Four hyperplanes do not always equipartition a mass in $\mathbb{R}^4$

Pablo Sober\'on

Four hyperplanes do not always equipartition a mass in $\mathbb{R}^4$

T0 review · 3 major / 5 minor · reviewed 2026-08-28 · deepseek-v4-flash

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Pith's one-line read This paper constructs a smooth strictly positive density in $\mathbb{R}^4$ that no four affine hyperplanes can divide into 16 equal parts, closing the last open case of Grünbaum's 1960 conjecture and disproving Ramos' general conjecture.

desk verdict Soberón settles the last open case of Grünbaum's conjecture with a Gaussian perturbation and an exact Bernstein certificate; the analytic reduction is solid and the only real risk is independent verification of the computer part. read the letter →

arxiv 2608.23312 v2 pith:VLODXHKN submitted 2026-08-24 math.CO

Pablo Soberón This is my paper · ORCID

classification math.CO MSC 52C3552A3768V05

keywords hyperplanemasspartitionsGrünbaum–Hadwiger–RamosproblemRamosconjectureGaussianperturbationspolynomialnonvanishingBernsteincoefficientssubdivisioncertificatescomputer-assistedproof

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

Grünbaum's 1960 conjecture asserted that every mass in $\mathbb{R}^d$ can be cut into $2^d$ equal parts by $d$ affine hyperplanes; before this paper it was known to be true in dimensions at most 3 and false in dimensions at least 5, leaving dimension 4 open. The paper constructs a smooth, strictly positive density in $\mathbb{R}^4$ that no four hyperplanes can equipartition into 16 pieces, and uses it to settle the value $\Delta(1,4)=5$. The proof reduces the geometric question to an algebraic one: five explicit polynomials in six variables have no common zero on $[-1,1]^6$. This is established by a computer-assisted subdivision argument with Bernstein coefficients and a checked integer-arithmetic certificate, and the same statement refutes the general conjectured formula for the numbers $\Delta(j,k)$.

What carries the argument

The load-bearing objects are two explicit symmetric tensors $A\in\mathrm{Sym}^3((\mathbb{R}^4)^*)$ and $B\in\mathrm{Sym}^4((\mathbb{R}^4)^*)$, viewed as the homogeneous polynomials $P_A(x)=A(x,x,x)$ and $P_B(x)=B(x,x,x,x)$. They are chosen so that the last five Walsh coefficients of the perturbed Gaussian are, at first order, the five evaluations of $A$ on three-member frames and $B$ on the full frame. The proof that those evaluations cannot all vanish has three components: the Walsh-coefficient reformulation that turns an equipartition into fifteen scalar equations; the quaternion double cover $\mathrm{Spin}(4)=S^3\times S^3\to SO(4)$, which parametrizes the frame space by six coordinates in $[-1,1]^6$; and the Bernstein-coefficient subdivision method, which shows the five polynomials are nonzero on each box of a partition. The subdivision ends in 38,857 terminal boxes, and for each box a separating vector is exhibited so that the convex hull of the Bernstein coefficient vectors avoids the origin, while a self-contained C program using integer arithmetic checks the whole certificate.

What would settle it

Run the supplied exact C checker on the full subdivision and independently verify every terminal box's separating vector against the published CSV: the certificate is false if any claimed vector $u$ fails to satisfy $\langle u,C_\alpha\rangle>0$ for all Bernstein vectors $C_\alpha$ in its box. Alternatively, a single point in $[-1,1]^6$ at which the five polynomials vanish, or an orthonormal frame in $O(4)$ at which all five tensor evaluations vanish, would refute Theorem 2 directly.

