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REVIEW 3 major objections 5 minor 25 references

Arrangements of small circles for Morse-Bott functions

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Pith's one-line read Adding a tiny circle centered on an existing circle changes the region's Poincaré-Reeb V-digraph in one of a specific, listed set of local ways.

desk verdict Useful construction and a plausible local picture, but the stated complete list for vertical poles is missing a degree-2 case, so the main classification theorem is not correct as written. read the letter →

arxiv 2412.03846 v3 pith:P45UK3SJ submitted 2024-12-05 math.AG math.COmath.GNmath.GT

Naoki Kitazawa This is my paper · ORCID
classification math.AGmath.COmath.GNmath.GT MSC 14P0514P1052C1557R4558C05
keywords arrangementsofcirclesMorse-BottfunctionsPoincaré-ReebgraphsReebrealalgebraicmapscircle-centeredlocalgraphchanges
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

This paper studies circle arrangements that arise as images of real algebraic maps and that carry Morse-Bott functions when composed with a projection. It aims to give a systematic construction—starting from disjoint circles and repeatedly adding a sufficiently small circle centered at a point of an existing circle—and to describe how the associated Poincaré-Reeb V-digraph changes at each step. The main result is a complete list of local graph changes for the generic cases: the new center can be a non-pole point, a vertical pole, a horizontal pole, or an intersection point of two existing circles, and each yields one of a short list of local moves. For vertical poles an extra genericity assumption is needed, and the general vertical case is posed as an open problem. If the list is correct, it gives a hands-on way to construct explicit real algebraic maps with prescribed Reeb-type graphs.

What carries the argument

The central object is the Poincaré-Reeb V-digraph of a bounded region surrounded by circles: the quotient space obtained by collapsing each connected component of each horizontal (or vertical) slice of the region to a point, with vertices at pre-vertices (vertical or horizontal poles and circle intersections) and with orientation and vertex labels inherited from the projection height. The operation that carries the argument is the inductive addition of a sufficiently small circle centered at a point of an existing circle, which locally removes the closed disk bounded by the new circle from the region and is then read off in the V-digraph by local observations of circles, tangent lines, and poles. The proof works by a case analysis organized around the type of the center point and the signs of the components of the two canonical angle segments at an intersection point; it relies on the fact that the new circle is small enough that all new intersections lie in a prescribed neighborhood and are transverse.

What would settle it

Construct a circle-centered arrangement and choose a center point for which no radius, however small, avoids triple intersections or pole intersections while keeping the new circle inside the region; then check whether the resulting Poincaré-Reeb V-digraph is isomorphic to one of the listed moves. For the vertical-pole case, take a vertical pole whose vertical segment contains another vertical pole and compute the graph change to see whether it matches the pattern in Theorem 5 or violates the list; a single mismatch would falsify the claimed completeness.

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Extended reading notes

Core claim

The paper claims that the operation of adding a sufficiently small circle centered at a point of an existing circle transforms the Poincaré-Reeb V-digraph of the region in one of the explicitly listed local ways. In the generic cases, the chosen point is either a non-pole point, a vertical pole, a horizontal pole, or an intersection point of two circles. Theorems 2 and 3 enumerate the possible local changes for each case—for example, a non-pole point adds a pendant edge to the interior of an edge or splits a vertex in one of two ways, while a horizontal pole either subdivides an edge into two vertices or splits a vertex and adds up to two pendant edges. The paper further shows that arrangements built inductively by this circle-centered procedure are always MBC arrangements (Theorem 1) and that, within that class, certain of the enumerated moves never actually occur (Theorem 4). The vertical-pole case is resolved only under the assumption that the vertical segment through the pole contains no other vertical poles or circle intersections; the general case is left open as Problem 1.

Load-bearing premise

The load-bearing premise is that the new circle can always be chosen 'sufficiently small' so that all new intersections are transverse, avoid poles and triple points, and lie in a prescribed neighborhood, and, for vertical poles, that the vertical segment through the pole contains no other vertical poles or circle intersections—conditions that are never quantified, with the general vertical case left open.

