{"id":"d8144b4e-d0fd-4643-936c-528933c5bd42","arxiv_id":"2508.08440","paper_version":1,"verdict":"UNVERDICTED","confidence":"LOW","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"For every real x > 1 the q-real series [x]_q converges in the disk |q| < 3−2√2 to a nonvanishing holomorphic function, partially proving the radius-convergence conjecture.","lead":"The paper claims that q-real numbers, q-deformed versions of ordinary real numbers, converge as infinite series for small values of q, a partial proof of an open conjecture. Only the abstract could be reviewed: the supplied manuscript body belongs to a different arXiv paper.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Supplied manuscript is a different paper (arXiv:2508.08447), so the central convergence claim for [x]_q has no supporting proof in the text; the claim is unverifiable as presented.","rationale":"The reader's verdict was UNVERDICTED with low confidence, driven by the mismatch between the abstract and the supplied full text. The reader's formal weakest_assumption, however, focused on the uniform convergence of the expansion. My stress-test agrees with the overall verdict but identifies the more fundamental issue: the supplied manuscript contains none of the claimed mathematics at all. That makes the convergence claim completely uncheckable. I do not raise a mathematical objection to the abstract's content (e.g., whether the radius 3−2√2 or the positivity interval is correct); rather, the text as provided cannot support any assessment of the proof. The reader already noted this in the rationale, but their weakest_assumption field points to a secondary mathematical fragility that cannot even be evaluated without the actual text. Thus 'partial' agreement: we converge on UNVERDICTED, but the primary load-bearing concern is not the expansion's convergence per se—it is that no expansion is present in the manuscript. The recommended verdict remains UNVERDICTED, because the central claim is unverified, not because it is false. Should the correct manuscript be obtained, the concrete test above would then allow a proper check of the expansion and its estimates.","tokens_in":3720,"tokens_out":2600,"duration_ms":29547,"concrete_test":"Retrieve the actual manuscript for arXiv:2508.08440 from arXiv and compare its full text with the supplied body. If the true paper is the q-real/q-complex paper, verify the central analytic step: that the claimed expansion of 1/[x]_q converges absolutely and uniformly on compact subsets of the explicit region D containing |q| < 3−2√2, with estimates valid for all real x > 1, including badly approximable x. If the expansion is absent or the body remains the Moles–Khan paper, the central claim is unverified.","verdict_should_be":"UNVERDICTED","load_bearing_attack":"The full text supplied for arXiv:2508.08440 is not the advertised paper: it is stamped 'arXiv:2508.08447v3 [math.AC] 2 Dec 2025' and titled 'Locally Associated Orders in Real Quadratic Number Fields' by Grant Moles and Talha Khan. The abstract promises a proof that the q-real series [x]_q converges for |q| < 3−2√2 to a nonvanishing holomorphic function for all real x > 1, via an expansion of 1/[x]_q into a q-adically convergent series of rational functions with uniform convergence on compact sets in an explicit region D. None of this appears in the body: there is no q-deformed continued fraction, no expansion of 1/[x]_q, no region D, no uniform-convergence estimates, no hypergeometric functions of modular functions, and no example such as x = cotan(1). The central claim therefore rests entirely on an abstract that the provided manuscript does not substantiate. This is a load-bearing concern because the main theorem cannot be checked: the proof is not merely flawed but absent. The problem is not an internal mathematical contradiction but an evidentiary one—there is nothing in the supplied text that would allow a reviewer to verify the claimed result or its key estimates.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","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.","tokens_in":3892,"tokens_out":3328,"duration_ms":39636,"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":[{"comment":"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.","section":"Abstract vs. full text"},{"comment":"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.","section":"Abstract, fourth sentence"},{"comment":"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.","section":"Abstract, final sentence"}],"minor_comments":[{"comment":"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.","section":"Title and metadata"},{"comment":"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).","section":"References"}],"recommendation":"major_revision","confidential_remarks":"The submitted PDF is not the advertised manuscript. Before any further review, the correct file for arXiv:2508.08440 must be supplied and the abstract must be backed by a complete proof. If the correct manuscript is not available, the submission would effectively be void."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"You should know two things at once: (1) the abstract promises a genuinely interesting result, and (2) the \"full text\" I was given is not this paper. It is arXiv:2508.08447 by Moles and Khan, on locally associated orders in real quadratic fields. The body is even stamped with that arXiv ID. So there is no proof of the q-real convergence claim in front of me—no estimates, no explicit region D, no uniform-convergence argument. Everything hangs on the abstract alone.