{"id":"f837c4fc-c6c5-4870-be63-edae7c3571f0","arxiv_id":"2508.00729","paper_version":2,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":4.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":3,"one_line_summary":"Every tree, and certain graphs assembled from single edges and small circles, is the Reeb graph of a Morse-Bott real algebraic function defined by degree-1 and degree-2 polynomials.","lead":"This paper constructs explicit real algebraic functions, built from lines and circles, whose Reeb graphs are any tree or certain cactus-like graphs. It is a specialized advance in real algebraic geometry and singularity theory, continuing the author's own program on prescribed Reeb graphs.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The proof of Theorem 1 never shows the added cap disks keep D a thickened tree; a cap that disconnects D or creates a hole would add an extra edge or cycle to the Reeb graph.","rationale":"The reader's conditional verdict identifies the same broad gap: the global Reeb-graph bookkeeping is asserted rather than demonstrated. I agree that this is the load-bearing point, but I make it more concrete: the unverified step is not just a smoothness check near curve intersections, but the global effect of deleting the disks bounded by the added circles. If any of those disks detaches a component of D or encloses a hole, the Reeb graph of any function built from D via Theorem 3 must contain an extra edge or cycle, so Theorem 1 would be false. The text only states that the disks are mutually disjoint and each new circle meets the old union in two points; it does not prove that these deletions are boundary caps rather than separating cuts. This is exactly the kind of global verification that a figure can suggest but cannot establish. I do not claim the theorem is false; the thickened-tree construction is very plausible and likely correct. The concern is that the paper as written lacks the argument that its own circle arrangement never creates holes or disconnections. If a full verification is supplied, the result would be acceptable; until then CONDITIONAL is the right verdict. A secondary issue, not used in my headline, is that the definition of tree decomposition in Definition 1 appears inconsistent with Corollary 1 for trees with a degree-3 vertex, which undermines the stated combinatorial framework of Theorem 4 independently of the algebraic construction.","tokens_in":9863,"tokens_out":36744,"duration_ms":485577,"concrete_test":"Realize the construction for the 3-edge star with center value 0 and leaves at -1, +1, +2 using §2.3: explicitly place the four lines, the S_{4+j} circles, and the three remaining circles with I_j'=1, satisfying the stated incidence rules. Compute, by cylindrical algebraic decomposition or interval arithmetic, the connected components of the fiber (π_{2,1}∘F)^{-1}(t) for t in each interval between successive vertical tangencies and curve intersections of the arrangement. If the component count changes at any t other than -1, 0, +1, +2 (plus the outer S1/S2 levels), or if D is not simply connected, then Theorem 1 fails; if the count matches the star's contour, this supplies the missing verification and the proof should be expanded to a general tree.","verdict_should_be":"UNCHANGED","load_bearing_attack":"Theorem 1's construction adds circles to the four-line rectangle and then applies Theorem 3. The Reeb graph of π_{2,1}|M is the vertical contour tree of the planar region D (tensored with connected spheres), so it is controlled by the connected components of D∩{x=t}. Section 2.3 asserts the non-singularity of M via Theorem 3's Cases 1–3, but it never proves the global topology of D: that each remaining circle, whose disk is deleted from D, is attached to the current boundary along one arc, does not disconnect D, does not create a hole, and gives vertical-slice component counts changing exactly at the vertex values of g_{GT}. The bullets specify pairwise disjoint disks and a two-point intersection of each new circle with the old union, but two-point contact is compatible both with harmless boundary caps and with disks that span across D and split it. Since every hole or new component of D would appear as an extra 1-cycle or branch in the Reeb graph, the central identification 'Reeb graph ≅ G_T' is not established by the text; it is only the plausible picture behind Figure 2.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper studies the realization of finite graphs as Reeb graphs of real algebraic functions. Its main result, Theorem 1, asserts that for every finite tree G_T there exists a Morse-Bott real algebraic function, obtained as the composition of a real algebraic map with the canonical projection π_{2,1}, whose Reeb graph is isomorphic to G_T and whose defining polynomials have degree at most 2; the construction uses only straight lines and circles in the plane. Theorem 4 extends the construction to a class of cactus-like graphs built from a tree by replacing certain vertices of the SCT decomposition with circles. The proof is based on a reconstruction theorem, Theorem 3, quoted from the author's preprints, together with a local case analysis in Section 2.2.","tokens_in":10012,"tokens_out":5577,"duration_ms":65522,"significance":"If the construction is completed rigorously, the paper would provide a very