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REVIEW 4 major objections 4 minor 49 references

Fundamental examples of height functions on closed manifolds and their 1st derivatives

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Pith's one-line read This paper constructs explicit closed manifolds whose height functions have '1st-derivative-on-P' functions that are smooth, with critical sets made of spheres and products of spheres, and determines exactly when those derivative functions

desk verdict A useful explicit construction, but the derivative-function theorem is not well-posed as printed; send to peer review only if the author fixes the definition and the gradient-flow claim. read the letter →

arxiv 2608.00556 v1 pith:QV6QCJUE submitted 2026-08-01 math.DG math.AGmath.DSmath.MG

Naoki Kitazawa This is my paper · ORCID
classification math.DGmath.AGmath.DSmath.MG MSC 53A0757R4558C0558C2526B10
keywords heightfunctionsMorse-BottroundsingularitytheoryofdifferentiablemapsclosedmanifoldsfirstderivativesonPgradientflowsimplicitfunctiontheorem
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

The paper sets out to show that an explicit family of closed manifolds—cut out by three simple equations in Euclidean space—carries height functions for which a derivative variant, the '1st derivative on P', is a smooth function with fully controlled critical behavior. This derivative variant is obtained by measuring the first-coordinate component of the height function's gradient flow and applying P1(x)=x² to its square; the paper shows its critical set is always a union of spheres and products of spheres. Depending on whether the bounding curve's slope vanishes at the origin and on a parameter t0, the derivative function is Morse-Bott (critical set a union of submanifolds with nondegenerate normal Hessian) or round (critical set a finite union of points and spheres), with exact conditions in terms of the dimensions m1 and m2. This matters because height functions are basic building blocks of Morse theory and singularity theory: explicit examples whose first derivative is also understood provide test cases for constructing functions with prescribed critical sets and Reeb graphs.

What carries the argument

The carrying object is the closed submanifold X_{D_{S,a,b,c_+,a,t0},m1,m2}⊂R^{m+3}, cut out by three equations; the implicit function theorem shows it is a smooth closed manifold and controls where the height function π_{m+3,1} has critical points. The second mechanism is the paper's '1st derivative on P': at each non-critical point, look at the unit tangent vector along the gradient flow of the height function, take its first-coordinate component a_v, and evaluate P(a_v²), with critical points assigned value 0. Taking P1(x)=x² turns the height function into a new smooth function whose critical points occur where that component of the gradient flow is zero. The monotonicity assumptions on |c

What would settle it

Fix t=t1 and y2 values, set x1=0 in X_{D_{...}}, and compute the value of π_{m+3,1}|X^{(1,P1)} from the defining equations and the gradient-flow vector; the paper predicts 1/((t1 c_+'(0))²+1). Any other value refutes the classification. Alternatively, apply the Section 2.2 definition literally to the unit sphere with P1(x)=x²: the result would be sin^4(θ), with extra critical points at θ=π/2, 3π/2, unlike Example 1's sin²θ.

Watch

Extended reading notes

Core claim

The central result, Theorem 3, concerns a closed manifold X_{D_{S,a,b,c_+,a,t0},m1,m2} cut out by three quadric equations: one in x1 and y1, one graph equation tc_+(x1)=x2, and one in t and y2. The paper shows its 1st-derivative-on-P function is smooth, with critical set over x1=−a,0,a. If |c_+'(0)|=0, the critical set is two S^{m2} components plus one S^{m1−1}×S^{m2}, and the function is round exactly for m1=1. If |c_+'(0)|>0 and t0<0, the critical set is two S^{m2}, two S^{m1−1}, and one S^{m1−1}×S^{m2−1}, and the function is round exactly when at least one of m1,m2 is 1. If |c_+'(0)|>0 and t0≥0, the product component vanishes, and the function is always round.

Load-bearing premise

The classification rests on the Section 2.2 definition of the '1st derivative on P' function; the paper explicitly says it omits rigorous exposition there, and Example 1's computed sin²θ conflicts with a literal reading of the stated composition P1(av²), so if that definition is not fixed, Theorem 3's critical-set statement is not well posed.

