Existence of a constant-mean-curvature hypertorus in $S^4$ via computer assistance

Oscar Perdomo

REVIEW 3 major objections 5 minor 1 cited by

Existence of a constant-mean-curvature hypertorus in $S^4$ via computer assistance

T0 review · 3 major / 5 minor · reviewed 2026-08-15 · deepseek-v4-flash

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Pith's one-line read The paper proves that the unit four-dimensional sphere contains a previously unknown embedded three-dimensional torus with constant mean curvature -3, via an exact rational-arithmetic computer-assisted argument.

desk verdict A promising computer-assisted proof with a real gap: the a=0.5204 trajectory starts outside the verified box, so the Poincare-Miranda left edge is unproved. read the letter →

arxiv 2506.19555 v1 pith:P3345YOP submitted 2025-06-24 math.DG

Oscar Perdomo This is my paper · ORCID

classification math.DG MSC 53C4265G3065L0534A12

keywords constantmeancurvaturehypertorus4-spherecomputer-assistedproofRoundTaylormethodPoincaré–Mirandatheoremrationalarithmeticordinarydifferentialequations

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 proves the existence of a new compact embedded constant-mean-curvature hypersurface of dimension three, called a hypertorus, inside the unit four-dimensional sphere $S^4$, with mean curvature $H=-3$. The proof is computer-assisted but fully rigorous: it integrates a three-dimensional ordinary differential equation with the Round Taylor method, which keeps all arithmetic exact by rounding to rational numbers and includes a built-in error bound, and then applies the Poincaré–Miranda theorem to force parameter values $a^*$ and $t^*$ where the profile curve closes up. Such examples are hard to certify because floating-point rounding can spoil the argument, and the known families do not include this hypertorus. The paper thereby converts numerical exploration into a theorem about the geometry of $S^4$.

What carries the argument

The load-bearing machinery is the Round Taylor Method (RTM), a Taylor-method integrator whose iterates are rounded to a fixed rational grid with the floor function, so every computed point is an exact rational number and round-off is inside the error analysis. Its error theorem states that if all computed points $z_j$ stay in a box $U_1$, if $f$ and its partials are controlled on a slightly larger box $U_2$, and if the relevant Taylor coefficients are bounded there, then the true solution exists and is within $\tilde R<0.0003048$ of the computed points. The second mechanism is the Poincaré–Miranda theorem, a two-variable intermediate value theorem, applied on the rectangle $[0.5204,0.5244]\times[0.3966,0.3991]$ to the functions $F=\alpha(a,t)-\pi/2$ and $G=\theta(a,t)-\pi/4$. The third is the Gronwall comparison theorem (Theorem 1.4), which controls how much $\alpha$ and $\theta$ change when the initial value $a$ moves by at most $0.002/15$. Together these components force the existence of the $(a^*,t^*)$ pair.

What would settle it

Run the RTM recurrence (3) with $m=1$, $R=10^{-10}$, and $h=0.3966/25000$ or $0.3991/25000$ for the 16 starting values $a_j$, and check whether any rational iterate $z_j$ leaves $U_1=[1.321,1.571]\times[0.522,0.786]\times[1.57,3.142]$; a single violation would invalidate the proof of Theorem 3.1. Independently, a high-precision integrator that found $\alpha(a,t)$ not crossing $\pi/2$ between $t=0.3966$ and $t=0.3991$, or $\theta$ failing to cross $\pi/4$, would contradict the corollary.

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Extended reading notes

Core claim

The central result is Theorem 3.1: there exist real numbers $a^*\in(0.5204,0.5244)$ and $t^*\in(0.3966,0.3991)$ such that the solution $(r(t),\theta(t),\alpha(t))$ of the system $\dot r=\cos\alpha$, $\dot\theta=\sin\alpha/\sin r$, $\dot\alpha=2\cot(2\theta)\cos\alpha/\sin r-3\cot r\sin\alpha-3$, with initial data $r(0)=\pi/2$, $\theta(0)=a^*$, $\alpha(0)=\pi$, is defined on $[0,0.3991]$ and satisfies $\alpha(t^*)=\pi/2$ and $\theta(t^*)=\pi/4$. The symmetries of the differential equation imply the solution continues to $[0,4t^*]$, and the curve $\beta(t)=(\sin r(t)\cos\theta(t),\,\sin r(t)\sin\theta(t),\,\cos r(t))$ closes smoothly. By the construction of [3], the resulting set $M=\{(\sin r(t)\cos\theta(t)\,y,\,\sin r(t)\sin\theta(t)\,z,\,\cos r(t)): t\in[0,4t^*],\ y,z\in S^1\}$ is an embedded hypersurface in $S^4$ with constant mean curvature $H=-3$.