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Extended reading notes

Core claim

The central claim is Theorem 1: there exists a smooth strictly positive density on $\mathbb{R}^4$ that admits no equipartition by four hyperplanes into 16 equal parts. The density is a small perturbation of the standard centered Gaussian measure, $\mathrm{d}\mu=(1+\varepsilon\,\eta(\|x\|^2)(P_A(x)+P_B(x)))\,\mathrm{d}\gamma$, where $P_A$ and $P_B$ are explicit homogeneous polynomials of degrees 3 and 4 and $\eta$ is a radial cutoff supported near the origin. An equipartition of the Gaussian by four hyperplanes must consist of hyperplanes through the origin whose normal vectors form an orthonormal frame, an element of $O(4)$. The paper computes that, to first order in the small parameter $\varepsilon$, the last five of the fifteen Walsh coefficients of such a candidate frame are the five values $A(q_1,q_2,q_3)$, $A(q_0,q_2,q_3)$, $A(q_0,q_1,q_3)$, $A(q_0,q_1,q_2)$, and $B(q_0,q_1,q_2,q_3)$ evaluated at the frame. Theorem 2 states that for the two explicit tensors $A$ and $B$ exhibited in Section 3 these five values have no common zero on $O(4)$, and the paper proves this by a computer-assisted nonvanishing certificate on $[-1,1]^6$. A compactness argument then shows that for sufficiently small $\varepsilon$ no equipartition can exist, which also gives $\Delta(1,4)=5$ and contradicts the conjectured formula $\Delta(j,k)=\left\lceil\frac{2^{k-1}}{k}j\right\rceil$.

Load-bearing premise

The proof depends on the computer-generated certificate that the five explicit polynomials have no common zero on $[-1,1]^6$; if the subdivision, the separation vectors, or the exact integer checker contain an error, Theorem 2 and with it the counterexample collapse.

Editorial extensions

If this is right

Grünbaum's conjecture is now resolved in every dimension: counterexamples exist in dimension 4 and in all higher dimensions, so it fails whenever $d\ge 4$.
The exact value $\Delta(1,4)=5$ follows: one mass in $\mathbb{R}^4$ does not always admit a four-hyperplane equipartition into 16 parts, while dimension 5 suffices through the known bound $\Delta(1,4)\le\Delta(2,3)=5$.
The conjectured general formula $\Delta(j,k)=\left\lceil\frac{2^{k-1}}{k}j\right\rceil$ is false; its value at $(j,k)=(1,4)$ is disproved.
The counterexample is a smooth positive density, not a singular or curve-concentrated measure, so the obstruction is caused by a small local perturbation of a Gaussian rather than by pathological mass concentration.
The proof gives a checkable route to similar statements: a finite subdivision certificate, with all 38,857 boxes and separating vectors published in a CSV file, can be re-verified independently.

Reading between the lines

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

Inference: the same tensor-plus-certificate strategy could be used at other entries of the $(j,k)$ table, where Gaussian equipartitions are again parametrized by a compact frame space and a local perturbation converts existence into a polynomial nonvanishing problem.
Inference: because the nonvanishing certificate is finite and explicit, it should be possible to translate the C checker output into a formal proof object, making the computational part machine-checked down to the axioms.
Inference: if one searches over tensors $A,B$ by optimization rather than by the paper's component-inspection heuristic, the setup may yield further counterexamples or evidence for positive results in nearby dimensions; the separation-vector search on the Bernstein hull is the natural scoring function.

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Formalized claims in Lean

Claim #1: The central claim is Theorem 1: there exists a smooth strictly positive density on $\mathbb{R}^4$ that admits no equipartition by four hyperplanes into 16 equal parts. The density is a small perturbation of the standard centered Gaussian measure, $\mathrm{d}\mu=(1+\varepsilon\,\eta(\|x\|^2)(P_A(x)+P_B(x)))\,\mathrm{d}\gamma$, where $P_A$ and $P_B$ are explicit homogeneous polynomials of degrees 3

/-- @claim 1 The central claim is Theorem 1: there exists a smooth strictly positive density on $\mathbb{R}^4$ that admits no equipartition by four hyperplanes into 16 equal parts. The density is a small perturbation of the standard centered Gaussian measure, $\mathrm{d}\mu=(1+\varepsilon\,\eta(\|x\|^2)(P_A(x)+P_B(x)))\,\mathrm{d}\gamma$, where $P_A$ and $P_B$ are explicit homogeneous polynomials of degrees 3  -/
def central_claim : Prop :=