Editorial extensions

If this is right

  • If the list is correct, every generic circle-centered arrangement built from disjoint circles has a Poincaré-Reeb V-digraph obtained from the initial graph by a finite sequence of the listed local moves.
  • The construction yields explicit real algebraic maps whose compositions with a projection are Morse-Bott functions with prescribed Reeb V-digraphs, at least within the generic class.
  • The non-occurrence result (Theorem 4) restricts which local moves can appear in circle-centered arrangements, so any graph requiring the excluded moves cannot arise from this construction.
  • The vertical-pole case, once the extra genericity condition is dropped, is described explicitly in Theorem 5 for one non-generic configuration, giving a concrete pattern for further classification.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • The smallness and genericity assumptions are never quantified, so a natural next step is to bound the radius of the new circle in terms of the existing circles' radii and distances; a failure of such a bound would narrow the scope of the enumeration.
  • The list of local moves resembles a rewriting system on V-digraphs; if it is confluent or terminating in some measure, it could give a normal form for graphs arising from circle-centered arrangements.
  • The same inductive 'small circle centered on existing circle' construction could be tested in higher dimensions with spheres or other hypersurfaces, where the analogous local moves would be larger in number but likely follow the same case structure.
  • The excluded moves in Theorem 4 suggest a characterization problem: which V-digraphs admit a circle-centered realization? This is not asked in the paper, but the list of moves gives a first obstruction.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

3 major / 5 minor

Summary. The paper introduces MB circle arrangements (MBC arrangements), circle-centered (MBCC) arrangements, and their Poincaré-Reeb V-digraphs. The central results (Theorems 1–5) claim that circle-centered arrangements are MBC arrangements and that adding a sufficiently small circle centered at a point of an existing circle changes the Poincaré-Reeb V-digraph in one of a short list of explicitly described local ways, depending on whether the center is a non-pole point, a vertical pole, a horizontal pole, or an intersection point of two circles. The proofs are given as local observations supported by figures; one case is deferred as an open problem and a supplementary discussion (Theorem 5) is added afterward.

Significance. If the claimed classification were complete, the paper would provide a concrete and elementary bridge between arrangements of circles and Reeb graphs of explicit real algebraic Morse-Bott functions, extending the author's earlier construction in [11] and connecting to the Poincaré-Reeb graph literature [2, 21, 22]. The paper is self-contained in its definitions and is honest in labeling some unresolved cases. However, the completeness of the local list is the main contribution, and the current proof does not establish that completeness; moreover, Theorem 2(2) appears to miss a case even under its stated hypotheses. The underlying idea is valuable, but the main claim is not yet verified in the form presented.