Load-bearing premise

The proof assumes that every one of the 25,000 rational-arithmetic integration steps, for all 16 starting values, stays inside the coordinate box $U_1$ required by the error theorem; the manuscript justifies this with a monotonicity statement but gives no formal containment proof, and if any step left that box the error bound and the existence conclusion would fail.

Editorial extensions

If this is right

The profile curve $\beta(t)$ closes smoothly after time $4t^*$, so the construction yields a compact embedded constant-mean-curvature hypertorus inside $S^4$ with $H=-3$.
The proof certifies that the ODE solution exists on the whole interval $[0,0.3991]$, not just at the sampled points, so no assumption of continued existence is hidden behind the numerics.
Every estimate in the argument is a rational number with an explicit error bound, so a reader can reproduce the certificate exactly without relying on a particular floating-point implementation.
The combination of RTM with Poincaré–Miranda is a general template for turning numerically observed ODE-generated geometry into a rigorous theorem, and this paper applies it to CMC hypertori.

Reading between the lines

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

I would infer that the parameter interval for $a^*$ is not sharp: the proof's recorded gaps between the computed values and $\pi/2$ or $\pi/4$ exceed the combined error, so nearby starting values should also close up; the paper does not claim a maximal window.
I would expect the same rational-arithmetic pipeline to transfer to the higher-dimensional CMC families of the form considered in [3,4], since only the explicit vector field and the boxes would change; that transfer is not carried out here.
The point most worth rechecking is the monotonicity assertion used to place all $z_j$ inside $U_1$; a second independent implementation of the RTM recurrence would turn that assertion into a checked fact.

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Editorial analysis

A structured set of objections, weighed in public.

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

Desk Editor's Note

read the letter

The headline: this paper claims a new embedded CMC hypertorus in S^4, proved by a rational-arithmetic Taylor method plus Poincare-Miranda. I think the conclusion may be true, but the proof as written has a hole at the left edge of the search rectangle, and the stress-test note is right.

What's new: the specific parameter range leading to alpha=pi/2, theta=pi/4 is not in the prior minimal/CMC hypertorus literature. The strategy is sensible: use exact rational arithmetic to control round-off and truncation, then verify boundary inequalities on a rectangle. The RTM error bounds and the comparison theorem are standard, and the constants in Section 2 are worked out carefully. That part is solid.

The problem: Theorem 2.1 requires all RTM points z_j to lie in U1, and the true solution to start in U2. For the trajectory with initial theta = 0.5204, the initial point is z0 = (pi/2, 0.5204, pi). But U1's second-coordinate interval is [0.522, 0.786], and U2 starts at 0.521. So z0 is outside both. The monotonicity assertion in Section 3 can't fix this: later iterates being larger doesn't help if the initial point is already below the lower bound. Thus the displayed error bound 0.0003048 is not established for that trajectory, and the inequality theta(0.5204,t) < pi/4 on [0.3966,0.3991] is unsupported. Since that is exactly the left-edge sign condition for Poincare-Miranda, the existence conclusion doesn't follow from the given proof.

Other issues are secondary but worth mentioning: no code or certificate is shipped, several 'direct computation' bounds are not fully auditable, and the claimed distinctness from the Wei et al. examples is not proved by the stated theorem. These are fixable. The z0 issue is also fixable — e.g., enlarge U1/U2 or start the verified integration after a short non-rigorous step — but as it stands, the proof doesn't close.

For whom: readers interested in computer-assisted proofs in geometry will find the RTM implementation and the error analysis instructive. The existence result itself should not be cited as a theorem yet. If I were the editor, I would send it to a careful referee, not desk reject, because the machinery is real and the flaw is localized. But I would expect a revised version before accepting.