explicit geometric realization: every finite tree is the Reeb graph of a Morse-Bott real algebraic function defined using only straight lines and circles, with polynomials of degree at most 2. That would be a concrete and appealing contribution to the study of Reeb graphs of real algebraic functions, going beyond general existence and approximation results. The constructive nature and the precise degree bounds are genuine strengths. However, the significance is conditional: the central identification of the Reeb graph with the prescribed tree is not established in the text, and the paper's main theorem therefore remains a plausible but unproven claim as written.","major_comments":[{"comment":"The proof does not establish the global topology of the planar region D, which is the load-bearing step for identifying the Reeb graph with G_T. The bullets describing the remaining circles only impose local intersection conditions; a circle whose disk is deleted from D can split a strip into two components or create a hole while still intersecting the previous union in two points. Since every extra component or 1-cycle in D would appear as an extra branch or cycle in the Reeb graph, the text needs a step-by-step verification that each attached disk is a boundary cap, that D remains a thickened tree, and that the vertical slices D ∩ π_{2,1}^{-1}(t) have component counts prescribed by g_{G_T}. Figure 2 illustrates the intended configuration but does not prove these facts.","section":"§2.3, proof of Theorem 1"},{"comment":"Theorem 1 reduces to Theorem 3, but Theorem 3 is only sketched here and is quoted from the author's preprints [7,9]. The local Cases 1–3 in the sketch describe normal forms around points of D and its boundary, but they do not verify the global Reeb graph of the resulting function on M. Since Theorem 3 is the central tool and does not by itself state or prove the Reeb-graph identification, the paper needs either a complete proof of Theorem 3 or a direct argument showing that the global Reeb graph of π_{2,1}|_M is the prescribed graph. Without this, the main theorem is not established by the text.","section":"§2.2, Theorem 3 and its use"},{"comment":"The proof of Theorem 4 inherits the same gap. For the first case it says only 'we can check' the preimages, and for the second case it asserts that omitting certain circles produces the desired SCT decomposition. No computation shows that the omitted circles convert exactly the prescribed vertices into circles in the Reeb graph and that no other vertices or edges are created. This needs an explicit verification for the same global reasons as Theorem 1.","section":"§2.3, Theorem 4"}],"minor_comments":[{"comment":"There are numerous typos and spacing errors ('singl e', 'algbraic', 'W e', 'differentiable', 'different'), and the abstract's sentence about a 'graph with exactly one edge and two edges' is inconsistent with the later 'two vertices' wording; these should be corrected.","section":"Abstract and throughout"},{"comment":"The notation 'D − D' should be written as the set-theoretic difference of the closure and the open set, e.g., \\overline{D} \\setminus D; as printed it is confusing. Also, the condition 'D = {x | f_j(x) > 0}' appears to depend on j and should be stated precisely as a single defining inequality system.","section":"Theorem 3"},{"comment":"The variables l, l1, and I_{g_{G_T},j} are used without a complete list of definitions, and the same quantity appears under different names ('l' and 'l1') in the bullets; a uniform notation would improve readability.","section":"Proof of Theorem 1"},{"comment":"Figure 2 is helpful, but the caption does not explain how the colored curves correspond to the subsequent labeling by ml(j); a more detailed caption or a short table of the labels would make the construction easier to follow.","section":"Figures"}],"recommendation":"major_revision","confidential_remarks":"The paper's main result is plausible, but the proof relies heavily on the author's own unreviewed preprints (references [7] and [9]) for the central Theorem 3, and the global Reeb-graph verification is deferred to figures and preprints. I would encourage the editor to require a fully self-contained proof of the key step before publication. The citation concentration in references [5]–[10] is not itself a flaw, but in this case it directly affects the verifiability of the main theorem."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Short version: the paper gives a concrete way to realize every tree (and certain cactus-like graphs) as the Reeb graph of a real algebraic Morse-Bott function, with defining curves of degree at most 2. The result is new and the approach is sensible. But the proof of Theorem 1 doesn't actually show the Reeb graph is the desired tree. It states a few bullets about how the added circles intersect the planar region and then says 'by Theorem 3' we're done. The key global topology check—that the region D remains a thickened tree after adding all circles, with slice component counts changing exactly at the vertex levels—is deferred to figures and the author's preprints. I think the stress-test concern is on target: a circle meeting the existing boundary in two points could be a harmless cap, but it could also split a vertical slice or create a hole, either of which adds a branch or cycle to the Reeb graph. The text does not rule that out. Figure 2 makes the intended arrangement clear, but the argument is not written.