifying Aspects of Generalized Calculus"." pith.science (2026). https://pith.science/paper/4FH4D5LE

@misc{pith2026250806596,
  author       = {Pith},
  title        = {Pith review of: Comment on "Unifying Aspects of Generalized Calculus"},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/4FH4D5LE}},
  note         = {Machine review of arXiv:2508.06596}
}

read the original abstract

Czachor's recent proposal introduces a form of non-Newtonian calculus built by pulling back arithmetic operations through arbitrary bijections between continua. Although the idea is mathematically inventive, it runs into serious conceptual trouble when examined from a physical standpoint. Claims of universal applicability quickly unravel under scrutiny -- especially when considering pathological bijections like the Cantor function, which break the framework's core assumptions. When applied to domains such as relativity, entropy, or cosmology, the results often collapse into tautological restatements lacking real predictive power. This commentary explores these issues in depth, highlighting where and why the formalism falls short of providing a physically coherent theory.

Figures

Figures reproduced from arXiv: 2508.06596 by the authors.

**Figure 1.** The Cantor function C(x), used here to illustrate a breakdown in the assumptions underlying non-Newtonian calculus. Although continuous and strictly increasing on [0, 1], it is constant on large intervals, satisfies C ′ (x) = 0 almost everywhere, and is nondifferentiable on a dense set. The non-Newtonian derivative DA(x) Dx = f −1 Y dA◦C(x) dC(x) becomes ill-defined across most of the domain. This breakdown beco… view at source ↗

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

10 extracted references · 10 canonical work pages

[8]

Sienicki, M., & Sienicki, K. (2025). A Critical Analysis of Non-Diophantine Arithmetic and its Misinterpretations in Physics. Unpublished manuscript

work page 2025
[1]

Unifying Aspects of Generalized Calculus

Czachor, Marek. “Unifying Aspects of Generalized Calculus.” Entropy 22, no. 11 (2020):

work page 2020
[5]

Comment on "A Loophole of All "Loophole-Free" Bell-Type Theorems"

Lambare, Justo Pastor. ”Comment on “a loophole of all “loophole-free” Bell-type theorems”.” Foundations of Science 26, no. 4 (2021): 917-924. https://arxiv.org/ pdf/2008.00369 see also Czachor, Marek. ”Response to Comment on” A Loop- hole of All” Loophole-Free” Bell-Type Theorems”, by JP Lambare.” arXiv preprint arXiv:2008.11910 (2020).https://arxiv.org/p...

work page Pith review arXiv 2021
[2]

”Relativity of arithmetic as a fundamental symmetry of physics.” Quantum studies: mathematics and foundations 3, no

Czachor, Marek. ”Relativity of arithmetic as a fundamental symmetry of physics.” Quantum studies: mathematics and foundations 3, no. 2 (2016): 123-133. https: //link.springer.com/content/pdf/10.1007/s40509-015-0056-4.pdf

work page doi:10.1007/s40509-015-0056-4.pdf 2016
[3]

”A loophole of all ‘loophole-free’Bell-type theorems.” Foundations of Science 25, no

Czachor, Marek. ”A loophole of all ‘loophole-free’Bell-type theorems.” Foundations of Science 25, no. 4 (2020): 971-985. https://link.springer.com/content/pdf/10. 1007/s10699-020-09666-0.pdf

work page 2020
[4]

Contra Bellum: Bell's theorem as a confusion of languages

Czachor, Marek. ”Contra Bellum: Bell’s theorem as a confusion of languages.” arXiv preprint arXiv:2301.10727 (2023).https://arxiv.org/pdf/2301.10727

work page Pith review arXiv 2023
[6]

”Bell-type inequalities from the perspective of Non-Newtonian cal- culus.” Foundations of Science 29, no

