The critical dimension of the one-phase Bernoulli problem

Source and verification

This article explains family 367 of the October 6, 2026 OpenAI collection. Research conclusions are attributed to its manuscripts. The review covers statements, introductions, and selected proof mechanisms; it is not an independent verification of the complete proofs.

Overview

A free boundary is an interface determined by an energy minimization problem. Family 367 claims that dimension seven is the first dimension permitting a singular minimizing cone for the one-phase Bernoulli energy. The proposed low-dimensional exclusion combines geometric stability with an exact finite arithmetic certificate.

🏷️ Preliminaries: the energy and its free boundary

For a nonnegative function $u$ on a ball $B\subset {\mathbb{R}}^d$, set

$$J(u;B)={\int }_B(|\nabla u{|}^2+1_{\{u>0\}}).$$

The positivity set pays a volume cost, while steep transitions pay gradient energy. A global minimizer minimizes this energy on every ball against all nonnegative Sobolev competitors with the same boundary trace. The free boundary is $\partial \{u>0\}$.

Inside the positive phase, compactly supported value variations give $\Delta u=0$. At a smooth free-boundary point, the corresponding domain variation gives the Bernoulli condition $|\nabla u|=1$. Solving these equations is weaker than minimizing against all competitors.

A function is one-homogeneous if $u(rx)=ru(x)$. The flat models are $u(x)=(x\cdot e{)}_{+}$ with $|e|=1$. Blow-ups $u(x_0+rx)/r$ at a free-boundary point lead to homogeneous minimizing models; their classification governs regularity.

🏷️ The claimed dimension threshold

Family 367, as stated in the manuscript

Every nonzero one-homogeneous global minimizer in dimensions at most six is flat. A nonflat one-homogeneous global minimizer exists in dimension seven. Consequently, local minimizing free boundaries are smooth through dimension six, have locally finite singular sets in dimension seven, and have singular-set Hausdorff dimension at most $n-7$ in dimension $n\ge 7$. (OpenAI, 2026)

The manuscript uses the known seven-dimensional cone for the existence side. Its new burden is excluding nonflat minimizers in the remaining lower dimensions. The existence half alone does not prove that seven is the first possible dimension.

🏷️ Reducing a cone to a spherical equation

Write a one-homogeneous function as $u(r\omega )=rg(\omega )$. The polar-coordinate Laplacian gives

$$\Delta u=r^{-1}({\Delta }_{S^{d-1}}g+(d-1)g).$$

Thus the positive spherical region carries an eigenfunction with eigenvalue $d-1$. Its spherical Hessian defect is

$$T={\nabla }_{S^{d-1}}^{2}g+gI.$$

The eigenfunction equation makes $T$ trace-free. For a flat solution, $g(\omega )=\omega \cdot e$ on a hemisphere and $T=0$. More generally, the ambient Hessian has tangential block $r^{-1}T$ and zero radial block. The tensor $T$ therefore measures precisely the failure of the cone to be linear in its positive phase.

This calculation explains the proof’s target: force the Hessian defect to vanish using the stability inequality inherited from minimization.

🏷️ Stability, certificates, and removal of cutoffs

The manuscript reduces a first nonflat example to a spherical problem in dimension six, allowing suspended configurations with two exceptional tips. It derives differential identities for $T$, its covariant derivative, and the spherical gradient of $g$.

An exact finite-dimensional certificate supplies rational tensor inequalities with strictly positive margins. Substituting the normalized geometric tensors turns these inequalities into a flux comparison that is incompatible with the minimizing stability inequality after integration.

Two analytic issues remain after the arithmetic comparison. Normalizing by $|T|$ creates singular expressions where the Hessian defect vanishes. Suspension introduces exceptional tips as well. The source uses weighted integrability estimates and cutoffs to justify passing to the full spherical domain. Without that passage, a pointwise inequality away from exceptional sets would not establish the classification.

The role of computation

The repository supplies exact-arithmetic reproduction scripts for coefficient comparisons. A successful arithmetic run would certify those finite comparisons, not the equivalence between the encoded expressions and every analytic step of the proof. These notes review the stated certificate interface and its geometric use; they do not report running the certificate engines or verifying the full proof.

Once the cone classification is available, dimension reduction yields the singular-set bound. Taking the seven-dimensional singular cone times ${\mathbb{R}}^{n-7}$ explains the claimed sharpness: its vertex becomes an $(n-7)$-dimensional singular set.

🏷️ Key steps and proof outlines

Normalize the Hessian while retaining its size separately

On the regular spherical working domain $\Omega \subset S^5$, write $w=|T|$ and $B=\nabla T$. Where $w>0$, set $A=T/w$, $X=B/w$, and $q_l=X_{ljk}{A}_{jk}$. If $p=\nabla g$, quotient differentiation gives

$$\begin{array}{rlrl}{\nabla }_{l}w& =w{q}_l,& {\nabla }_l{A}_{ij}& =X_{lij}-q_l{A}_{ij},\\ {\nabla }_l{p}_i& =w{A}_{li}-g{\delta }_{li},& {\nabla }_{l}g& =p_l.\end{array}$$

The normalized coefficient variables lie in a compact set: $A$ is trace-free with $|A|=1$, and $g^2+|p{|}^2\le 1$. The quantities $w$ and $X$ are not bounded by this normalization and must remain explicit in subsequent estimates.

