{"id":"8accd392-db1c-4328-9386-692713c28688","arxiv_id":"2607.04565","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":5.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":3,"one_line_summary":"Explicit real-analytic projections on manifolds in Euclidean space realize prescribed normal graph diagrams for NF of their Reeb digraph compactifications.","lead":"The paper constructs real-analytic manifolds and projections whose Reeb spaces, after natural compactification, realize prescribed 1-dimensional cell complexes with controlled non-finite points. It supplies explicit algebraic and analytic examples that extend earlier graph-realization results to non-graph Reeb spaces.","discovery_kind":"extension","skeptic_critique":{"model":"grok-4.5","headline":"Asymptotic oscillation of q_{p,s,i} is asserted to produce exactly the prescribed non-finite points, but the proof supplies no local analysis ruling out extra accumulation or coalescence after rotation and compactification.","rationale":"The Reader correctly isolates the weakest link: the asymptotic oscillation claims of Subsection 3.3 are used as a black box to guarantee the exact combinatorial type of the graph diagram for NF. The proof of Theorem 1 never supplies a local model or an explicit estimate showing that the limsup/liminf oscillations survive the simultaneous rotations and do not create extra non-finite points. Because that combinatorial exactness is precisely the content of the strongest claim, the concern is load-bearing. No stronger internal contradiction appears; the constructions are concrete and the general Reeb-space background is standard. Hence the Reader’s CONDITIONAL verdict with medium correctness risk remains appropriate; the concrete numerical check above would either confirm the asymptotics or force a revision of the parameter regime.","tokens_in":17845,"tokens_out":655,"duration_ms":6772,"concrete_test":"Fix the smallest non-trivial parameters m=2, n_d=1, n_e=0, n_c=0 and an explicit elementary choice (e.g. p1=1, p2=x^{2}+1, s1=s2=1, t=π/3). Compute (numerically or symbolically) the critical set of π_{3,1}|_X inside a large disk and track its image under the quotient map after the one-point compactification to S^{2}. If any critical value accumulates at a point other than the single intended non-finite point, or if the resulting GDNF contains more than one vertex, the asymptotic control asserted in 3.3 is insufficient.","verdict_should_be":"CONDITIONAL","load_bearing_attack":"Theorem 1’s existence claim rests on the construction in the proof (pp. 9–11): rotate the graphs of p_{h,t} and the perturbed functions p_{h,t,q_{p,s,i}} by multiples of 2(π/2-t), then form the zero set X_{m,{S_j,f_j}}. The claim that the resulting Reeb-D-C has a normal GDNF consisting of precisely one vertex plus exactly n_d outgoing rays, n_e incoming rays and n_c circles depends on the limsup/liminf statements of Subsection 3.3 guaranteeing that the critical loci of the q-perturbed branches accumulate only at the desired non-finite points and create no additional vertices or edges after one-point or circle compactification. The text merely asserts “Remember arguments on critical points of p_{h,t} and p_{h,t,q…}” and cites general Reeb-space theorems; it never verifies that the oscillations remain transverse to the radial directions after rotation, nor that distinct rotated branches do not produce extra accumulation points inside any compact disk D_{R,o}. If even one extra accumulation occurs, the normal one-vertex diagram fails and the strongest claim collapses.","agreement_with_reader":"agree"},"referee_report":{"model":"grok-4.5","summary":"The paper formulates compactifications of continuous (mainly real-analytic) maps and studies the resulting Reeb spaces, which are 0- or 1-dimensional cell complexes. It introduces pre-digraphs and the graph diagram for NF (GDNF) to represent Reeb spaces that need not be finite graphs. Theorem 1 asserts that for every m>1 and non-negative integers n_d, n_e, n_c there exists an m-dimensional real-analytic submanifold X_{m,n_d,e,c} of R^{m+1} whose projection admits a Reeb digraph compactification whose normal GDNF consists of a single vertex with exactly n_d outgoing rays, n_e incoming rays and n_c circles. Theorem 2 constructs a real-analytic manifold diffeomorphic to R^m and a smooth compactification diffeomorphic to S^m such that the Reeb digraph of the compactification is a Reeb-D-C of the original function but is not isomorphic to its Reeb digraph. The constructions rely on rotated components of hyperbolas, auxiliary oscillating functions q_{p,s,i}, and zero-set manifolds of the form X_{m,{S_j,f_j}} or X_{m1,m2,{S_j',g_j},I}.","tokens_in":18243,"tokens_out":1144,"duration_ms":8740,"significance":"If the existence claims hold, the paper supplies the first systematic real-analytic realizations of Reeb spaces whose combinatorial type is controlled by an arbitrary finite collection of rays and circles attached to a single non-finite point. This extends the classical reconstruction programme (Sharko, Masumoto–Saeki, Michalak, and the author’s earlier algebraic constructions) beyond finite graphs and gives concrete examples of compactifications that alter the Reeb digraph. The local rank calculations (Propositions 2–4) are standard and correctly establish that the zero sets are smooth manifolds of the expected dimension; the novelty lies in the global combinatorial