Editorial extensions

If this is right

  • The construction yields an explicit infinite family of closed manifolds for which both a height function and its 1st-derivative-on-P function have completely known critical sets.
  • In the |c_+'(0)|=0 regime, roundness of the derivative function is equivalent to m1=1; otherwise the S^{m1−1}×S^{m2} component prevents the critical set from being a union of spheres and points.
  • In the |c_+'(0)|>0, t0<0 regime, roundness is equivalent to at least one of m1,m2 being 1.
  • In the |c_+'(0)|>0, t0≥0 regime, the derivative function is always round, with critical set two S^{m2} components and two S^{m1−1} components.
  • The explicit formula for the derivative function's value at x1=0, namely 1/((t1 c_+'(0))²+1), gives a direct quantitative signature any realization must reproduce.

Reading between the lines

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

  • Because Remark 1 allows replacing (t−t0)(1−t) by any function F(t) with the same sign and nonzero endpoint derivative, the same sphere-factor critical sets should persist under that replacement; this would confirm that the quadratic normalization is not essential to the classification.
  • Iterating the '1st derivative on P' operation, or applying it to the higher canonical projections π_{m+3,k}, could produce a hierarchy of functions whose critical sets are built from the same sphere factors; the paper does not explore this.
  • The m1/m2 roundness conditions suggest a recipe for designing round functions with prescribed sphere components: choose one of the dimensions equal to 1 to eliminate product critical components, then shape |c_+'| to control where the remaining components sit.
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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

4 major / 4 minor

Summary. The paper proposes an explicit family of closed manifolds X_{D_{S,a,b,c_+,a,t0},m1,m2} ⊂ R^{m+3}, defined as the intersection of an S^{m1}-type equation in (x1,y1), a sphere equation in (t,y2), and the graph condition x2 = t c_{+,a}(x1). It studies the height function π_{m+3,1} restricted to these manifolds, and then introduces a '1st derivative on P' function c_{eX,k}^{(1,P)} in Section 2.2. Theorem 2 asserts that the manifolds are smooth and that the height function is Morse-Bott, with specified critical components and indices. Theorem 3, the main result, describes the critical set of c_{eX,k}^{(1,P1)} for P1(x)=x^2, in three cases depending on |c'(0)| and t0, and gives roundness criteria. The argument relies on previously published construction methods and on the author's unpublished preprints [21] and [25].

Significance. If the main theorem were correct, the paper would provide rare explicit examples of closed manifolds for which the critical set of the derivative of a height function can be completely described and controlled, with potential applications to the study of round functions, Reeb graphs, and singularity theory. The algebraic construction of X_{D,...,m1,m2} is concrete, and the implicit-function-theorem strategy for proving smoothness is plausible and checkable. However, the central object c_{eX,k}^{(1,P1)} is not defined consistently, and the proof of Theorem 3 uses a false statement about the gradient flow of the induced metric. The main theorem is therefore not well-posed as stated. The paper also depends on unpublished preprints for load-bearing ingredients, despite a statement in the final section that it does not. No machine-checked proofs, reproducible code, or parameter-free derivations are supplied; the contribution is conceptual rather than computationally verified.