major comments (3)
  1. [§3.2, Theorem 2(2)] Theorem 2(2) is not exhaustive even under its stated hypothesis. For a vertical pole p whose vertical fiber is a single interval, the pre-vertex v0 = q_{DS,1}(p) has degree 2 in the Poincaré-Reeb V-digraph, because vertical poles are not critical points of the projection π_{2,1,1} and the graph is locally a single edge with an artificial vertex at the x-coordinate of p. The hypothesis that the vertical segment contains no other vertical poles or circle intersections does not exclude this situation. Concretely, take C1: x^2+y^2=4, C2: (x-1/2)^2+(y+3)^2=4, DS = interior(C1) ∩ exterior(C2), and p=(0,2). The vertical segment {0} × (-3+√15/2, 2] contains no vertical poles or circle intersections apart from p itself, so the hypothesis of Theorem 2(2) is satisfied, yet the graph has a degree-2 vertex at x=0. Neither subcase (2a) (degree 1) nor (2b) (degree 3) applies, and no degree-2 vertical-pole move is listed anywhere in Theorem 2. Thus the claimed complete list of local moves is incomplete; the later discussion in §3.4 (Problem 1, Theorem 5) is not incorporated into Theorem 2(2) and does not repair this gap.
  2. [§3.2, proof of Theorems 1–3] The proof of Theorems 2 and 3 does not establish the completeness of the enumeration. The proof states: 'FIGURE 2 shows a complete list of local observations ... We assume the list.' Most subcases are then justified only by phrases such as 'By considering the location of the circles' (e.g., cases (1a), (2a), (2b), (3a), (3b), and Theorem 3(5)). For a classification theorem whose content is exactly that every generic addition produces one of the listed moves, a proof of exhaustiveness is required; relying on an assumed list and on figures leaves the central claim unverified. This is a load-bearing gap, not a presentational one.
  3. [§3.3–§3.4, Theorem 4 and Theorem 5] The relationship between the main theorem and the supplementary statements is not clear enough to compensate for the missing case. Theorem 4 is proved by referring back to 'our proof of Theorems 1–3' and to Figure 2, but since the completeness of that proof is in question, Theorem 4 inherits the same uncertainty. Theorem 5 is stated after the proof, explicitly for the case where the condition in Theorem 2(2) is dropped, and it is not used to state a corrected version of Theorem 2(2). The paper should either restate Theorem 2(2) with a complete list that includes the degree-2 vertical-pole move, or clearly mark Theorem 2 as a partial result and move the complete statement to a later section with a full proof.
minor comments (5)
  1. [Abstract and throughout] There are several typos in the abstract, including 'attr act us' and 'circl es', and the spelling 'Poinar´e-Reeb' appears in §2.2 alongside 'Poincaré-Reeb' elsewhere; the text needs copyediting.
  2. [§3.2, Theorem 1(2)] The condition 'p = xj′ ∈ Sxj,rj ⋂ DS' is problematic because DS is an open connected component of the complement of the circles and is disjoint from them; the intended condition is likely p ∈ Sxj,rj ∩ closure(DS).
  3. [§2.2, definition of pre-vertex] For i=1, vertical poles are included among pre-vertices even though they are not critical points of π_{2,1,1}; this is the source of the degree-2 artificial vertices discussed above, and a remark clarifying this point would help the reader.
  4. [Figure 2 caption] The caption says 'the remaining four is for a case where p is a vertical pole or a horizontal pole'; the grammar and the distinction between the two cases should be clarified, and the figure would be easier to use if each subcase were labeled by the corresponding part of Theorems 2 and 3.
  5. [§3.2, Theorem 2(1a) and (1b)] The inequalities defining the labels of the new vertices are written in a verbose way; they could be summarized with a small diagram, and the orientation of the new edge e should be stated explicitly in a single unambiguous sentence.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the local-change theorems are derived by direct geometric inspection, and the author's prior work is used only as background.

full rationale

The central claims (Theorems 1–3) are not equivalent to their inputs. The MB and MBCC classes are defined independently (Definitions 2 and 3), and Theorem 1 is proved from the local geometry of a small circle centered on an existing circle rather than from the definition itself. Theorem 2's proof explicitly re-derives case (1) ("However, we argue this case again here"), and Section 2.2 explicitly sets the real-algebraic-map construction aside ("Hereafter, we do not need or use such arguments on such Morse-Bott functions"), so the citations to [11] are background, not load-bearing. No parameter is fitted and no quantity is renamed and then predicted. The paper openly states that the general vertical-pole case is unsolved (Problem 1), and the proof relies on figures and "local observations"; those are completeness or rigor limitations, not circularity. Therefore no circular step is present.

Assumptions & free parameters 0 free parameters · 4 assumptions · 0 invented entities

No free parameters fitted to data appear; this is a pure mathematical construction. The axioms are standard theorems plus the paper's own smallness and vertical-line assumptions. No new physical entities are introduced; the new objects (MB/MBCC arrangements, Poincaré-Reeb V-digraphs, canonical angle segments) are mathematical definitions, not entities requiring independent empirical evidence.

assumptions (4)
  • standard math Ehresmann's theorem applies to the cornered manifold DS and the projection π_{2,1,i}|_DS.
    Used in the proof of Proposition 1 to show the quotient space is a graph; the paper cites [19, Theorem 2.12].
  • standard math Reeb graph theory for Morse-Bott functions gives the isomorphism between the Poincaré-Reeb graph and the Reeb graph of the composed Morse-Bott function.
    Used to identify the graph with a Reeb graph; based on [18, 19, 17].
  • domain assumption The newly added small circle can be chosen small enough that all new intersections stay in a prescribed small neighborhood of its center, with no triple intersections and no pole intersections.
    Central to the local-change proofs; 'sufficiently small' is never quantified (Definition 3, Theorem 1(2)).
  • ad hoc to paper For a vertical pole, the vertical segment through the pole contains no other vertical poles or circle intersections (I-type and II-type classification).
    Explicitly assumed in Theorem 2(2); the cases where it fails are deferred as Problem 1.

