REVIEW 3 major objections 2 minor 2 cited by
On $q$-real and $q$-complex numbers
T0 review · 3 major / 2 minor · reviewed 2026-08-05 · deepseek-v4-flash
Pith's one-line read This paper proves that for every real \(x>1\), the \(q\)-real series \([x]_q\) converges in the disk \(|q|<3-2\sqrt2\) to a nonvanishing holomorphic function, partially proving the conjecture that the common radius of convergence should be
desk verdict The abstract advertises a credible partial resolution of the Morier-Genoud–Ovsienko conjecture, but the supplied full text is a different paper, so the proof is absent and the claims are unverifiable as presented. read the letter →
The pith
A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.
The reading
What carries the argument
The load-bearing object is the expansion of the reciprocal \(1/[x]_q\) into a \(q\)-adically convergent series of rational functions. This expansion carries the argument because it turns a formal Laurent series whose convergence is delicate into a sum of explicit rational functions whose absolute and uniform convergence on compact subsets can be estimated, and it is what yields nonvanishing and positivity.
What would settle it
For \(x=\sqrt2\), compute the partial sums of the reciprocal expansion at \(q=0.17\); if they fail to form a Cauchy sequence while the original series coefficients remain controlled, the claimed uniform convergence in \(|q|<3-2\sqrt2\) is false.
Extended reading notes
Core claim
The central claim is that convergence of \(q\)-real numbers is not a special property of rationals. For every real \(x>1\), the Laurent series \([x]_q\) converges to a nonvanishing holomorphic function on \(|q|<3-2\sqrt2\); the mechanism is an explicit expansion of \(1/[x]_q\) into rational functions that converges \(q\)-adically and uniformly on compact sets. The paper also shows the expansion converges to a positive analytic function on \((-(3-\sqrt5)/2,1)\), so \([x]_q\) has a real meaning for those \(q\), and reports that a cited result implies convergence on the larger disk \(|q|<2-\sqrt3\). Finally, a \(q\)-complex version \([\tau]_q\) is proposed as a meromorphic function of \(\tau\in
Load-bearing premise
The proof rests on the claimed absolute and uniform convergence of the expansion of \(1/[x]_q\) into a series of rational functions for every real \(x>1\); if that uniformity fails for even one real \(x\), the main conclusion collapses.
Editorial extensions
If this is right
- For every \(x>1\), \([x]_q\) becomes a genuine holomorphic function on \(|q|<3-2\sqrt2\), so zeros, derivatives, and special values become meaningful questions.
- Nonvanishing in the disk makes \(1/[x]_q\) holomorphic there as well, opening reciprocal identities and integral formulas.
- The positive extension on \((-(3-\sqrt5)/2,1)\) gives a real-valued interpolation of \(q\)-real numbers for a whole interval of \(q\), not just a formal power series.
- The reported corollary broadens the uniform convergence disk to \(|q|<2-\sqrt3\), which is numerically useful.
- If the proposed \(q\)-complex number is coherent, it connects continued-fraction \(q\)-deformations to modular and hypergeometric functions of a complex parameter.
Reading between the lines
- The method suggests that the conjectured optimal radius \((3-\sqrt5)/2\) could be reached by tracking the boundary of the region \(D\); the golden ratio is the natural extremal case where the expansion may first fail.
- The positive analytic extension on the real interval may admit an integral representation, which would make numerical evaluation and possibly combinatorial interpretations more direct.
- The \(q\)-complex construction hints that \(q\)-deformed continued fractions may be specializations of modular or hypergeometric objects; testing special values at arithmetic points could reveal structure.
- The full text supplied alongside this record is a different manuscript on locally associated orders in quadratic fields; none of its statements concern \(q\)-real or \(q\)-complex numbers, so this summary is drawn from the abstract and the cited prior works.
Editorial analysis
A structured set of objections, weighed in public.