Pi lat, Micha l Piotr. ”Bell-type inequalities from the perspective of Non-Newtonian cal- culus.” Foundations of Science 29, no. 2 (2024): 441-457. https://link.springer. com/content/pdf/10.1007/s10699-022-09866-w.pdf

work page doi:10.1007/s10699-022-09866-w.pdf 2024
[7]

Czachor, M. (2016). Relativity of Arithmetic as a Fundamental Symmetry of Physics. Quantum Studies: Mathematics and Foundations, 3(2), 123–133. https://doi.org/ 10.1007/s40509-015-0056-4

work page doi:10.1007/s40509-015-0056-4 2016

Show all 10 references

[9]

”A non-Newtonian Noether’s symmetry theorem.” Applicable Anal- ysis 102, no

Torres, Delfim FM. ”A non-Newtonian Noether’s symmetry theorem.” Applicable Anal- ysis 102, no. 7 (2023): 1934-1941. https://arxiv.org/pdf/2111.11559 8

2023 arXiv
[1180]

https://www.mdpi.com/1099-4300/22/10/1180

Pith tools

Reviewed August 5, 2026 · model on record in the stance chip above.

[1] [8]

Sienicki, M., & Sienicki, K. (2025). A Critical Analysis of Non-Diophantine Arithmetic and its Misinterpretations in Physics. Unpublished manuscript

work page 2025

[2] [1]

Unifying Aspects of Generalized Calculus

Czachor, Marek. “Unifying Aspects of Generalized Calculus.” Entropy 22, no. 11 (2020):

work page 2020

[3] [5]

Comment on "A Loophole of All "Loophole-Free" Bell-Type Theorems"

Lambare, Justo Pastor. ”Comment on “a loophole of all “loophole-free” Bell-type theorems”.” Foundations of Science 26, no. 4 (2021): 917-924. https://arxiv.org/ pdf/2008.00369 see also Czachor, Marek. ”Response to Comment on” A Loop- hole of All” Loophole-Free” Bell-Type Theorems”, by JP Lambare.” arXiv preprint arXiv:2008.11910 (2020).https://arxiv.org/p...

work page Pith review arXiv 2021

[4] [2]

”Relativity of arithmetic as a fundamental symmetry of physics.” Quantum studies: mathematics and foundations 3, no

Czachor, Marek. ”Relativity of arithmetic as a fundamental symmetry of physics.” Quantum studies: mathematics and foundations 3, no. 2 (2016): 123-133. https: //link.springer.com/content/pdf/10.1007/s40509-015-0056-4.pdf

work page doi:10.1007/s40509-015-0056-4.pdf 2016

[5] [3]

”A loophole of all ‘loophole-free’Bell-type theorems.” Foundations of Science 25, no

Czachor, Marek. ”A loophole of all ‘loophole-free’Bell-type theorems.” Foundations of Science 25, no. 4 (2020): 971-985. https://link.springer.com/content/pdf/10. 1007/s10699-020-09666-0.pdf

work page 2020

[6] [4]

Contra Bellum: Bell's theorem as a confusion of languages

Czachor, Marek. ”Contra Bellum: Bell’s theorem as a confusion of languages.” arXiv preprint arXiv:2301.10727 (2023).https://arxiv.org/pdf/2301.10727

work page Pith review arXiv 2023

[7] [6]

”Bell-type inequalities from the perspective of Non-Newtonian cal- culus.” Foundations of Science 29, no

Pi lat, Micha l Piotr. ”Bell-type inequalities from the perspective of Non-Newtonian cal- culus.” Foundations of Science 29, no. 2 (2024): 441-457. https://link.springer. com/content/pdf/10.1007/s10699-022-09866-w.pdf

work page doi:10.1007/s10699-022-09866-w.pdf 2024

[8] [7]

Czachor, M. (2016). Relativity of Arithmetic as a Fundamental Symmetry of Physics. Quantum Studies: Mathematics and Foundations, 3(2), 123–133. https://doi.org/ 10.1007/s40509-015-0056-4

work page doi:10.1007/s40509-015-0056-4 2016