major comments (4)
  1. [§2.2 and Example 1] The definition of c_{eX,k}^{(1,P)} is internally inconsistent. The text says the value at a noncritical point is P(av_{p,c,eX,k,+}^2). With P1(x)=x^2, this would be (av)^4. Example 1, however, computes the value as (sin θ)^2, and its second derivative 2 cos 2θ is the second derivative of sin^2 θ, not of sin^4 θ. Thus the function whose critical set is classified in Theorem 3 is never given a consistent definition. The sentence 'omit related rigorous exposition on this' cannot replace a definition of the central object. Theorem 3 is not well-posed until this is fixed.
  2. [§3, proof of Theorem 3, gradient-flow paragraph] The proof asserts that 'the gradient flow associated to the height function π_{m+3,1}|X is seen to be along fixed numbers t = t1, by our construction.' This is false for the induced Euclidean metric. In local coordinates (x1,t) on the generic part of X, the induced metric has cross term g_{x1t}= t c_{+,a}(x1)c'_{+,a}(x1), which is generally nonzero. Therefore the unit vector orthogonal to ker dπ_{m+3,1} has a nonzero t-component, and the stated value 1/((t1 c'(0))^2+1) corresponds to a computation that ignores this cross term. Consequently the determination of S(c^{(1,P1)}) and the roundness criteria in Theorem 3 do not follow from the stated definitions.
  3. [Theorem 2(3)] The listed 'preimages of graphs' are not critical sets of π_{m+3,1}|X and have the wrong diffeomorphism type. For the graph x2 = t0 c_{+,a}(x1), the equations force t=t0 and y2=0, leaving S^{m1-1} fibers over x1; the preimage is diffeomorphic to S^{m1}, not S^{m2}. The same holds for the graph x2 = c_{+,a}(x1), where t=1 and y2=0. Since these fibers are regular fibers of the height function, the Morse-Bott index is not defined on them. This invalidates the supporting statement and the way it is used in the proof of Theorem 3.
  4. [§3, proof of Theorems 2 and 3; final section] The proof of Theorem 2 says 'Some main ingredients of the proof are first in [21], and some are first in [25].' The final section says 'We do not assume non-trivial arguments in formally unpublished preprints.' References [21] and [25] are arXiv preprints, not formally published. The submanifold construction and the flow assertions used in Theorem 3 therefore cannot be verified from the present text. The paper needs either to make these ingredients fully self-contained or to cite published sources; the present reliance on unpublished work is not adequate for the central claims.
minor comments (4)
  1. [Throughout] There are many notation inconsistencies, e.g., 'Sc+.a,1', 'c+a,t0', 'c+.a|', and 'Gc+a,t0'. These should be regularized; as printed they make the already technical statement harder to read.
  2. [§3, Theorem 3 statement] The sets written as π_{m+3,2}^{-1}({(0,c+(0)) | x2 ∈ R}) are confusing: the set notation suggests a single point in R^2 rather than a vertical line. The intended preimage is likely {(0,y): y∈R} (or {(0,x2): x2∈R}), and the notation should be corrected.
  3. [Abstract and Introduction] The phrasing 'We including the author' and 'We the author' is grammatically odd and should be edited. The paper would also benefit from a precise statement of the assumptions on c_{+,a} in Theorem 3 in displayed form.
  4. [§2.2] The quantities 'av_{p,c,eX,k,+}^2 ≤ 1' and 'length av ≤ 1' are never precisely defined. In particular, the relation between the normalized tangent vector and the value of the projected differential needs a rigorous definition before any theorem about c^{(1,P)} can be proved.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: the central claims are explicit construction-based computations; the author's self-citations are contextual provenance rather than load-bearing inputs.

full rationale

Theorem 3 is a conditional statement about an explicitly defined manifold X_{DS,a,b,c+,a,t0,m1,m2} and an explicitly defined function c^{(1,P1)}, computed by inspecting the gradient flow of the height function. The proof chain is: Theorem 1 (reviewed self-contained via implicit function theorem) constructs smooth submanifolds; Theorem 2 applies it to the concrete region and computes the critical set of the height function by partial derivatives; Theorem 3 then computes the critical set of c^{(1,P1)} under the stated monotonicity assumptions on |c'|. No parameter is fitted to a subset of data and then renamed a prediction. The unpublished preprints [21] and [25] are cited for provenance ('Some main ingredients of the proof are first in [21], and some are first in [25]') and for comparison ('Compare this to [25, Theorems 4, 5 and 6]'), but the proof does not import a nontrivial conclusion from them and the paper explicitly says it does not assume non-trivial arguments in formally unpublished preprints. No uniqueness theorem from the author's prior work is invoked to force a choice. The internal inconsistency in §2.2 (P1(av^2) with P1(x)=x^2 would give (sin θ)^4 whereas Example 1 states (sin θ)^2) is a well-posedness/correctness defect, not a circular reduction of the conclusion to its inputs; likewise the asserted gradient-flow direction along fixed t is a geometric claim that can be checked from the metric, not a definitional equivalence. Thus there is no circular step warranting a nonzero score.

