De Giorgi rigidity in dimension eight

Source and verification

This article explains family 375 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

The Allen–Cahn equation models transitions between two preferred phases. Family 375 claims that a bounded entire solution in eight dimensions, strictly increasing in one direction, has a flat transition layer. Its central analytic claim is a stronger classification of stable solutions in seven dimensions without an assumed energy-growth bound.

🏷️ Preliminaries: wells, interfaces, and stability

Set $W(u)=(1-u^2{)}^2/4$. The energy and Euler–Lagrange equation are

$$\begin{array}{c}\mathcal{E}(u;\Omega )={\int }_{\Omega }\left(\frac{1}{2}|\nabla u{|}^2+W(u)\right),\\ \Delta u=u^3-u.\end{array}$$

The wells $u=\pm 1$ have zero potential energy. Transition layers interpolate between them. A solution is stable if its second variation is nonnegative:

$$\begin{array}{c}{Q}_u(\phi )\\ =\int \left(|\nabla \phi {|}^2+(3u^2-1){\phi }^2\right)\\ \ge 0\end{array}$$

for all compactly supported tests. A global minimizer satisfies this inequality, but stability only tests infinitesimal variations and is weaker than full minimization.

The planar transition is $g(s)=\tanh (s/\sqrt{2})$. Direct differentiation gives $g^{\prime }=(1-g^2)/\sqrt{2}$ and $g^{\prime \prime }=g^3-g$, so $u(x)=g(e\cdot x-c)$ solves the equation for any unit vector $e$.

🏷️ The two rigidity claims

Stable rigidity claimed in dimension seven

Every stable entire $C^2$ solution $v:{\mathbb{R}}^7\to [-1,1]$ is one of the constant wells or a planar transition. No energy-growth hypothesis is imposed. (OpenAI, 2026)

Monotone rigidity claimed in dimension eight

Every $C^2$ solution $u:{\mathbb{R}}^8\to (-1,1)$ satisfying ${\partial }_{8}u>0$ has the form

$$\begin{array}{c}u(x)=\tanh \left(\frac{e\cdot x-c}{\sqrt{2}}\right),\\ e\in S^7,e_8>0.\end{array}$$

Neither minimization nor directional limits at infinity are assumed. (OpenAI, 2026)

These are statements for the specified scalar quartic equation. Stability, monotonicity, and minimization should not be interchanged without an argument.

🏷️ A direct proof that monotonicity gives stability

Let $w={\partial }_{8}u>0$. Differentiating the equation gives

$$(-\Delta +3u^2-1)w=0.$$

For compactly supported $\phi =w\eta$, expand $|\nabla (w\eta ){|}^2$ and integrate the cross term by parts. The equation for $w$ cancels the terms containing ${\eta }^2|\nabla w{|}^2$ and the potential. What remains is

$$\begin{array}{rl}{Q}_u(\phi )& =\int w^2|\nabla \eta {|}^2\\ & =\int w^2{\left|\nabla (\phi /w)\right|}^2\ge 0.\end{array}$$

Positivity of $w$ makes the division legitimate on the compact support. This proves the stability implication independently of the new classification.

Monotonicity and boundedness also give limits as $x_8\to \pm \infty$. Elliptic compactness turns them into stable entire solutions in seven variables. The manuscript’s seven-dimensional theorem therefore controls these end profiles.

🏷️ From stable end profiles to a flat interface

The classified end profiles are minimizers. The source then uses clipping between them and sliding translates to prove that the original eight-dimensional solution is itself a global minimizer. Comparing with a constant phase inside a large ball gives the surface-order energy bound $\mathcal{E}(u;B_R)\le C{R}^7$.

Under large dilations, energy concentrates on an area-minimizing boundary. The normalization is the one-dimensional surface tension

$${\int }_{-1}^1\sqrt{2W(t)}dt=\frac{2\sqrt{2}}{3}.$$

Identifying all the limiting energy requires a recovery comparison, not merely convergence to two phases. Monotonicity constrains the orientation of the limiting cone; the proposed geometric reduction forces a hyperplane. A near-planar rigidity theorem then returns from the limiting interface to the original solution.

🏷️ Removing the energy-growth assumption

The new analytic burden lies earlier, in stable rigidity. The manuscript first treats stable solutions with finite normalized energy density. It then has to prove that an arbitrary stable seven-dimensional solution has that property.

A scale measuring both curvature of a zero-level sheet and proximity to neighboring sheets organizes the argument. Local regularity at this scale, together with stability, controls curvature moments and deviations from the preferred transition profile. The source compares an inequality on the sheets with an ambient stability test built from the same radial amplitude.

Joining amplitudes across neighboring sheets costs energy. The required estimate keeps that cost below a strict positive margin. Unbounded surface density would then force more interface area than the stability inequality permits. This produces the missing growth bound and completes the proposed classification. Assuming the growth bound at the start would omit the central difficulty of the strongest claim.

🏷️ Key steps and proof outlines

An exact defect identity detects failure of a planar profile

For a nonconstant solution $v$ with $|v|<1$, write

$$g(s)=\tanh (s/\sqrt{2}),U=g^{-1}(v),q=1-v^2,D=1-|\nabla U{|}^2.$$

Modica’s gradient estimate gives $D\ge 0$. Direct differentiation of the equation gives

$$\begin{array}{rl}\Delta U& =-\sqrt{2}vD,\\ (-\Delta +3v^2-1)q& =-q^{2}D.\end{array}$$

Inserting $q\eta$ into the stability quadratic form and integrating by parts therefore yields the exact identity

$$0\le Q_v(q\eta )=\int q^2|\nabla \eta {|}^2-\int q^{3}D{\eta }^2.$$

The second term measures nonplanarity, including near irregular zeros where a surface description is unavailable. There is also an equation $\mathcal{L}D=2|{\nabla }^{2}U{|}^2$, with nonnegative zeroth-order coefficient. If $D$ vanishes anywhere, the strong minimum principle makes it identically zero; then ${\nabla }^{2}U=0$ and $v$ is a planar transition.