control after compactification.","major_comments":[{"comment":"Proof of Theorem 1 (pp. 9–11): the claim that the normal GDNF consists of precisely one vertex plus exactly n_d outgoing rays, n_e incoming rays and n_c circles rests on the limsup/liminf oscillation of the derivatives of q_{p,s,i} (Subsection 3.3) after rotation by multiples of 2(\\pi/2-t). The text only says “Remember arguments on critical points of p_{h,t} and p_{h,t,q…}” and cites general Reeb-space theorems; it never verifies that the oscillations remain transverse to the radial directions after rotation, nor that distinct rotated branches produce no extra accumulation points inside any compact disk D_{R,o}. Without a local analysis ruling out coalescence or additional non-finite points, the prescribed combinatorial type is not established.","section":null},{"comment":"Theorem 2, STEP 2-4 and STEP 2-5: the assertion that the Reeb space of π_{m+2,1}|_{X_{m,2}} is a Peano continuum with exactly one non-finite point q(0) relies on the same asymptotic oscillation of the non-analytic branch g_{2,1}. The argument that every neighbourhood of this point contains infinitely many discrete critical values is sketched via Proposition 4, but no estimate is given showing that these critical values accumulate only at 0 and do not create additional non-finite points or edges after the compactification. The non-isomorphism claim in (4) therefore remains incomplete.","section":null}],"minor_comments":[{"comment":"Subsection 3.3: the phrase “π/2 is divisible by 2(π/2-t)” is unclear; the intended meaning is that the angle 2(π/2-t) divides 2π an integer number of times. A precise statement would help.","section":null},{"comment":"Definition 3 and Definition 4 introduce “compactification in C'” and “Reeb digraph compactification” with several nested embeddings; a short commutative diagram would make the relations transparent.","section":null},{"comment":"Throughout: numerous typographical slips (e.g., “fintiely”, “roation”, “preimagec”, missing spaces after punctuation) and inconsistent notation for the same projection (π_{m+1,1} versus π_{m+2,1,2}) should be cleaned.","section":null},{"comment":"The paper repeatedly refers to the author’s own arXiv preprints [15–19] for both definitions and technical lemmas. A self-contained appendix summarizing the needed asymptotic estimates would improve readability for readers unfamiliar with that series.","section":null}],"recommendation":"major_revision","confidential_remarks":"The manuscript is the latest in a long sequence of closely related arXiv notes by the same author. While the constructions appear new, the heavy dependence on unpublished prior work and the incomplete verification of the asymptotic claims make the paper borderline for a first-tier journal; a carefully revised version with full local analysis of the critical loci would be more suitable."},"author_rebuttal":null,"desk_editor":{"model":"grok-4.5","letter":"This paper continues Kitazawa's sequence of arXiv notes on realizing non-graph Reeb spaces by explicit real-analytic (or nearly analytic) projections. The new piece is Theorem 1: for any m>1 and any triple of non-negative integers (n_d, n_e, n_c) one can arrange rotated hyperbolas, some of them mildly perturbed by the oscillating factors q_{p,s,i}, so that a compactification of the projection has normal graph-diagram-for-NF consisting of a single vertex plus exactly those numbers of outgoing rays, incoming rays and circles. Theorem 2 gives a related pair of manifolds, one non-compact analytic and one compact smooth, whose Reeb digraphs are related by compactification but are not isomorphic. Both constructions are concrete and sit inside the same zero-set framework the author has used before.\n\nWhat works: the local rank calculations (Props. 2–4) are standard and clean; the combinatorial language of pre-digraphs and GDNF is a useful bookkeeping device once one accepts the earlier notes; and the examples really do produce Reeb spaces that are not finite graphs. The paper is honest about its place in the literature and does not claim more than a flexible family of models.\n\nThe soft spot is exactly the one the stress-test flags. The proof of Theorem 1 (pp. 9–11) asserts that the limsup/liminf oscillation of the q-functions, together with a “sufficiently large” angle t, produces precisely the prescribed non-finite points and no extras after rotation and compactification. There is no local analysis ruling out accidental accumulation or coalescence inside a compact disk, nor a verification that the rotated critical loci remain transverse to the radial directions. The text simply says “remember the critical-point arguments” and cites general Reeb-space theorems. That gap is real but not fatal; it is the sort of thing a careful referee can demand be filled in a few pages of estimates. Circularity with the author’s own unpublished notes is present but not circular in the logical sense—the existence statements are new constructions.