Referee Report

3 major / 5 minor

Summary. The paper claims to prove, by a computer-assisted argument, that there exist a* in (0.5204, 0.5244) and t* in (0.3966, 0.3991) such that the solution of the ODE system for (r, theta, alpha) with initial data (pi/2, a*, pi) satisfies alpha(t*) = pi/2 and theta(t*) = pi/4. From this, Corollary 1.2 constructs an embedded constant-mean-curvature hypertorus in S^4 with H = -3. The proof combines the rational-arithmetic "Round Taylor Method" (RTM) of the author with the Poincaré–Miranda theorem: it estimates alpha on a 16-point grid in the a-direction and theta on two boundary trajectories in the t-direction, then applies the two-variable intermediate-value theorem. Several lemmas give interval bounds for the vector field and its derivatives on a box U2, and the computational core is presented as large tables of rational values together with the assertion that the RTM error is below 0.0003048.

Significance. If the result were fully established, it would provide a new embedded CMC hypertorus in S^4, complementing the recent constructions of Carlotto–Schulz and Huang–Wei, and it would demonstrate the RTM as a viable rigorous tool for low-dimensional ODE existence proofs. The paper's strengths are its explicit rational arithmetic, the transparent error formula, the use of the Poincaré–Miranda theorem, and the concrete, checkable constants in Lemmas 2.4–2.7. However, the proof as written contains a load-bearing failure: the RTM theorem is invoked for an initial condition that lies outside the required box, and the key computational verification is asserted rather than supplied in verifiable form. The central claim is therefore not established in the present version, although the approach appears potentially repairable.

major comments (3)

[Section 3, Proof of Theorem 3.1] The RTM error estimate is invoked for the trajectory with a_0 = 0.5204, but the hypothesis z_j in U1 of Theorem 2.1 fails already at j=0. Definition 2.2 gives b_2 = 261/500 = 0.522, whereas z_0 = (r(0), theta(0), alpha(0)) = (pi/2, 0.5204, pi); the second component is strictly below b_2. Moreover, y_0 is outside U2 because b_2 - epsilon = 0.521 > 0.5204, so the containment conclusion of Theorem 2.1 cannot hold at t=0. Consequently, the displayed bound |Y(jh)-z_j| < 0.0003048 is not proved for the a_0 trajectory, and the inequality theta(0.5204,t) < pi/4 for all t in [0.3966, 0.3991] is unsupported. The sentence "Since all entries ... are monotone, it is not hard to show z_j in U1" cannot repair this because the failure is at the initial iterate. This left-edge sign condition is required by the Poincaré–Miranda theorem, so Theorem 3.1 and Corollary 1.2 are not established as written.
[Section 3, computational verification] The proof delegates essential verification to unverifiable assertions: that z_j lies in U1 for 25,000 steps and 16 trajectories, that the sequences alpha_tilde(a_j, 0.3966) and alpha_tilde(a_j, 0.3991) are monotone, and that the theta_i sequences are monotone. No code, pseudocode, machine-checkable certificate, or detailed derivation is supplied for these claims. Because the RTM error bound depends exactly on these hypotheses, and because one of the asserted containments is false for a_0, the computational part must be made independently verifiable, for example by providing exact rational arithmetic code or a proof of the monotonicity and containment claims.
[Section 3, theta estimates and grid indexing] The indexing in the theta estimates is inconsistent. The quantities theta_i(0.5204) are defined as RTM approximations at i * 0.3991/25000, but in the Lipschitz comparison they are used together with theta(0.5204, i * 0.3966/25000). The two grids have different step sizes, so as written the estimate |theta(0.5204,t) - theta(0.5204,i * 0.3966/25000)| <= ... is not tied to the computed theta_i values. This needs correction; likely 0.3991 should replace 0.3966. In addition, the text contains the apparent typos "for every t in [0.3966, 0.3944]" and "theta_i(0.5004)". Since these estimates carry the right-edge and left-edge sign conditions for theta, they must be fixed before the Poincaré–Miranda step is valid.

minor comments (5)