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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 is true: this paper constructs a smooth strictly positive density in R^4 with no four-hyperplane equipartition, which gives Δ(1,4)=5 and refutes Ramos' conjecture. That is a genuine resolution of a decades-old open case, not a minor incremental step. The idea is clean: perturb the Gaussian by a cubic plus quartic symmetric tensor, reduce equipartitions to nonvanishing of five explicit polynomials on O(4), then verify the nonvanishing with a subdivision certificate. The analytic chain—Walsh coefficient reduction, implicit function theorem with the explicit diagonal Jacobian, compactness argument, quaternion parametrization—is standard and I found no gap by inspection. The paper is honest about what is heuristic: Section 3.1 explicitly says the search for the tensors is not a proof, and the proof is the later certificate. That is the right structure.

The weak spot is exactly where the reader puts it: the 38,857-leaf Bernstein certificate. The manuscript includes a self-contained C checker, a CSV with all leaf boxes and separating vectors, a manifest with hashes, and the code is public. That is far better than most computer-assisted proofs in the literature. But the manuscript does not prove the checker is correct, does not bound the integer sizes, and does not show a transcript or expected output. The coverage condition—that the terminal boxes exactly cover [0,1]^6 with no gaps—is verified by the checker's tree structure, but the reader cannot see that from the text alone. I disagree with the stress-test note's stronger framing: a checker logic error or a missing region is a real but standard risk in any large computer certificate, and the provided CSV makes independent reimplementation easy. This is a minor-to-moderate concern about reproducibility, not a mathematical flaw.

The circularity concern is unfounded. The tensors were found by search aimed at the conclusion, but the nonvanishing is an independent exact certificate. The proof does not fit parameters inside the theorem; the coefficients are fixed and checked. Citation pattern looks fine: Grünbaum, Hadwiger, Avis, Ramos, the BFHZ survey, and the recent positive results in dimension four are all there. The self-disclosure about AI use is unusual but not a problem.

Who is this for? Anyone working in mass partitions, topological combinatorics, or computer-assisted proofs in discrete geometry. It deserves a serious referee: the result is important, the analytic part is checkable by hand, and the computational part is well-packaged for independent verification. My recommendation: send it out. A referee should run the checker or at least spot-check the CSV against a reimplementation, and the author should be asked to add a short proof or bound for the checker's integer arithmetic and to document the coverage check more explicitly.

Referee Report

3 major / 5 minor

Summary. The paper constructs a smooth strictly positive density in R^4 that, according to Theorem 1, admits no equipartition by four hyperplanes into 16 equal parts. The proof proceeds by perturbing the standard Gaussian measure and reducing the existence of an equipartition to the nonvanishing of five polynomial components on O(4) (Theorem 2). The nonvanishing is established by a computer-assisted subdivision argument: the five polynomials, obtained from two explicit tensors A and B via a quaternion parametrization of SO(4), are shown to have no common zero on [-1,1]^6 using 38,857 terminal boxes and separating vectors, with an exact integer checker supplied in the repository. The analytic parts use Walsh coefficients, an implicit-function theorem argument to track zeros of the first ten Walsh coefficients, and a compactness argument to exclude solutions far from the Gaussian equipartitions.

Significance. If correct, this is a major result: it settles the last open case d=4 of Grünbaum's 1960 conjecture, proves Δ(1,4)=5, and disproves Ramos' conjectured formula Δ(j,k)=ceil(2^{k-1}/k · j). The paper is notable for the clarity of the analytic reduction and for shipping a self-contained exact checker, a full CSV of the 38,857-leaf certificate, and a reproducibility manifest; the theorem's statement itself is parameter-free, since the tensors A and B are fixed explicit objects. The main risks are computational reproducibility and one step in the geometric parametrization that is insufficiently justified.

major comments (3)

