REVIEW 3 major objections 5 minor 9 references
On $q$-deformed real numbers
T0 review · 3 major / 5 minor · reviewed 2026-08-14 · deepseek-v4-flash
Pith's one-line read Every irrational number gets a unique q-deformed power series with integer coefficients.
desk verdict A novel q-deformation of real numbers with attractive examples, but the main proof has a parity mistake that needs fixing before publication. 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 engine is the weighted Farey graph, on which q-rationals are defined by the recursive rule assigning weights $q^\ell$ to edges, together with the equivalent $2\times 2$ matrix representation of convergents. The decisive identity is the determinant formula $R_n S_{n-1} - S_n R_{n-1} = q^{a_1+\cdots+a_n-1}$, which controls exactly how many Taylor coefficients two neighboring convergents share. Lemma 3.1 transfers this control to every rational between two convergents by drawing a vertical line in the Poincaré half-plane and collecting the crossed Farey triangles, enabling the stabilization proof.
What would settle it
Compute the q-deformed Taylor series of a rational lying strictly between two consecutive convergents of some irrational, for instance between the convergents of $\sqrt{2}$, and check whether its first $a_1+\cdots+a_m-1$ coefficients equal those of both convergents; a single counterexample would falsify Lemma 3.1 and with it Theorem 1.
Extended reading notes
Core claim
The central discovery is a stabilization theorem: for every irrational $x\ge 1$, the Taylor coefficients of the q-deformed convergents $[x_n]_q$ eventually agree coefficient by coefficient, and the resulting power series $[x]_q$ has integer coefficients independent of the rational sequence approximating $x$. The proof gives a quantitative form: two consecutive convergents share every term up to $q^{a_1+\cdots+a_n-1}$, and every rational between them shares those same initial terms. For quadratic irrationals the paper derives explicit functional equations; for the golden ratio the coefficients are, up to alternating signs, the generalized Catalan numbers.
Load-bearing premise
The proof of the key lemma assumes that every rational between two consecutive convergents can be reached from both by a finite chain of adjacent Farey triangles, justified by a vertical-line argument that is stated rather than fully proved; if this connectivity assertion fails, the stabilization theorem lacks proof.
Editorial extensions
If this is right
- For any irrational $x\ge 1$, the formal power series $[x]_q$ is well defined and has integer coefficients, extending the quantization map from rationals to reals.
- Truncating at the $n$th convergent gives an approximation of $[x]_q$ that is accurate up to degree $a_1+\cdots+a_n-1$, so stabilization comes with a quantitative convergence rate.
- The translation rules $[x+1]_q=q[x]_q+1$ and $[x-1]_q=([x]_q-1)/q$ extend the definition to all real numbers; for $x<1$ these are Laurent series, and Theorem 2 describes the initial string of $1$s and the first gap.
- The q-deformations of quadratic irrationals satisfy explicit quadratic functional equations, such as $q[\varphi]_q^2-(q^2+q-1)[\varphi]_q-1=0$ for the golden ratio.
- For the golden ratio, the coefficients of $[\varphi]_q$ are, up to sign, the generalized Catalan numbers, connecting the construction to a known combinatorial sequence.
Reading between the lines
- The direction asymmetry noted for rational limits—right-approaching sequences recover the q-rational while left-approaching sequences do not—suggests that q-deformation carries an ordering or orientation datum; one could test whether this matches a canonical direction in the Farey tessellation.
- The functional equations for $\sqrt{2},\sqrt{3},\sqrt{5},\sqrt{7}$ all have the same shape, $q^d[\sqrt{n}]_q^2$ minus a polynomial times $[\sqrt{n}]_q$ equals a palindromic polynomial; presumably every quadratic irrational admits such an equation, and searching directly from the period of its continued fraction would settle the authors' open question.
- The apparent 7-periodicity in the coefficients of $[e]_q$ and the isolated vanishing coefficient in $[\pi]_q$ may be artifacts of few terms or signs of deeper structure; extending the computations and comparing with other constants whose continued fractions obey periodic patterns would distinguish these possibilities.
Signed reviews
Editorial analysis
A structured set of objections, weighed in public.