\n\nWho it is for: people already working on Reeb graphs of real-analytic or definable maps who want concrete models with controlled non-finite points. Outside that circle the paper is hard to read and the novelty is incremental. I would send it to peer review; the constructions are explicit enough and the gap is fixable. I would not put it in next week’s reading group unless someone is already deep in this literature, and I am unlikely to cite it myself in the next year.","headline":"Solid constructive extension of the author's own Reeb-space program; the combinatorial claims are plausible but rest on sketched asymptotics that a referee will need to check carefully.","tokens_in":18772,"tokens_out":632,"would_cite":false,"duration_ms":6721,"reading_group":"no","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["26C05","26E05","54C30","54F15","57R45","58C05"],"pacs":[],"model":"grok-4.5","headline":"Real-analytic height functions can be compactified so their Reeb spaces have any prescribed mix of rays and circles around one non-finite vertex.","keywords":["Reeb spaces","compactifications of maps","real analytic functions","graph diagram for NF","pre-digraphs","hyperbolas and asymptotes","cell complexes"],"falsifier":"Exhibit a concrete triple (n_d,n_e,n_c) for which every real-analytic height function built from the rotated-hyperbola construction either acquires an extra non-finite point or fails to realise one of the required rays or circles after compactification.","tokens_in":18752,"feed_emoji":"📐","tokens_out":712,"duration_ms":11345,"temperature":0.7,"pith_summary":"The paper constructs families of real-analytic height functions on non-compact manifolds so that, after a natural compactification, the associated Reeb space collapses to a simple combinatorial object: a single non-finite vertex together with any chosen numbers of outgoing rays, incoming rays and circles. Reeb spaces record how level sets of a function change; they are usually graphs when the domain is compact, but become more complicated when the domain is open. By carefully arranging rotated hyperbolas whose asymptotes and oscillatory perturbations control critical points, the author realises every triple of non-negative integers as the edge counts of a normal graph diagram for the non-finite part. A second construction exhibits a smooth compactification of an analytic function on Euclidean space whose Reeb digraph is not isomorphic to the original, showing that compactification can alter the combinatorial type. The results give the first systematic real-analytic realisations of Reeb spaces that are not finite graphs, answering a reconstruction problem the author has pursued in a sequence of earlier notes.","feed_headline":"Analytic height functions compactify to any mix of rays and circles","feed_subtitle":"One non-finite vertex realises every prescribed combination of outgoing, incoming and circular edges","key_machinery":"The graph diagram for NF of a pre-digraph: the 0- or 1-dimensional cell complex obtained by collapsing each equivalence class of ordinary edges that share the same ascending or descending non-finite point, thereby encoding the combinatorial type of a possibly non-finite Reeb space by a simpler oriented cell complex.","core_discovery":"For every dimension m>1 and every triple of non-negative integers (n_d,n_e,n_c) there exists an m-dimensional real-analytic submanifold of Euclidean space whose natural height function admits a compactification whose Reeb digraph has a normal graph diagram for non-finite points consisting of exactly one vertex, n_d outgoing rays, n_e incoming rays and n_c circles.","pith_inferences":[],"forward_implications":[],"fun_headline_variants":["Any mix of rays and circles from one-vertex Reeb digraphs of analytic heights","Compactified real-analytic heights realise every prescribed ray-circle triple","One non-finite vertex yields arbitrary outgoing, incoming rays and circles","m-dimensional analytic submanifolds compactify to any ray-circle combination","Prescribed Reeb digraphs with rays and circles via analytic height compactifications"],"cache_read_input_tokens":128,"weakest_assumption_plain":"The oscillatory auxiliary functions and the angle of the rotated hyperbolas are assumed to produce exactly the prescribed numbers of non-finite points and no extra vertices after compactification, without a complete local analysis of every possible accumulation pattern.","fun_headline_variants_meta":{"raw":{"variants":["Any mix of rays and circles from one-vertex Reeb digraphs of analytic heights","Compactified real-analytic heights realise every prescribed ray-circle triple","One non-finite vertex yields arbitrary outgoing, incoming rays and circles","m-dimensional analytic submanifolds compactify to any ray-circle combination","Prescribed Reeb digraphs with rays and circles via analytic height compactifications"]},"model":"grok-4.5","effort":"low","cost_usd":0.00598,"raw_usage":{"total_tokens":1498,"prompt_tokens":704,"num_sources_used":0,"completion_tokens":103,"cost_in_usd_ticks":59800000,"prompt_tokens_details":{"text_tokens":704,"audio_tokens":0,"image_tokens":0,"cached_tokens":128},"completion_tokens_details":{"audio_tokens":0,"reasoning_tokens":691,"accepted_prediction_tokens":0,"rejected_prediction_tokens":0}},"tokens_in":704,"tokens_out":103,"duration_ms":6182,"temperature":1.0,"reasoning_tokens":691,"cache_read_input_tokens":128,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-07-11T17:11:06.486336+00:00","model_set":{"reader":"grok-4.5"},"falsifier":"Exhibit a concrete triple (n_d,n_e,n_c) for which every real-analytic height function built from the rotated-hyperbola construction either acquires an extra non-finite point or fails to realise one of the required rays or circles after compactification.","supporting_citations":[],"review_version":1}