[Section 3.2, quaternion parametrization] The reduction to the hypercube [-1,1]^6 is not fully justified. The text asserts that 'using this operation we can assume that the first coordinate of p and r is the one with largest absolute value', and then parametrizes p=(1,x1,x2,x3), r=(1,y1,y2,y3). This is load-bearing because the Bernstein certificate is checked only for this single chart. The operations described (scaling and sign-permuting columns of Q(p,r)) do not make it obvious that every element of SO(4), modulo the symmetries of the zero set, has a representative with both first coordinates nonzero and of maximal absolute value; pairs with p_0=0 or r_0=0 would be excluded. Please provide a precise proof of this claimed surjectivity, or extend the certificate to cover the missing charts.
[Section 4, computer-assisted certificate] Because Theorem 2 rests entirely on the 38,857-leaf certificate, the manuscript should state explicitly that verify_grunbaum_tensors.c verifies not only the per-box positive dot-product condition but also that the terminal boxes form an exact cover of [0,1]^6 with no gaps or overlaps that could leave a region unchecked. The paper currently describes the CSV as containing 'a subdivision of the hypercube [0,1]^6' but does not spell out which component of the checker proves coverage. The integer checker should also specify its integer types and overflow bounds, or confirm that arbitrary-precision arithmetic is used, and the expected output or a verification transcript should be included so that a reader can compare results without independently re-running the code. Without these details, the correctness of the certificate is not fully auditable from the manuscript.
[Section 2.1, final paragraph] The claim that 'all bisecting hyperplanes for μ_ε have uniformly bounded offsets when ε is small' is asserted without proof. This is load-bearing because it is the only step that excludes equipartitions far from the centered orthogonal configurations handled by the implicit-function theorem. A short argument using a uniform lower bound on the density with respect to γ would make the compactness step rigorous; as written, the exclusion of far-away solutions is incomplete.

minor comments (5)

[Section 3.2] The notation tilde q_i and tilde f is used freely; it would help to state explicitly that the simultaneous vanishing of the normalized maps is equivalent to the simultaneous vanishing of f_i because the common scalar factor (|p||r|)^{-deg} is nonzero.
[Section 4] There is a typo 'subidvision' for 'subdivision' in the paragraph describing Figure 1.
[Figure 1] The caption of Figure 1 would benefit from explaining what the colors or shading represent and whether the slice shown is a terminal-box pattern or an intermediate subdivision state.
[Section 4, table] The table explains that the C checker uses 'B and 5A' but the factor 5 is not mentioned in the proof of Theorem 2; adding a sentence there would help the reader connect the polynomial PA in the text with the integer coefficients used in the checker.
[Section 2.1] The notation X_ε(Q,z) and Y_ε(Q,z) is introduced but the dependence on the choice of signs of the columns of Q is implicit; a remark that the Walsh coefficients depend on ordered oriented hyperplanes would remove ambiguity.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: the proof reduces to an independent exact nonvanishing certificate for five explicit polynomials, with no fitted parameter or self-referential definition in the load-bearing chain.

full rationale

The claimed derivation is self-contained and non-circular. Sections 1–2 reduce Theorem 1 to Theorem 2 by an exact Gaussian-integration computation: the leading term of the remaining five Walsh coefficients is explicitly shown to be proportional to the five tensor evaluations, and the implicit-function theorem controls the first ten coefficients. No definition of A or B is made in terms of the desired conclusion; Section 3.1 explicitly states that the candidate tensors were found by heuristic sampling and separation and that this is 'not a complete argument' for correctness. The actual proof of Theorem 2 is the Bernstein-coefficient subdivision certificate: Lemma 1 gives a convex-hull sufficient condition, and Section 4 supplies an exact integer-arithmetic checker plus a CSV listing all 38,857 terminal boxes and separating vectors, allowing independent reconstruction of the polynomial system. The checker's correctness is not formally proved in the manuscript and was not independently re-executed here, but that is a computational-reliability risk, not circularity: no fitted parameter is renamed as a prediction, no uniqueness theorem is imported from the author's prior work, and no equation reduces to its own input. Self-citations such as [MS24] and [RPS22] are background references and are not load-bearing in the argument.