Referee Report
Summary. The paper proposes a q-deformation of positive real numbers as the coefficient-wise limit of Taylor expansions of q-deformed rational convergents. For an irrational x ≥ 1, the authors associate a formal power series [x]_q with integer coefficients, claim that the Taylor coefficients of [x_n]_q stabilize independently of the approximating sequence (Theorem 1), and extend the construction to negative reals by translation to obtain Laurent series. The paper also gives explicit series for quadratic irrationals, identifies the coefficients of the golden ratio with generalized Catalan numbers, presents conjectural functional equations for square roots, and discusses computational data for e and π. The main theoretical content is Section 3, where Proposition 1.1 and Lemma 3.1 are used to prove the stabilization theorem, and Section 6, where translation properties and the gap theorem are proved.
Significance. If the stabilization theorem is correct, the paper introduces a genuinely new and simple-looking construction: a q-analogue of real numbers with integer Taylor coefficients, extending the authors' earlier q-rationals. The concrete examples are a real strength: the golden-ratio series identified with alternating generalized Catalan numbers, the silver-ratio series, and the detailed computational expansions for e and π give the reader something to test and explore. The paper does not ship machine-checked proofs or code, but the displayed series and functional equations are explicit enough to be checked independently. The central claim is important enough to justify publication once the proof is repaired, but the current manuscript contains a false stated proposition in the proof of the main theorem, so the result as written is not established.
major comments (3)
- [§3.2, Proposition 1.1 and Eq. (11)] Proposition 1.1 is false as stated for odd n. For x = φ with a1 = a2 = a3 = 1, the convergents are x2 = 2 and x3 = 3/2. The proposition asserts that [2]_q and [3/2]_q agree through degree a1+a2+a3−1 = 2, but [2]_q = 1+q and [3/2]_q = (1+q+q^2)/(1+q) = 1+q^2−q^3+..., so their q^1-coefficients differ. Accordingly, the determinant identity (11) is not correct in this case: R3 S2 − S3 R2 = −q, not q^2. Since the proof of Theorem 1 in §3.3 invokes Proposition 1.1 for every m, the proof as written is invalid. The theorem is likely salvageable by working only with even m, because the interval [x_{m−1}, x_m] still contains x for even m and the corrected agreement lengths still tend to infinity, but Proposition 1.1 and every use of it in §3.3 must be revised.
- [§3.3, proof of Lemma 3.1] The proof that every rational between x_{m−1} and x_m can be joined to both endpoints by a finite chain of Farey triangles is only sketched: the vertical-line argument in the Poincaré half-plane is plausible but not a proof. This step is load-bearing, since Lemma 3.1 is the bridge from consecutive convergents to arbitrary approximating sequences. The authors should either provide a rigorous construction of the chain, including the cases where the vertical line passes through a vertex or along an edge, and verify that each triangle in the chain supports the induction step with the same q-exponent a, or cite a precise statement from the theory of the Farey tessellation.
- [§4.3, Proposition 4.5] The four functional equations (17)–(20) are asserted with the sentence “The calculations are quite long but straightforward, so we omit the details.” If these equations are intended as results, the paper should include proofs or at least a reproducible verification from the defining continued fractions; otherwise the statements should be labeled as conjectures. As written, a reader cannot distinguish a transcription error from a genuine identity, especially since the displayed equations are central to Section 4's claims about square-root q-deformations.
minor comments (5)
- [§4.2] In the proof of Proposition 4.4, “Fromula” should read “Formula”.
- [§5.2] The convergents of π are misnumbered: [3,7,15,1] = 355/113 is the fourth convergent, not π5, and the next convergent [3,7,15,1,292] is the fifth. This makes the reported accuracy of the approximations harder to follow.
- [§5.1] The sentence about the coefficients of q^{2+7k} is vague; please specify precisely which coefficients are observed to be smaller than their neighbors and how the comparison is made.
- [§3, Remark 3.2] The claim that sequences approaching a rational from the left stabilize to a different series is described only from experimental computation; state explicitly that this is a observation/conjecture or provide a proof.
- [Figure 1] The labels in the weighted Farey graph are difficult to read in the printed version; a larger figure or a separate table of the relevant q-rationals would improve clarity.