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Cite this review

Pith. "Pith review of Arrangements of small circles for Morse-Bott functions." pith.science (2026). https://pith.science/paper/P45UK3SJ

@misc{pith2026241203846,
  author       = {Pith},
  title        = {Pith review of: Arrangements of small circles for Morse-Bott functions},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/P45UK3SJ}},
  note         = {Machine review of arXiv:2412.03846}
}
read the original abstract

As a topic of mathematics, "arrangements", systems of hyperplanes, circles, and general (regular) submanifolds, attract us strongly. We present a natural elementary study of arrangements of circles. It is also a kind of new studies. Our study is closely related to geometry and singularity theory of Morse(-Bott) functions. Regions surrounded by circles are regarded as images of real algebraic maps and composing them with projections gives Morse-Bott functions: this observation is natural, and surprisingly, recently presented first, by the author. We present a systematic way of constructing such arrangements by choosing small circles centered at existing circles inductively. We are interested in graphs the regions surrounded by the circles naturally collapse. We have studied local changes of the graphs in adding these circles. These graphs are essentially so-called {\it Reeb graphs} of the previous Morse-Bott functions: they are spaces of all components of preimages of single points for the functions.

Figures

Figures reproduced from arXiv: 2412.03846 by the authors.

Figure 1
Figure 1. The case of Example 1. Blue dots show vertical poles and red dots show horizontal poles. Graphs are Poinar´e-Reeb graphs of DS . 2.3. Some observations on circle arrangements. Some elementary transfor￾mations produce MBC arrangements which are different from the original ones and which are ”congruent” to them. We present this property as follows. We can easily check from fundamental properties of the transformations… view at source ↗
Figure 2
Figure 2. Local observations for p = xj ′ ∈ Sxj′ ,rj′ . Blue colored lines show vertical lines and horizontal lines. Dots are for vertical poles, horizontal poles, and the centers of circles. Dotted lines are for straight lines tangent to circles at points contained in exactly two circles Sxj,rj ∈ S and Sxj′ ,rj′ ∈ S′ . Furthermore, each of the first four is for a case where p is not a vertical pole or a horizontal pole. In a… view at source ↗
Figure 3
Figure 3. A case for the case (1b). The blue circle shows the circle Sxj′ ,rj′ and the change of our graph is depicted in blue. Hereafter, we use colored objects in this way. Vertices and edges are shown by the notation with dotted segments [PITH_FULL_IMAGE:figures/full_fig_p014_3.png] view at source ↗
Figures from the paper (6 more)
Figure 4
Figure 4. Figure 4: The case (2a): vertices and edges are shown by the notation [PITH_FULL_IMAGE:figures/full_fig_p014_4.png]
Figure 5
Figure 5. Figure 5: The case (2b): vertices and edges are shown by the notation [PITH_FULL_IMAGE:figures/full_fig_p014_5.png]
Figure 6
Figure 6. Figure 6: Explicit cases for the case (3). We omit the notation on vertices and edges. cases show simplest global examples. We consider the case where after a suitable operation of Proposition 2 (1), we have the case p is in the circle centered at the origin 0 and p = (0, p2) wi…
Figure 7
Figure 7. Figure 7: A local picture around a point p ∈ Sxj1 ,rj1 T Sxj2 ,rj2 (for the case (5a)). The set of all points of DS which are also contained in the interior of the closed disk bounding the new circle SXj′ ,rj′ , depicted partially as an arc colored in blue, is colored in green. …
Figure 8
Figure 8. Figure 8: Example 2. Dots are for horizontal poles and vertical poles of circles and vertices of Poincar´e-Reeb graphs for example. Blue colored lines show horizontal lines and vertical lines related to this case [PITH_FULL_IMAGE:figures/full_fig_p018_8.png]
Figure 9
Figure 9. Figure 9: A simplest case for general cases of Theorem 2 (2). By deleting vertices colored in blue and collapsing the edge depicted as a small segment in red, we have a graph isomorphic to our original Poincar´e-Reeb V-digraph. In Theorem 5, it is discussed again and we add the …

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