[Section 1] The sentence "Let us state the our main theorem" contains a grammatical error; it should read "Let us state our main theorem."
[Section 2, Eq. (3)] The displayed definition of the RTM sequence has an indexing typo: "z0 = R floor(1/R yi+1)" should define z_{i+1}; the text should clarify that z_0 = y_0.
[Section 2, Lemmas 2.5–2.7] Several bounds are justified only by "a direct computation" (e.g., the bounds for g7, g8, g9). These are elementary and likely correct, but a short derivation using monotonicity of the trigonometric functions on the stated intervals would make the lemmas easier to verify.
[Section 3, tables] The two large rational tables are hard to check by eye; providing decimal approximations with a clear statement of the guaranteed error or moving the tables to supplementary material would improve readability.
[References] Reference [4] is listed as "Personal communication"; since Corollary 1.2 relies on the embedding result from that line of work, a public reference or a proof sketch of the specific statement being used would be preferable.

Circularity Check

0 steps flagged · score 0.0 of 10

The derivation is not circular: the existence of (a*, t*) is obtained by independent boundary checks with error bounds, not by fitting or by a self-citation chain.

full rationale

The proof is self-contained in the relevant sense. The target existence of (a*, t*) is concluded from Poincaré–Miranda after proving four boundary inequalities: alpha(a, 0.3966) > pi/2 and alpha(a, 0.3991) < pi/2 for all a in [0.5204, 0.5244], and theta(0.5204, t) < pi/4 and theta(0.5244, t) > pi/4 for t in [0.3966, 0.3991]. The constants h, R, U1, U2, and the bounds M_i, K0 are uniform bounds on the vector field and its derivatives; they are not fitted to the desired intersection alpha = pi/2, theta = pi/4. The RTM error bound 0.0003048 is applied to the computed tables, and every boundary value is separated from pi/2 or pi/4 by a margin (0.000375 to 0.00264) larger than that error, so the target is checked rather than assumed. The author's self-citations [8] and [6,7] do not make the result circular: [8] supplies a general parameter-free error theorem whose hypotheses do not include the target result, and [6,7] merely motivate the search. The genuine weakness is a non-circular correctness gap: for a0 = 0.5204, the initial point z0 = (pi/2, 0.5204, pi) has second component below b2 = 0.522, so the containment hypothesis z_j in U1 of Theorem 2.1 fails at j = 0, and the asserted monotonicity cannot repair that. This is an unproved hypothesis, not an equivalence between the input and the claimed derivation.

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

The central claim rests on standard intermediate value and ODE comparison theorems, on an external embeddedness result from [3], and on an unverified computational certificate. No auxiliary physical entities are introduced.

free parameters (4)

Rounding radius R = 10^{-10}
Chosen in Section 3 to balance exact arithmetic size against the RTM error term; not derived from first principles.
Step sizes h = 0.3966/25000 and 0.3991/25000
Chosen so that k*h equals the target times 0.3966 and 0.3991; these are proof parameters rather than physical constants.
Search rectangle = [0.5204,0.5244] x [0.3966,0.3991]
Selected from the author's prior numerical exploration of CMC families (references [6,7]), not generated by the proof.
Epsilon width of U2 = 10^{-3}
Hand-chosen slack in Definition 2.2 to satisfy the RTM containment hypotheses.

assumptions (4)

standard math Poincare-Miranda theorem in two variables
Used as Theorem 1.3 to convert boundary inequalities into existence of (a*, t*).
standard math Gronwall/comparison estimate for nearby solutions
Used as Theorem 1.4 to pass from 16 sampled initial angles a_j to all a in [0.5204,0.5244].
domain assumption Rotational CMC profile construction of [3]
The paper states that a closed profile curve beta produced by the ODE generates an embedded CMC hypertorus in S^4; the proof in [3] is cited but not reproduced.
domain assumption Symmetry extension of solution to [0,4t*]
The paper asserts 'Due to the symmetries of the differential equation...' and cites [2,3] for the fact that the curve closes up and the hypersurface is smooth and embedded.