The certificate defines a fourth-order coefficient tensor $G_{lijk}={\delta }_{li}{U}_{jk}+C_{lijk}$ with $C_{iljk}=-C_{lijk}$. When its flux is differentiated, the apparent fourth derivatives of $g$ occur in $G_{lijk}{\nabla }_l{B}_{ijk}$. The trace part is determined by the tensor Laplacian identity; the skew part is determined by commuting covariant derivatives on the sphere. Thus the certificate does not need to control arbitrary fourth derivatives. The prescribed contraction eliminates them exactly.

A completed square turns the polynomial certificate into energy

The source chooses $a=29/50$ and a coefficient polynomial $f\ge {ϵ}_{\ast }>0$. This $f$ is an auxiliary polynomial, unrelated to the original minimizing function. It constructs

$$h=w^a\sqrt{f},Z_l=w^{2a}(G_{lijk}{X}_{ijk}+w{M}_l+P_l).$$

The positivity margin makes the square root uniformly legitimate. If ${\mathcal{F}}_l=D_{l}f+2a{q}_{l}f$, where $D_l$ is the coefficient derivative specified by the preceding identities, then

$$|\nabla h{|}^2=\frac{w^{2a}}{4f}|\mathcal{F}{|}^2.$$

Minimizing the certificate’s auxiliary quadratic expression $f|\ell {|}^2-\mathcal{F}\cdot \ell$ at $\ell =\mathcal{F}/(2f)$ yields

$$\mathrm{div}Z\ge 4h^2+|\nabla h{|}^2+{ϵ}_{\ast }{w}^{2a}.$$

This is the exact analytic use of the rational polynomial inequality. On the regular side boundary, reflection in the normal direction removes the uncontrolled even-normal components of $B$ from the relevant contraction. The boundary certificate then gives $N\cdot Z+H{h}^2\ge 0$, with $N$ the inward normal. The outward normal in the divergence theorem is $-N$, so the sign must be retained consistently.

Stability cancels the energy and leaves three cutoff errors

For a smooth cutoff $\xi$ avoiding Hessian zeros and suspension tips, integrate the flux inequality and use the boundary comparison. The spherical stability inequality bounds the resulting boundary term by $\int (|\nabla (\xi h){|}^2+4{\xi }^2{h}^2)$. Expanding that square cancels the common bulk energy terms, leaving

$$\begin{array}{rl}{ϵ}_{\ast }\int {\xi }^2{w}^{2a}\le \int (& h^2|\nabla \xi {|}^2+2\xi h⟨\nabla h,\nabla \xi ⟩\\ & -2\xi ⟨Z,\nabla \xi ⟩).\end{array}$$

All cancellation is done before removing the cutoff. The proof therefore needs an integrable bound for these three errors, rather than separate limiting bounds for possibly singular energy and boundary integrals.

A weighted Bochner estimate handles arbitrary Hessian zero sets

On the original smooth link, regularize $w_o^2$ by $W_{\delta }=w_o^2+\delta$. Its identities give

$$\Delta W_{\delta }\ge 2|B_o{|}^2,|\nabla W_{\delta }{|}^2\le 4W_{\delta }|B_o{|}^2.$$

Multiply by a compact cutoff times $W_{\delta }^{a-1}$ and integrate by parts. The adverse chain-rule term costs $4(1-a)$ times the desired weighted integral, whereas the Laplacian supplies coefficient two. The remaining coefficient is

$$2-4(1-a)=4a-2=8/25>0.$$

Absorption and monotone convergence prove $\int w_o^{2a-2}|B_o{|}^2<\infty$. No dimension or regularity estimate for the zero set is required.

Cutoffs ${\alpha }_j=\vartheta (j{w}_o)$ satisfy $|\nabla {\alpha }_j|\le C|B_o|/w_o$ and have derivatives eventually zero at every point with $w_o>0$. In a suspension, add ${\beta }_j=\vartheta (j\sin t)$. The exact scaling is

$$w=\frac{w_o}{\sin t},|B{|}^2=\frac{|B_o{|}^2+3{\cos }^{2}t{w}_o^2}{{\sin }^{4}t},dV={\sin }^{4}tdtd{V}_o.$$

The three error densities are dominated by an integrable multiple of ${\sin }^{2-2a}t{w}_o^{2a}(1+|B_o|/w_o{)}^2$. Here $2-2a=21/25>-1$. Dominated convergence sends every error to zero, while the positive mass on the left remains nonzero for a nonflat cone. This contradiction excludes the two remaining candidate dimensions, using the stated lower-dimensional and seven-dimensional results as inputs. The coefficient certificate itself remains a separately attributed finite computation. (OpenAI, 2026 s “From the algebraic certificate to a geometric flux” and “Hessian zeros, suspension tips, and the contradiction”)

See Also

• Family 366 uses blow-up classification to describe planar cracks.
• Family 375 connects diffuse interfaces, stability inequalities, and dimensional rigidity.

🏷️ Manuscripts and verification scope

• The critical dimension for one-phase Bernoulli minimizers — pinned manuscript and companion source.

The family scope page describes selected formalization coverage. The scope page covers the existence of a nonflat homogeneous minimizer in dimension seven. It explicitly leaves out flatness in dimensions at most six and the resulting regularity and singular-set bounds. This existence statement alone does not certify the claimed critical dimension. No Lean build, axiom audit, or manuscript-to-formal-statement equivalence check was performed for these notes.

OpenAI collection provides prerequisites and the reading sequence; coverage register records completed and pending families.

🐻  OpenAI 2026. The critical dimension for one-phase Bernoulli minimizers.