Assumptions & free parameters 0 free parameters · 6 assumptions · 1 invented entities

No data fitting or fitted constants appear: a, t0, and c_+,a are arbitrary inputs satisfying explicit inequalities, not fitted to data. The heavy reliance is on the author's unpublished preprints [21] and [25] and on a new derivative definition with an internal inconsistency. No new particles or forces are introduced.

assumptions (6)
  • standard math Implicit function theorem applies to the three defining equations at every point of the constructed manifold.
    Invoked throughout the proof of Theorem 2 in the three cases (Cases 2-1, 2-2, 2-3) to show the solution set is an m-dimensional submanifold.
  • standard math Standard Morse-Bott theory identifies indices by Hessians on normal bundles.
    Used in Example 1 and in Theorem 2(3) to assign indices 0 and m1 to critical components, and in Theorem 3 to call the function round.
  • standard math Mather's singularity theory for smooth maps gives the local trivial family of Morse functions around the critical set.
    The proof of Theorem 2(3) invokes [15] and [29-36] to describe neighborhoods of critical fibers.
  • ad hoc to paper The normalized gradient-flow section v_{p,c,e,+} is a well-defined smooth section of the tangent bundle away from critical points.
    Section 2.2 defines the first-derivative function using this vector but says it omits rigorous exposition on this. The main theorem inherits this definitional gap.
  • domain assumption The function c_+,a is positive and its first derivative absolute value is minimal at 0 and monotone toward the endpoints.
    Theorem 3 assumes these hypotheses to force the critical set of the derivative function to lie over x1=0 and the boundary fibers.
  • ad hoc to paper The constructions and estimates of the author's preprints [21] and [25] are valid.
    The proof of Theorem 2 says some main ingredients of the proof are first in [21] and some are first in [25]. These are author's own preprints, not formalized or independently reproduced here.
invented entities (1)
  • the first-derivative-on-P function c_{eX,k}^{(1,P)}
    purpose: Assigns to each point the value of P applied to the squared size of the normalized gradient-flow component of the height function; this is the object whose critical set Theorem 3 classifies.
    New definition in Section 2.2, specific to this paper. Its consistency is questionable because Example 1 computes sin^2(θ) under P1(x)=x^2 rather than the stated P1(sin^2(θ))=sin^4(θ).

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Cite this review

Pith. "Pith review of Fundamental examples of height functions on closed manifolds and their 1st derivatives." pith.science (2026). https://pith.science/paper/QV6QCJUE

@misc{pith2026260800556,
  author       = {Pith},
  title        = {Pith review of: Fundamental examples of height functions on closed manifolds and their 1st derivatives},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/QV6QCJUE}},
  note         = {Machine review of arXiv:2608.00556}
}
read the original abstract

Height functions are fundamental and important in mathematics, especially in geometry, more explicitly, singularity theory of differentiable maps and applications to differential topology and differential geometry. We including the author are interested in constructing explicit maps onto regions in Euclidean spaces and functions represented as compositions of them with the canonical projections. The author has been interested in and succeeded in constructing these maps with the functions being height functions systematically. This is important in singularity theory of differentiable maps. We study global behaviors of the 1st derivative of such a function. The author is also interested in topological properties and combinatorial ones of such functions. For so-called Morse functions and functions of generalized classes, these properties are still studied actively mainly by Gelbukh and Michalak, and their 1st derivatives are of a kind of new challenges.

Figures

Figures reproduced from arXiv: 2608.00556 by the authors.

Figure 1
Figure 1. A great circle in the unit sphere S m. A local coordi￾nate θ and the tangent vector of length 1, which is regarded to be a vector along the gradient flow associated to πm+1,1|Sm, are presented, for example. Note that the orientation of the tangent vector is different from that of the (so-called positive) gradient flow associated to πm+1,1|Sm. Theorem 1. Let {Sj} l j=1 be a family of mutually disjoint l > 0 smooth su… view at source ↗
Figure 2
Figure 2. DS,a,b,c+,a,t0 and associated objects such as curves and graphs, in Theorems 2 and 3. No data other than the present file is generated, related to our paper. We do not assume non-trivial arguments in formally unpublished preprints. Referring to these preprints to some extent is no problem. Motivated by [25], the author has presented this paper. More precisely, the author has been motivated by the news that this has … view at source ↗

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Pith tools

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