\n\nThe abstract itself is worth taking seriously. It claims a partial resolution of the Morier-Genoud–Ovsienko conjecture: for every real x > 1, the q-real series [x]_q converges in |q| < 3−2√2 to a nonvanishing holomorphic function, via a q-adic expansion of 1/[x]_q. That would extend the known rational case to all real x, and the smaller radius (3−2√2 vs. the conjectured golden-ratio radius) is a conservative, plausible intermediate step. The abstract also mentions an analytic continuation to negative q in an interval and a proposed q-complex number expressed through hypergeometric functions of modular functions. Relative to the cited literature (2102.00891, 2405.15970), these appear to be new moves. The citation pattern looks honest, and the abstract explicitly says the full conjecture remains open—no overclaiming.\n\nThe soft spot is not in the mathematics but in the submission itself. With no corresponding text, I cannot check whether the expansion converges uniformly for badly approximable x, which is the fragile part. The reader's scores are all low-confidence estimates, and they should stay that way. This is a load-bearing evidentiary gap, not a mere formatting glitch.\n\nMy recommendation: a serious editor should contact the author and ask for the correct manuscript. If the actual paper matches the abstract, it deserves a careful peer review by someone in q-series or continued fractions. If the correct file never materializes, then there is nothing to review. For now, I would not cite it, but I would not dismiss the underlying idea either. Treat this as an unverified but promising lead, not a finished paper.","headline":"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.","tokens_in":4546,"tokens_out":2179,"would_cite":false,"duration_ms":25060,"reading_group":"maybe","serious_thinker":"unclear","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["11A55","30B10","11F03","33C05"],"pacs":[],"model":"deepseek-v4-flash","headline":"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","keywords":["q-real numbers","q-rational numbers","continued fractions","Laurent series","convergence radius","golden ratio","modular functions","hypergeometric functions"],"falsifier":"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.","tokens_in":3459,"feed_emoji":"🧮","tokens_out":12399,"duration_ms":131228,"temperature":0.7,"pith_summary":"The paper sets out to establish that the \\(q\\)-real number \\([x]_q\\), a formal Laurent series attached to the continued fraction of a real number \\(x>1\\), is actually convergent: for every real \\(x>1\\) the series converges in the fixed disk \\(|q|<3-2\\sqrt2\\) to a nonvanishing holomorphic function. This is a partial proof of a conjecture that predicted the same property with the larger radius \\((3-\\sqrt5)/2\\), attained at the golden ratio. The proof proceeds by writing \\(1/[x]_q\\) as a \\(q\\)-adically convergent series of rational functions and proving absolute, uniform convergence on compact subsets of a region containing that disk. The same expansion gives a positive analytic extension of \\([x]_q\\) on the real interval \\((-(3-\\sqrt5)/2,1)\\), and the paper also proposes a \\(q\\)-complex number associated to points in the upper half-plane.","feed_headline":"q-real series converge for every real x>1","feed_subtitle":"A partial proof of the golden-ratio radius conjecture, plus a first definition of q-complex numbers.","key_machinery":"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.","core_discovery":"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","pith_inferences":["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."],"forward_implications":["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."],"supporting_citations":[{"why":"Introduces \\(q\\)-rational numbers by \\(q\\)-deforming continued fractions; these are the objects being generalized.","marker":"arXiv:1812.00170"},{"why":"Introduces \\(q\\)-real numbers as Laurent series; this is the series whose convergence is studied.","marker":"arXiv:1908.04365"},{"why":"Proves convergence for rational \\(x>1\\) in \\(|q|<3-2\\sqrt2\\) and states the golden-ratio radius conjecture.","marker":"arXiv:2102.00891"},{"why":"Establishes the rational case via Kleinian groups; the paper derives the larger disk \\(|q|<2-\\sqrt3\\) from this result.","marker":"arXiv:2405.15970"}],"fun_headline_variants":["q-real convergence proven for every real x>1","All real x>1 now get convergent q-series","q-real series converge for all real x>1; q-complex defined","Golden ratio radius: partial proof for all real x"],"cache_read_input_tokens":2816,"weakest_assumption_plain":"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.","fun_headline_variants_meta":{"raw":{"variants":["q-real convergence proven for every real x>1","All real x>1 now get convergent q-series","q-real series converge for all real x>1; q-complex defined","Golden ratio radius: partial proof for all real x"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.001426,"raw_usage":{"total_tokens":5759,"prompt_tokens":1082,"completion_tokens":4677,"prompt_tokens_details":{"cached_tokens":256},"prompt_cache_hit_tokens":256,"prompt_cache_miss_tokens":826,"completion_tokens_details":{"reasoning_tokens":4609}},"tokens_in":826,"tokens_out":4677,"duration_ms":44561,"temperature":1.0,"reasoning_tokens":4609,"cache_read_input_tokens":256,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-05T21:32:30.317563+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"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.","supporting_citations":[],"review_version":1}