Referee Report
Summary. The abstract advertises a proof that, for every real x > 1, the q-real series [x]_q converges in the disk |q| < 3−2√2 to a nonvanishing holomorphic function, via an expansion of 1/[x]_q into a q-adically convergent series of rational functions, uniformly on compact sets in an explicit region D. It further claims convergence to a positive analytic function on (−(3−√5)/2, 1), a strengthening to |q| < 2−√3 using results from arXiv:2405.15970, explicit computations such as x = cotan(1), and a definition of a q-complex number [τ]_q. However, the supplied full text is a completely different paper, arXiv:2508.08447, titled "Locally Associated Orders in Real Quadratic Number Fields" by Grant Moles and Talha Khan. None of the advertised content — no definition of [x]_q, no expansion of 1/[x]_q, no region D, no convergence estimates, no examples, no q-complex construction — appears in the body of the submitted manuscript.
Significance. If the advertised results are correct, they would constitute a substantial step toward the Morier-Genoud–Ovsienko conjecture: establishing convergence of [x]_q for all real x > 1 in a disk of radius 3−2√2 is a partial resolution with an explicit uniform radius, and the proposed q-complex numbers would be a new construction linking q-deformations to modular and hypergeometric functions. The claimed connection to the Kleinian-group result of arXiv:2405.15970 would also be noteworthy. However, because the submitted manuscript does not contain the advertised theorems or any supporting argument, the significance cannot be assessed from the provided text. The claimed results are contingent on a proof that is absent.
major comments (3)
- [Abstract vs. full text] The full text supplied for this submission is arXiv:2508.08447, an unrelated paper on locally associated orders in real quadratic number fields. The abstract promises a proof of convergence of [x]_q and an expansion of 1/[x]_q, but the body contains no such statement, no definition of [x]_q, no region D, no uniform-convergence estimates, and no derivation. This is not a local gap: the central theorem of the advertised paper has no visible proof in the manuscript.
- [Abstract, fourth sentence] The claim that the result of arXiv:2405.15970 implies convergence of [x]_q for |q| < 2−√3 ≈ 0.27 is asserted with no argument. Since the body does not even state the relevant theorem, this implication cannot be verified. This is a load-bearing claim of the abstract, not a mere aside.
- [Abstract, final sentence] The proposed q-complex number [τ]_q, described as a meromorphic function on the upper half-plane expressed via hypergeometric functions of modular functions, is not defined anywhere in the supplied full text. No formula, domain of meromorphy, or relationship to the q-real series is given.
minor comments (2)
- [Title and metadata] The manuscript title and arXiv identifier in the abstract do not match the body's title and arXiv identifier. This mismatch will cause confusion and must be corrected in any resubmission.
- [References] The bibliography of the supplied full text is for the unrelated factorization paper; it contains none of the q-deformation references cited in the abstract (arXiv:1812.00170, arXiv:1908.04365, arXiv:2102.00891, arXiv:2405.15970).
Circularity Check
No circularity is demonstrable: the supplied full text is a different arXiv paper (2508.08447), so the claimed q-real derivation is absent rather than circular.
full rationale
The manuscript supplied under arXiv:2508.08440 is not the advertised paper: after the q-real abstract, the body is arXiv:2508.08447v3 [math.AC], 'Locally Associated Orders in Real Quadratic Number Fields', by Grant Moles and Talha Khan. Within the supplied text there is no q-deformed continued fraction, no expansion of 1/[x]_q, no explicit region D, no uniform-convergence estimates, no positive analytic extension to (-(3−√5)/2,1), and no hypergeometric/modular construction of [τ]_q. The claimed derivation chain is therefore missing, not present in a form that could be checked for circularity. The abstract's convergence claim is presented as anchored to external prior work (arXiv:2102.00891 and arXiv:2405.15970) and to the Morier-Genoud–Ovsienko definition of [x]_q; no equation in the abstract or body reduces the theorem to its own inputs by construction, and no fitted parameter is renamed as a prediction. The problem is evidentiary—the central proof is absent—rather than circular. Under the stated circularity definition (equivalence by construction, fitted input called prediction, or a load-bearing self-citation chain), no circular step can be identified, so the circularity score is 0.
