{"id":"56a7889f-178c-4151-9e11-cbca4295ac22","arxiv_id":"2508.06596","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":3.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"Czachor's non-Newtonian calculus is criticized as unfalsifiable and physically inert, but the criticism is undermined by a mischaracterization of the Cantor function and unsupported claims.","lead":"This comment argues that a proposed framework called non-Newtonian generalized calculus cannot be a predictive physical theory because it allows arbitrary mathematical choices. It gives counterexamples, but one key example uses a function that is not actually a bijection, weakening the critique.","discovery_kind":"incremental","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Counterexamples violate the framework's own definition: f_X=β^3 is not a map X→R, so the superluminal result is an undefined operation, not a valid prediction; the underdetermination charge is unsupported.","rationale":"The reader's weakest assumption was that the comment never engages Czachor's actual constraints on admissible bijections. My stress test identifies a more specific, internal defect: the comment's own counterexamples violate the formal definition it states. f_X(β)=β^3 is not surjective onto R, so the non-Newtonian sum is undefined for inputs whose f-values sum outside the range; the quoted superluminal value is thus not a result of Czachor's calculus. The Cantor function is not a bijection at all and therefore irrelevant. Because these are the paper's primary demonstrations of physical inconsistency, the central conclusion is not supported as written. This does not prove Czachor's framework is physically sound; it shows the comment fails to make its case. A repaired comment would use admissible bijections and directly examine Czachor's selection principles. Thus the paper should be rejected in its current form, even though a revision might salvage the broader critique.","tokens_in":5713,"tokens_out":8817,"duration_ms":107242,"concrete_test":"Recompute the velocity-composition counterexample with an admissible bijection: take X=(-1,1), f_X(β)=(artanh β)^3, which is a bijection X→R with real cube roots. For β1=β2=0.9, compute f_X^{-1}(f_X(β1)+f_X(β2)). If the result lies in (-1,1), the 'superluminal' outcome is an artifact of using a non-surjective f_X rather than a property of the non-Newtonian calculus. Also inspect Czachor (2020), Definition 1, to confirm f_X is required to map X onto R; if so, the comment's cube and Cantor examples are outside the formalism.","verdict_should_be":"REJECT","load_bearing_attack":"The paper's central claim—that Czachor's calculus permits physically absurd results because bijections are arbitrary—rests on two key examples, and both fail the framework's stated requirements. The comment itself defines the construction via bijections f_X:X→R (Section 1). For velocity addition, f_X(β)=β^3 is not such a bijection: it maps (-1,1) onto (-1,1), not onto R. Consequently, when β1=β2=0.9, f(β1)+f(β2)=1.458, which lies outside the range of f_X, so f_X^{-1} is undefined and Eq. (4) has no value in the calculus. β≈1.133 is not a prediction; it is a domain error. The Cantor function example is similarly inadmissible: it is not a bijection, is not strictly increasing, and is constant on intervals. Since these are the demonstrations meant to show 'dynamics that violate well-established empirical boundaries', the conclusion that the framework is physically incoherent is unsupported. The underdetermination claim may still be true, but this paper does not establish it. An admissible bijection such as f(β)=(artanh β)^3 yields a different but always subluminal addition law, so the specific failure mode advertised in Eqs. (3)-(4) disappears.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","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.","tokens_in":6003,"tokens_out":5590,"duration_ms":66109,"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":[{"comment":"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.","section":"Velocity addition, Eqs. (3)–(4)"},{"comment":"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.","section":"Cantor function, Eq. (1) and Figure 1"},{"comment":"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'.","section":"Entropy, Eq. (6)"},{"comment":"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.","section":"Cosmology paragraph"},{"comment":"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.","section":"Bell-type theorems and Appendix"},{"comment":"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.","section":"Introduction and conclusion, underdetermination"}],"minor_comments":[{"comment":"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.","section":"Eq. (8), Appendix"},{"comment":"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.","section":"Footnote 1"},{"comment":"The text says 'nowhere ⊙ denotes a non-Newtonian product'; this should be 'where'.","section":"Eq. (7)"},{"comment":"The caption repeats the incorrect statement that the Cantor function is 'strictly increasing' on [0,1]. The function is nondecreasing and constant on intervals.","section":"Figure 1 caption"},{"comment":"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.","section":"Abstract"}],"recommendation":"reject","confidential_remarks":"The manuscript is a comment on a published paper, but its two main counterexamples are mathematically invalid under the framework's own definitions, and the cosmology claim is unsubstantiated. The paper also leans on an unpublished manuscript [8] for key arguments. The errors are not local typos; they undermine the central thesis. I would recommend rejection, though a substantially rewritten version—with valid admissible-bijection examples, a rigorous treatment of any actual constraints in Czachor's framework, and a derivation of the cosmology claim—might be worth considering as a new submission."