Circularity Check
No significant circularity: the stabilization theorem is derived from the q-rational definitions rather than being equivalent to its inputs.
full rationale
The construction of [x]_q is explicitly defined as the coefficient-wise limit of Taylor series of q-deformed convergents, with the underlying q-rationals taken from the authors' earlier paper [6] but independently recalled in Section 2 via the weighted Farey graph, the continued-fraction formula (8), and the matrix formula (9); this is a definitional dependency, not a circular argument. Theorem 1 — stabilization, integrality, and independence of the approximating sequence — is attacked directly through Proposition 1.1 and Lemma 3.1, which rest on determinant identity (11) and Farey-triangle plumbing; no parameter is fitted from the target coefficients, and no 'prediction' is a renamed input. The numerical identifications (golden ratio, silver ratio, square roots) are supported by explicit functional equations derived from the q-continued fractions, so they are not renaming known results. Two non-circular weaknesses should be flagged: Proposition 1.1 is stated for every n, but the determinant identity (11) appears to be parity-dependent (for x = φ and n = 3, [2]_q and [3/2]_q differ at q^1), which invalidates the proof of Theorem 1 as written; and Lemma 3.1's vertical-line/Farey-triangle chain is asserted rather than proved. These are correctness gaps, not circularity, so the circularity score remains 0.
Assumptions & free parameters
assumptions (3)
- domain assumption q-deformed rationals [r/s]_q = R/S defined via the even continued fraction (7) and matrix formula (9) are well-defined and satisfy the Farey recursion (6).
- standard math Every rational between two consecutive convergents x_{m-1} and x_m is connected to both by a finite chain of Farey triangles.
- standard math Taylor expansion at q = 0 of a rational function with integer coefficients and constant term 1 lies in Z[[q]].
invented entities (1)
-
q-deformed real number [x]_q (and its Laurent extension to negative x)
independent evidence
Cite this review
Pith. "Pith review of On $q$-deformed real numbers." pith.science (2026). https://pith.science/paper/RKPUBD5T
@misc{pith2026190804365,
author = {Pith},
title = {Pith review of: On $q$-deformed real numbers},
year = {2026},
howpublished = {\url{https://pith.science/paper/RKPUBD5T}},
note = {Machine review of arXiv:1908.04365}
}
abstract
We associate a formal power series with integer coefficients to a positive real number, we interpret this series as a "$q$-analogue of a real." The construction is based on the notion of $q$-deformed rational number introduced in arXiv:1812.00170. Extending the construction to negative real numbers, we obtain certain Laurent series.
Figures
Reference graph
Works this paper leans on
-
[2]
J. Borwein, A. van der Poorten, J. Shallit, W. Zudilin, Ne verending fractions. An introduction to continued fractio ns. Australian Mathematical Society Lecture Series, 23. Cambr idge University Press, Cambridge, 2014
work page 2014
- [1]
-
[3]
Automated Proofs of Many Conjectured Recurrences in the OEIS made by R.J. Mathar
S.B. Ekhad, M. Yang, D. Zeilberger, Automated Proofs of Many Conjectured Recurrences in the OEI S made by R.J. Mathar, arXiv:1707.04654
-
[4]
G. H. Hardy, E. M. W right, An introduction to the theory of numbers. Sixth edition. Revised by D. R. Heath-Brown and J. H. Silverman. With a foreword by Andrew Wiles. Oxford U niversity Press, Oxford, 2008, 621 pp
work page 2008
-
[5]
S. Morier-Genoud, V. Ovsienko, Farey boat. Continued fractions and triangulations, modul ar group and polygon dis- sections, Jahresber. Dtsch. Math.-Ver. 121 (2019), no. 2, 91–136
work page 2019
-
[6]
S. Morier-Genoud, V. Ovsienko, q-deformed rationals and q-continued fractions, arXiv:1812.00170
-
[7]
OEIS Foundation Inc., The On-Line Encyclopedia of Integ er Sequences, http://oeis.org
-
[8]
Stanley, Enumerative combinatorics
R. Stanley, Enumerative combinatorics. Volume 1. Secon d edition. Cambridge Studies in Advanced Mathematics, 49. Cambridge University Press, Cambridge, 2012. xiv+626 pp
work page 2012
Show all 9 references
-
[9]
Zeilberger, The C-finite ansatz , Ramanujan J
D. Zeilberger, The C-finite ansatz , Ramanujan J. 31 (2013), no. 1-2, 23–3. q-DEFORMED REAL NUMBERS 15 Sophie Morier-Genoud, Sorbonne Universit ´e, Universit ´e Paris Diderot, CNRS, Institut de Math ´ematiques de Jussieu-Paris Rive Gauche, F-75005, Paris, France V alentin Ovsie...
2013
Reviewed August 14, 2026 · model on record in the stance chip above.
Discussion (0). Continue with ORCID to comment.