\n\nI want to give credit where it's due. The construction is explicit and the algebraic data are very concrete. The SCT decomposition framing is a useful way to organize the class of graphs considered. The paper is honest about where the technical details live: Theorem 3, the author's reconstruction theorem from preprints [7] and [9], is the engine, and the paper gives a plausible local sketch of it. The self-citation load is heavy, but that's not inherently a flaw when the cited results are the actual foundation.\n\nThe soft spots are in proportion to the paper's ambition. Theorem 4's proof is even more of a sketch. Typos are rampant, but that's not the main issue. The main issue is that the central verification is missing. A specialist can probably fill it in, and I suspect the theorem is true, but the paper as written does not stand alone.\n\nWho is this for? People working on explicit construction of real algebraic maps with prescribed Reeb graphs, and anyone interested in Morse-Bott functions with simple polynomial data. It deserves a serious referee. My recommendation: send to peer review, but require the author to expand the proof of Theorem 1 with a genuine global bookkeeping argument, and either supply the details for Theorem 4 or move it to a separate paper. I would not cite the current arXiv version in my own work until that is done.","headline":"Plausible new construction with a real gap: the Reeb graph identification is asserted, not proved, and the paper leans heavily on the author's own preprints.","tokens_in":10691,"tokens_out":8489,"would_cite":false,"duration_ms":99517,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["05C05","05C10","14P05","14P10","14P25","57R45","58C05"],"pacs":[],"model":"deepseek-v4-flash","headline":"Every finite tree is the Reeb graph of a Morse-Bott real algebraic function whose defining curves are only lines and circles.","keywords":["Reeb graphs","real algebraic functions","Morse-Bott functions","tree decompositions","simple cactus tree decompositions","real algebraic manifolds","graphs","canonical projection"],"falsifier":"Take a concrete small tree, say a path with three edges or a tree with one vertex of degree 3, write out the lines and circles specified by the proof, and compute the number of connected components of the preimage $f^{-1}(t)$ for a dense set of values $t$ between the chosen critical levels. If any value of $t$ shows a component-count change that does not correspond to a vertex of the tree, the Reeb graph would acquire an extra vertex and Theorem 1 would be false.","tokens_in":9499,"feed_emoji":"🌳","tokens_out":5807,"duration_ms":55440,"temperature":0.7,"pith_summary":"The paper claims that every finite tree can appear as the Reeb graph of a real algebraic Morse-Bott function built from very simple ingredients: straight lines and circles of fixed radius, i.e., zero sets of polynomials of degree 1 or 2. The construction embeds the tree in the plane, surrounds it by such curves, and uses a reconstruction theorem to lift these curves into a real algebraic manifold whose projection back to the plane has the desired Reeb graph. A second theorem extends the result to a class of graphs with simple cactus tree decompositions, where each block is an edge or a circle. If true, this means the topological obstruction to realizing a graph as a Reeb graph in this algebraic setting is essentially the graph's tree-decomposition structure.","feed_headline":"Every finite tree is a Reeb graph of a real algebraic function","feed_subtitle":"The proof uses only straight lines and circles of fixed radius, with polynomials of degree 1 or 2.","key_machinery":"The load-bearing device is Theorem 3, a reconstruction theorem for real algebraic maps whose image is the closure of a non-empty open set $D \\subset \\mathbb{R}^2$ bounded by real algebraic curves $S_j$. The theorem builds a real algebraic manifold $M$ as the zero set of equations $\\prod_{j_I \\in \\{j \\mid ml_1(j)=j'\\}} f_{j_I}(x) - \\|y_{I,j'}\\|^2 = 0$ for each label $j'$, and the canonical projection to $\\mathbb{R}^2$ is a real algebraic map onto $D$. Around any point of $D$ the local preimage is a product of disks and spheres, so component counts can be controlled locally; the proof of the main theorem arranges the lines and circles so that these counts change only at levels corresponding to vertices of the tree, making the Reeb graph isomorphic to $G_T$.","core_discovery":"On the paper's own terms, the central discovery is Theorem 1: for any tree $G_T$, there is a real algebraic function obtained as the composition of a real algebraic map with the canonical projection $\\pi_{2,1}$, whose defining polynomials $f_j$ have degree 1 or 2, whose each $S_j$ is a circle of fixed radius or a straight line in $\\mathbb{R}^2$, which is a Morse-Bott function, and whose Reeb graph is isomorphic to $G_T$. Theorem 4 broadens the statement: certain graphs obtained by replacing selected vertices of the SCT decomposition with circles are also realizable in the same explicit way. The proof works by placing the tree inside a planar region bounded by lines and circles, assigning the curves labels and multiplicities according to Theorem 3, and then reading off