Assumptions & free parameters 4 free parameters · 6 assumptions · 1 invented entities

The proof rests on standard topological and analytic facts: implicit function theorem, convex separation, Bernstein basis properties, and the quaternion double cover. The ad hoc ingredients are the two tensors A and B, found heuristically, and the finite certificate data. No physical entities are postulated beyond these explicit mathematical constructions, and the certificate is checked rather than derived from the target result.

free parameters (4)

Coefficients of the cubic tensor A (polynomial PA) = Explicit coefficients listed in Section 3
Chosen during a heuristic computational search in Section 3.1 rather than derived; the proof requires that this particular tensor yields no common zero, which is then verified by the certificate.
Coefficients of the quartic tensor B (polynomial PB) = Explicit coefficients listed in Section 3
Also chosen by the heuristic search; the nonvanishing of the five-component map depends on this specific choice of B.
Radial cutoff scale R and cutoff function eta = R unspecified; any positive radius works
The construction needs a C-infinity radial cutoff so the perturbation is localized near the origin; the proof only requires positive constants c3,eta and c4,eta, so R is a free construction parameter not fitted to data.
Perturbation size epsilon = Sufficiently small, not explicitly quantified
The theorem constructs a density for some sufficiently small epsilon; the existence follows from compactness, but no numerical value or explicit bound is given.

assumptions (6)

standard math Bernstein basis nonnegativity and partition of unity
Used in Lemma 1 to contain f(z) in the convex hull of the Bernstein vectors; the whole subdivision certificate depends on this.
standard math Convex separation theorem
Used to justify that 0 outside the convex hull of Bernstein vectors is witnessed by a separating vector u; the algorithm searches for exactly such vectors.
standard math Quaternionic double cover Spin(4) to SO(4)
Used to parametrize SO(4) by a product of two projective spaces and thereby reduce the zero question to the cube [-1,1]^6.
standard math Implicit function theorem applied to the first ten Walsh equations
Used in Section 2.1 to obtain a unique small correction z_epsilon(Q) for each orthogonal frame after perturbing the Gaussian.
standard math Gaussian parity and independence identities for Walsh coefficients
Used to compute singleton and pair Walsh coefficients, to show the Jacobian D_z X_0 is diagonal with nonzero entries, and to show D_z Y_0 = 0.
standard math Compactness plus uniform convergence of Walsh coefficients as mu_epsilon tends to gamma
Used in the final paragraph of Section 2.1 to exclude equipartitions outside a small neighborhood of centered orthogonal frames.

invented entities (1)

Symmetric cubic tensor A and quartic tensor B
purpose: Define the perturbing polynomials PA and PB used to break all four-hyperplane equipartitions of the Gaussian measure.
New explicit mathematical objects introduced for this proof. Their required nonvanishing property is supported only by the internal subdivision certificate, not by any external falsifiable prediction.

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Pith. "Pith review of Four hyperplanes do not always equipartition a mass in $\mathbb{R}^4$." pith.science (2026). https://pith.science/paper/VLODXHKN

@misc{pith2026260823312,
  author       = {Pith},
  title        = {Pith review of: Four hyperplanes do not always equipartition a mass in $\mathbbR^4$},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/VLODXHKN}},
  note         = {Machine review of arXiv:2608.23312}
}

abstract

We construct a smooth strictly positive density in $\mathbb{R}^4$ that cannot be divided into $16$ parts of the same size by four affine hyperplanes. This settles the last open case of Gr\"unbaum's 1960 conjecture and disproves Ramos' general conjecture on hyperplane equipartitions. We reduce the construction to finding two homogeneous polynomials in four variables, of degrees three and four, whose multilinear coefficients cannot vanish simultaneously after any orthogonal change of coordinates. We give two proofs of this nonvanishing result. The first uses a local perturbation argument. The second reduces it to the absence of a common zero for five explicit polynomials on $[-1,1]^6$, verified by a computer-assisted Bernstein subdivision argument.

Figures

Figures reproduced from arXiv: 2608.23312 by the authors.

**Figure 1.** A 2-dimensional slice of the subdivision of [−1, 1]6 . This slice is obtained by fixing x2 = −1/3, x3 = 1/3, y2 = −1/3, y3 = −1/3. The horizontal axis is x1 and the vertical axis is y1. simultaneous zeros. The algorithm also imposes a tree structure on the subdivision of [0, 1]6 , where each subdivision creates two nodes, one for each new box, which are the children of the current box. A Python program was used to r… view at source ↗

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