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","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.\n\nThe 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.\n\nThe 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.\n\nWho 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.","headline":"The paper's broad worry about arbitrary bijections is real, but its new counterexamples violate the framework's own definitions, so the case collapses.","tokens_in":6499,"tokens_out":4785,"would_cite":false,"duration_ms":53988,"reading_group":"no","serious_thinker":"no","would_accept_peer_review":false},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":[],"pacs":[],"model":"deepseek-v4-flash","headline":"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.","keywords":["non-Newtonian calculus","generalized arithmetic","arbitrary bijections","physical underdetermination","relativistic velocity addition","Bell's theorem","measurement independence","Cantor function"],"falsifier":"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.","tokens_in":5581,"feed_emoji":"⚛️","tokens_out":14468,"duration_ms":124631,"temperature":0.7,"pith_summary":"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.","feed_headline":"Arbitrary bijections make generalized calculus scientifically inert","feed_subtitle":"Even smooth bijections like the cube function yield superluminal velocities, so an external physical rule is needed","key_machinery":"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.","core_discovery":"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","pith_inferences":["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."],"forward_implications":["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."],"supporting_citations":[{"why":"Presents the non-Newtonian calculus under critique: pullback arithmetic through bijections, the derivative formula, and the physical applications the comment attacks.","marker":"[1]"},{"why":"Earlier statement of the 'relativity of arithmetic as a fundamental symmetry' claim that the comment treats as ungrounded.","marker":"[2]"},{"why":"Czachor's Bell-type theorem papers that the comment argues sidestep the inequality by redefining expectation values.","marker":"[3, 4]"},{"why":"Lambare's independent comment showing the modified expectation values violate measurement independence, a key support for the Bell critique.","marker":"[5]"},{"why":"Pilat's critical reappraisal, used to acknowledge that the Bell violation is symbolic and assumption-shifting, reinforcing the main critique.","marker":"[6]"},{"why":"The authors' own companion manuscript supplying the Cauchy-additivity and harmonic-mean arguments in the appendix.","marker":"[8]"},{"why":"Torres's non-Newtonian Noether theorem, cited to distinguish formal mathematical generalization from physical content.","marker":"[9]"}],"fun_headline_variants":["Generalized calculus fails physics test: arbitrary bijections","Arbitrary bijections render non-Newtonian calculus physically useless","Comment: generalized calculus is tautological when applied to physics","Czachor's calculus: mathematically cute, physically inert","Non-Newtonian calculus unravels under physical scrutiny"],"cache_read_input_tokens":2816,"weakest_assumption_plain":"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.","fun_headline_variants_meta":{"raw":{"variants":["Generalized calculus fails physics test: arbitrary bijections","Arbitrary bijections render non-Newtonian calculus physically useless","Comment: generalized calculus is tautological when applied to physics","Czachor's calculus: mathematically cute, physically inert","Non-Newtonian calculus unravels under physical scrutiny"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000595,"raw_usage":{"total_tokens":2575,"prompt_tokens":652,"completion_tokens":1923,"prompt_tokens_details":{"cached_tokens":256},"prompt_cache_hit_tokens":256,"prompt_cache_miss_tokens":396,"completion_tokens_details":{"reasoning_tokens":1839}},"tokens_in":396,"tokens_out":1923,"duration_ms":16117,"temperature":1.0,"reasoning_tokens":1839,"cache_read_input_tokens":256,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-05T22:44:11.370165+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"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.","supporting_citations":[{"cited_title":"Unifying Aspects of Generalized Calculus","cited_arxiv_id":null,"evidence_quote":"Presents the non-Newtonian calculus under critique: pullback arithmetic through bijections, the derivative formula, and the physical applications the comment attacks."},{"cited_title":"”Relativity of arithmetic as a fundamental symmetry of physics.” Quantum studies: mathematics and foundations 3, no","cited_arxiv_id":null,"evidence_quote":"Earlier statement of the 'relativity of arithmetic as a fundamental symmetry' claim that the comment treats as ungrounded."},{"cited_title":"Comment on \"A Loophole of All \"Loophole-Free\" Bell-Type Theorems\"","cited_arxiv_id":"2008.00369","evidence_quote":"Lambare's independent comment showing the modified expectation values violate measurement independence, a key support for the Bell critique."},{"cited_title":"”Bell-type inequalities from the perspective of Non-Newtonian cal- culus.” Foundations of Science 29, no","cited_arxiv_id":null,"evidence_quote":"Pilat's critical reappraisal, used to acknowledge that the Bell violation is symbolic and assumption-shifting, reinforcing the main critique."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"The authors' own companion manuscript supplying the Cauchy-additivity and harmonic-mean arguments in the appendix."},{"cited_title":"A Non-Newtonian Noether's Symmetry Theorem","cited_arxiv_id":"2111.11559","evidence_quote":"Torres's non-Newtonian Noether theorem, cited to distinguish formal mathematical generalization from physical content."}],"review_version":1}