Assumptions & free parameters
assumptions (3)
- domain assumption The Morier-Genoud and Ovsienko definition of q-rational and q-real numbers [x]_q (q-deformation of continued fractions) provides a well-defined object with the algebraic identities used in the expansion of 1/[x]_q.
- domain assumption Prior convergence results for [x]_q are correct: rational x converge in |q| < 3−2√2 (arXiv:2102.00891) and the Kleinian-groups proof for rational x (arXiv:2405.15970) is valid.
- standard math For the q-complex proposal, standard analytic properties of hypergeometric functions and modular functions of tau hold as in the literature.
invented entities (1)
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q-complex number [tau]_q, a meromorphic function of tau in the upper half-plane
Cite this review
Pith. "Pith review of On $q$-real and $q$-complex numbers." pith.science (2026). https://pith.science/paper/UGLULHUB
@misc{pith2026250808440,
author = {Pith},
title = {Pith review of: On $q$-real and $q$-complex numbers},
year = {2026},
howpublished = {\url{https://pith.science/paper/UGLULHUB}},
note = {Machine review of arXiv:2508.08440}
}
abstract
In arXiv:1812.00170 and arXiv:1908.04365, Morier-Genoud and Ovsienko introduced $q$-rational numbers $[x]_q$, rational functions specializing to $x$ at $q=1$, and their extension to $q$-real numbers, Laurent series agreeing with $[x]_q$ for rational $x$. It was conjectured in arXiv:2102.00891 that for every real $x>1$, $[x]_q$ has positive radius of convergence, with optimal common radius $R_*=(3-\sqrt{5})/2$, attained at the golden ratio. We prove that $[x]_q$ converges to a nonvanishing holomorphic function for every real $x>1$ and $|q|<3-2\sqrt{2}$. The proof gives an expansion of $1/[x]_q$ as a $q$-adically convergent series of rational functions, converging absolutely and locally uniformly on an explicit region. It also defines a positive analytic function for $q\in(-R_*,1)$. Using arXiv:2405.15970, we further obtain convergence for $|q|<2-\sqrt{3}$. We compute $[x]_q$ explicitly for some transcendental $x$, including ${\rm cotan}(1)$ and $e$. We also establish sharp inequalities for numerators and denominators of $q$-rationals when $|q|=1$ and determine the closure of the set of $[x]_q$ when such $q$ is not a root of unity. Next, we show that coefficientwise reduction modulo every $m\ge2$ is injective on the Cantor line (the extended real line with doubled up rationals and the Cantor set topology); for $m=2$, it identifies the Cantor line with $\mathbb P^1(\mathbb F_2((q)))$. We describe the inverse map, extend rationality results modulo every prime, characterize quadratic series over $\mathbb F_2(q)$ corresponding to quadratic irrationals, derive criteria for eventual parity of coefficients, and compute the real numbers corresponding to $1+q^n$. Finally, we propose a definition of $q$-complex number $[\tau]_q$, a meromorphic function in $\tau\in\mathbb C_+$ expressed via hypergeometric functions evaluated at modular functions of $\tau$.
Forward citations
Cited by 2 Pith papers
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Coefficients of $q$-real numbers: their combinatorial meaning and growth
Every coefficient of every q-real number between 1 and 2 is bounded in absolute value by the corresponding coefficient of the q-deformed golden ratio, resolving the radius-of-convergence conjecture.
-
Dimers, filters, and $q$-deformed real numbers
Every positive real x gets a snake-graph dimer model whose distinguished-edge odds define a new q-deformation [[x]]_q, equal to q[x]_q for rational x.
Reviewed August 5, 2026 · model on record in the stance chip above.
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