Proof outline for small positive defect. Weak Harnack bounds an integral of $D$ by its value at the center. Testing its equation bounds the local integral of $|{\nabla }^{2}U{|}^2$. A differentiated equation and a mean-value estimate then give $|{\nabla }^{2}U{|}^2\le CD$. This makes the potential in the normalized defect equation bounded, enabling ordinary Harnack and derivative estimates for $D/D(z)$ and ${\nabla }^{2}U/\sqrt{D(z)}$. These relative estimates remain informative when the defect tends to zero. (OpenAI, 2026)

The unrestricted seven-dimensional classification has two separate branches

For finite energy density, the manuscript invokes precisely stated preliminary sheeting results from the literature, then argues against a critical integer-density solution with at least two layers. The nearest gap between layers is tested first by a concave square-root function, which absorbs gradient errors, and then by a sharper nearly linear function. A radial Green estimate and a Hardy estimate from diffuse stability yield incompatible logarithmic gap coefficients. The core-gradient lower bound and control of the full ambient derivative of the normal are required before applying the sheeting results.

The unbounded-density case cannot use that classification directly. The source constructs a penalized reach $d$: a local length measuring both curvature and the distance to another sheet. It obtains a fourth-moment bound for curvature, defect, and the interaction $e^{-\kappa d}$, with a density-dependent factor retained. This controls the exceptional regions before constructing the final test. These are substantial geometric inputs, not consequences of the elementary defect identity alone. (OpenAI, 2026)

An optimized interpolation prevents a factor-of-two loss

To compare surface geometry with ambient stability, sheet amplitudes must be joined through gaps. Near the middle of a gap of length $\ell$, the normal derivative cost has weight proportional to $e^{-\sqrt{2}\ell }{\cosh }^2(\sqrt{2}t)$. For an interpolation $F(-3)=1$, $F(3)=0$, Cauchy–Schwarz gives

$$1\le \left({\int }_{-3}^3{\cosh }^2(\sqrt{2}t)|F^{\prime }{|}^{2}dt\right)\left({\int }_{-3}^3{\mathrm{sech}}^2(\sqrt{2}t)dt\right).$$

Equality selects $F^{\prime }$ proportional to $-{\mathrm{sech}}^2(\sqrt{2}t)$. For amplitudes $a,b$, the cost is multiplied by $(a-b{)}^2$, so equal amplitudes incur no switching cost.

The source assigns each band portion to the sheet with larger uncut amplitude. On that sheet’s nearest gap, the reach comparison makes the amplitude ratio tend to one. Only its other side can incur the fixed positive loss. Counting both sides with the worst bound would destroy the needed margin. The resulting normalized loss is bounded by an explicit one-variable maximum,

$$\frac{3}{4}\underset{u\ge 0}{\sup }{e}^{-\sqrt{2}u}(1-e^{-1.1G_{1}u}{)}^2,1\le G_1\le \mathrm{1.001.}$$

After the source’s finite tolerance factors, the switching loss is below $0.100$. This retains enough defect mass to compare the radial stability and reach inequalities with strict coefficient margins. Normal-tube packing then supplies surface area contradicting the resulting small upper bound. The numerical budget depends on the preceding uniform geometric estimates; it is not an independent proof of them. (OpenAI, 2026)

Returning from stable end profiles to the monotone solution

For the eight-dimensional monotone solution, long-cylinder tests transfer stability to the seven-dimensional end profiles: the vertical cutoff error, after normalization, is $O(L^{-2})$. The claimed classification makes these profiles constant wells or planar transitions. Both are global minimizers, the latter by

$$\frac{1}{2}|\nabla w{|}^2+W(w)\ge \sqrt{2W(w)}{\partial }_{e}w={\partial }_e\left({\int }_{-1}^w\sqrt{2W(a)}da\right),$$

with equality on the oriented planar profile.

Sliding vertical translates against a minimizing competitor between the end profiles proves that the original solution minimizes too. A competitor equal to a well inside a large ball gives surface-order energy growth. In the blow-down, a perimeter lower bound is insufficient: the manuscript patches a recovery sequence to the original solution in one low-cost thin annulus, obtaining exact convergence of the whole energy measure to perimeter. Only then does it apply directed minimizing-cone geometry and unit-density rigidity. This order rules out hidden diffuse energy before the geometric conclusion is used. (OpenAI, 2026)

See Also

• Family 367 classifies minimizing cones to derive a dimension threshold for singularities.
• Family 370 similarly aims to obtain an energy bound from the equation rather than impose it.

🏷️ Manuscripts and verification scope

• A positive resolution of De Giorgi’s conjecture in dimension eight — pinned manuscript and companion source.

No family-specific scope page was available at lean/docs/375.md in this snapshot. This records the catalogue lookup, not a claim that no related Lean code exists. No Lean build or axiom audit was performed.

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

🐻  OpenAI 2026. A positive resolution of De Giorgi’s conjecture in dimension eight.