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Pith. "Pith review of Existence of a constant-mean-curvature hypertorus in $S^4$ via computer assistance." pith.science (2026). https://pith.science/paper/P3345YOP

@misc{pith2026250619555,
  author       = {Pith},
  title        = {Pith review of: Existence of a constant-mean-curvature hypertorus in \(S^4\) via computer assistance},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/P3345YOP}},
  note         = {Machine review of arXiv:2506.19555}
}

read the original abstract

The round Taylor method uses rational arithmetic, allowing control of both round-off and truncation errors in approximating solutions of differential equations. In this paper, we employ this method together with the Poincare-Miranda theorem to prove the existence of a new embedded constant mean curvature (CMC) hypertorus in the unit four dimensional sphere

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Forward citations

Cited by 1 Pith paper

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

New Explicit Eigenfunctions of the stability operator on some minimal hypersurfaces
math.DG 2026-07 accept novelty 5.0 of 10

For minimal immersions ϕ(y,z,t)=(f(t)y,f₂(t)z,f₁(t)), the functions ω(t)y_i z_j with ω=f^{-k}f₂^{-ℓ} are eigenfunctions of the stability operator for eigenvalue −n, implying index ≥ kℓ+3k+3ℓ+8.

Reference graph

Works this paper leans on

9 extracted references · 8 canonical work pages · cited by 1 Pith paper

[1]

E. A. Coddington,An Introduction to Ordinary Differential Equations, Dover Books on Mathematics, Dover Publications, New York, 1989

work page 1989
[2]

and Schulz M

Carlotto, A. and Schulz M. B. ,Minimal hypertori in the four-dimensional sphere, Ars Inven. Anal. Paper No. 8, 33 pp (2023)

work page 2023
[3]

and Wei, G

Huang, C. and Wei, G. New examples of constant mean curvature hypersurfaces in the sphere , ArXiv:2209.13236v1 (2022)

arXiv 2022
[4]

and Wei, G.Embedded contant mean curvature hypertori in the2n-sphere, Personal communi- cation

Lai, J. and Wei, G.Embedded contant mean curvature hypertori in the2n-sphere, Personal communi- cation

work page
[5]

Perdomo, O.Embedded constant mean curvature hypersurfaces on spheresAsian J. Math. 14, (2010), no 1, pp 73-108

work page 2010
[6]

Perdomo, O.Navigating the Space of Compact CMC Hypersurfaces in Spheres, Part I, arXiv:2503.13823 (2025)

work page Pith review arXiv 2025
[7]

Perdomo, O.Navigating the Space of Compact CMC Hypersurfaces in Spheres, Part II,arXiv:2505.09102 (2025)

work page Pith review arXiv 2025
[8]

Perdomo, O.The round Taylor method, Amer. Math. Monthly126 (2019), no. 3, pp 237-251

work page 2019

Show all 9 references

[9]

Perdomo, O.A small variation of the Taylor Method and periodic solutions of the 3-body problemArXiv 1507.01100 (4 Jul 2015) Department of Mathematical Sciences, Central Connecticut State University Email address: perdomoosm@ccsu.edu

2015 arXiv

Pith tools

Reviewed August 15, 2026 · model on record in the stance chip above.

[1] [1]

E. A. Coddington,An Introduction to Ordinary Differential Equations, Dover Books on Mathematics, Dover Publications, New York, 1989

work page 1989

[2] [2]

and Schulz M

Carlotto, A. and Schulz M. B. ,Minimal hypertori in the four-dimensional sphere, Ars Inven. Anal. Paper No. 8, 33 pp (2023)

work page 2023

[3] [3]

and Wei, G

Huang, C. and Wei, G. New examples of constant mean curvature hypersurfaces in the sphere , ArXiv:2209.13236v1 (2022)

arXiv 2022

[4] [4]

and Wei, G.Embedded contant mean curvature hypertori in the2n-sphere, Personal communi- cation

Lai, J. and Wei, G.Embedded contant mean curvature hypertori in the2n-sphere, Personal communi- cation

work page

[5] [5]

Perdomo, O.Embedded constant mean curvature hypersurfaces on spheresAsian J. Math. 14, (2010), no 1, pp 73-108

work page 2010

[6] [6]

Perdomo, O.Navigating the Space of Compact CMC Hypersurfaces in Spheres, Part I, arXiv:2503.13823 (2025)

work page Pith review arXiv 2025

[7] [7]

Perdomo, O.Navigating the Space of Compact CMC Hypersurfaces in Spheres, Part II,arXiv:2505.09102 (2025)

work page Pith review arXiv 2025

[8] [8]

Perdomo, O.The round Taylor method, Amer. Math. Monthly126 (2019), no. 3, pp 237-251

work page 2019