the Reeb graph from the component counts of preimages.","pith_inferences":["A plausible next step, not taken in the paper, is that every graph admitting a simple cactus tree decomposition is realizable, since the paper's obstruction appears to be combinatorial rather than algebraic.","Because the constructed functions are Morse-Bott with only degree-1 and degree-2 polynomials, they are concrete enough to be tested computationally; running the construction on small trees would provide numerical evidence for or against the local bookkeeping assumptions.","If every tree is realizable, then likely no additional topological restriction on Reeb graphs of real algebraic Morse-Bott functions arises from tree structure alone; the same construction may adapt to graphs built from more general polyhedral blocks.","The gap between Morse and Morse-Bott matters: the construction deliberately uses circles to create degenerate singularities, so the result does not immediately say anything about the classical Morse-function realization problem for trees."],"forward_implications":["Every finite tree is realizable as the Reeb graph of a Morse-Bott real algebraic function whose defining polynomials have degree at most 2.","The realizing function is explicit: the construction gives the actual lines and circles, so for any tree the equations are in principle written down.","The class of graphs with simple cactus tree decompositions whose blocks are edges or circles is also realizable, subject to the conditions in Theorem 4.","The result reduces a question about Reeb graphs of algebraic functions to a planar configuration problem about circles and lines."],"supporting_citations":[{"why":"Supplies Theorem 3, the reconstruction theorem for real algebraic maps onto planar regions that the proof applies to the chosen lines and circles.","marker":"[7]"},{"why":"Restates Theorem 3 in a self-contained way and gives the moment-like map construction used in the proof.","marker":"[9]"},{"why":"Establishes the base case of the reconstruction method when the curves are mutually disjoint, which Theorem 3 extends to intersecting curves.","marker":"[6]"},{"why":"Provides the singularity theory used to check that the resulting function is Morse-Bott.","marker":"[4]"},{"why":"Shows that the Reeb space of a smooth function on a closed manifold is a graph, fixing the terminology the theorem relies on.","marker":"[19]"},{"why":"Defines Morse-Bott functions, the class to which the constructed function is claimed to belong.","marker":"[3]"},{"why":"Supplies background results on real algebraic geometry used to verify that the zero set is a non-singular real algebraic manifold.","marker":"[11]"}],"fun_headline_variants":["Every tree is a Reeb graph","All trees as Reeb graphs","Trees realized as Reeb graphs","Line-and-circle Reeb graphs for every tree","Explicit construction: every tree is a Reeb graph"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The construction assumes that the local preimage bookkeeping around every intersection of the chosen lines and circles works exactly as asserted, so that no unintended critical values appear and the Reeb graph has precisely the vertices and edges of the given tree.","fun_headline_variants_meta":{"raw":{"variants":["Every tree is a Reeb graph","All trees as Reeb graphs","Trees realized as Reeb graphs","Line-and-circle Reeb graphs for every tree","Explicit construction: every tree is a Reeb graph"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000192,"raw_usage":{"total_tokens":1311,"prompt_tokens":871,"completion_tokens":440,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":487,"completion_tokens_details":{"reasoning_tokens":375}},"tokens_in":487,"tokens_out":440,"duration_ms":4520,"temperature":1.0,"reasoning_tokens":375,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-06T05:59:22.385128+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Take a concrete small tree, say a path with three edges or a tree with one vertex of degree 3, write out the lines and circles specified by the proof, and compute the number of connected components of the preimage $f^{-1}(t)$ for a dense set of values $t$ between the chosen critical levels. If any value of $t$ shows a component-count change that does not correspond to a vertex of the tree, the Reeb graph would acquire an extra vertex and Theorem 1 would be false.","supporting_citations":[{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Supplies Theorem 3, the reconstruction theorem for real algebraic maps onto planar regions that the proof applies to the chosen lines and circles."},{"cited_title":"Moment-like maps and real algebraic functions with prescribed preimages","cited_arxiv_id":"2506.17791","evidence_quote":"Restates Theorem 3 in a self-contained way and gives the moment-like map construction used in the proof."},{"cited_title":"Golubitsky and V","cited_arxiv_id":null,"evidence_quote":"Provides the singularity theory used to check that the resulting function is Morse-Bott."},{"cited_title":"Koll´ ar,Nash’s work in algebraic geometry , Bulletin (New Series) of the American Matem- atical Society (2) 54, 2017, 307–324","cited_arxiv_id":null,"evidence_quote":"Supplies background results on real algebraic geometry used to verify that the zero set is a non-singular real algebraic manifold."}],"review_version":1}