Lipschitz height fields and SLE4 interfaces

Source and verification

This article explains family 232 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 Lipschitz height surface constrains every neighboring height difference. Family 232 claims Gaussian field limits for two integer models and a real-valued model on the triangular lattice. Only the real-valued companion also claims the stated SLE₄ interface limit, at a tuned boundary amplitude.

🏷️ Preliminaries: hard constraints and generalized fields

An integer Lipschitz height assigns $h_v\in \mathbb{Z}$ to lattice vertices with $|h_u-h_v|\le 1$ across every edge. A weighted version assigns a factor $x$ for each edge on which the height changes. With fixed boundary values the finite configuration space is sampled proportionally to these weights.

A real Lipschitz surface has $h_v\in \mathbb{R}$ with the same bound. Its uniform law is normalized Lebesgue measure on the admissible polytope of interior heights. This is a hard-constraint model: the energy is constant for allowed gradients and infinite outside the allowed interval. It is not a smooth uniformly convex gradient model.

The Dirichlet Gaussian free field (GFF) is a random distribution with covariance $(-{\Delta }_D{)}^{-1}$, whose two-dimensional kernel has singularity $-(2\pi {)}^{-1}\log |z-w|$. One tests a field against smooth functions. Its pointwise variance need not stay bounded when lattice mesh tends to zero.

🏷️ Three model-specific statements

The weighted integer paper treats zero boundary values on bounded $C^2$ Jordan domains and inside approximations with uniformly convergent boundary parametrizations. For each fixed

$$x\in [1/\sqrt{2},1],$$

it claims convergence to a positive multiple of the zero-Dirichlet GFF. This includes uniform integer heights and the stated critical endpoint. (OpenAI, 2026)

The two-arc integer companion uses odd integer heights with neighboring differences zero or two, and boundary values $+1$ and $-1$ on the two marked arcs. After centering by the actual mean, it claims a universal multiple of the Dirichlet GFF and gives an absolutely convergent finite-volume formula for its normalization. A change between unit and two-unit jumps changes the field multiplier. (OpenAI, 2026b)

The real-valued companion claims both the centered GFF limit and convergence of the zero-height interface to chordal ${\mathrm{SLE}}_4$, on its smooth marked-domain approximations, when the boundary values are the tuned pair $\pm \lambda$. (OpenAI, 2026c) An arbitrary boundary amplitude is not included in that interface assertion.

🏷️ Stiffness and the boundary amplitude

For the real model the source defines a positive stationary tangent-flux coefficient $A$. With its triangular-lattice density factor $v=2/\sqrt{3}$, it obtains

$$\begin{array}{c}{\sigma }^2=(vA{)}^{-1},\\ {\lambda }^2=\frac{\pi }{8vA},0<\lambda \le \frac{1}{2}.\end{array}$$

Thus $\lambda /\sigma =\sqrt{\pi /8}$. This is a relation between the boundary height needed for the interface theorem and the variance multiplier of the limiting field. It does not give an elementary closed form for either constant separately.

The limiting mean is harmonic:

$$\begin{array}{c}{m}_D(z)\\ =\lambda [{\omega }_D(z,A_{+})-{\omega }_D(z,A_{-})],\end{array}$$

where harmonic measure ${\omega }_D(z,A)$ is the probability that Brownian motion started at $z$ exits through arc $A$. For a reflection-symmetric domain and opposite symmetric arcs, the midpoint has equal harmonic measures and hence mean zero. This checks the sign and boundary convention without proving fluctuation convergence.

🏷️ Integer fields: spectral identification and moment recursion

The integer arguments first obtain uniform moment bounds for averages of heights, including domains exposed by stopped explorations. Reflection positivity supplies a positive transfer operator, and its spectral representation expresses a directional covariance through a mixture of simple frequency kernels.

Triangular-lattice rotations impose additional angular identities. Together with positivity, these force a small-frequency covariance proportional to $|p{|}^{-2}$, the Fourier signature of a GFF. The coefficient must still be fixed through a spectral sum rule.

A covariance limit alone is not Gaussianity. The proof transports plane estimates into conditioned domains, makes separated limiting moments harmonic in each variable, and identifies their singularities when test locations meet. These collision rules recursively produce the Gaussian pairing formulas. Boundary comparison then selects the Dirichlet covariance.

🏷️ Real surfaces: conditioned polytopes and interfaces

For real heights, local volume comparisons and reflected dynamics control how a height polytope changes under boundary perturbations. A stationary tangent update identifies the effective flux and its response to small tilts. Conditional covariance estimates must remain valid after sign information has been exposed along an interface.

The interface part adds crossing barriers, separation estimates, and a height-gap calculation. A stopped conditional mean becomes the harmonic observable used to identify the Loewner driver as $2B_t$. Those geometric and conditional inputs are additional to the field theorem: smooth-test convergence alone cannot see all microscopic zero contours.

All three papers need uniformity over the conditioned domains actually produced by their explorations. Applying an estimate proved only for a fixed unobserved domain after an adaptive reveal would leave a gap.

🏷️ Key steps and proof outlines

The integer models and the real-valued model reach the GFF by different mechanisms. The integer arguments identify every moment through reflection and collision identities; the real argument identifies the response to a small external field. Keeping these routes separate explains which estimates are doing the work.

Reflection detects the continuum dispersion relation

For an integer height field, first work with zero-total tests: a smooth function $f$ satisfying $\int f=0$. The pairing $H(f)$ is unchanged by adding a constant to the height. This is the right observable for an infinite-plane law whose absolute height has no canonical origin. Shell-count estimates control all moments of these pairings; reflection positivity supplies a positive self-adjoint contraction describing translation normal to a lattice mirror.

The spectral theorem expresses the covariance as a positive mixture of Cauchy kernels. Write $k$ for tangential frequency, $l$ for normal frequency, and $s\ge 0$ for the mixture parameter. The continuum Laplacian corresponds to $s=|k|$. Why must the lattice mixture approach that relation? The useful test is the fourth angular harmonic

$$\chi (k,l)=\cos (4\arg (k+il))=1-\frac{8k^2{l}^2}{(k^2+l^2{)}^2}.$$

The Poisson formula gives the exact identity, for $a,s>0$,

$${\int }_{\mathbb{R}}\chi (a,l)\frac{sdl}{\pi (s^2+l^2)}={\left(\frac{a-s}{a+s}\right)}^2.$$

The right side is a nonnegative measure of the error in the dispersion relation. Its proof is particularly instructive: apply the upper-half-plane Poisson formula to the real part of the bounded analytic function $((a+iz)/(a-iz){)}^2$, evaluated at $z=is$. Periodizing the Cauchy kernel changes the answer by $O(a)$, uniformly in $s$.

On the other hand, averaging $\chi$ over the lattice’s six rotations gives zero. Comparing these two evaluations, controlling the small deleted angular caps by annular covariance bounds, proves that the defect has finite mass down to frequency zero. A rescaled annulus approaching zero therefore has vanishing defect. Together with the three mirror directions and the endcap estimates, this forces the limiting spectrum onto the Laplacian form. An additional angular sum rule fixes its scalar coefficient. Positivity is crucial: it converts a cancellation identity into control of an error that cannot cancel itself. (OpenAI, 2026a)

Rare conditioning must cancel its own denominator

A plane covariance calculation does not yet apply to heights with prescribed boundary arcs. The manuscript transfers it through separated pins, meaning specified spin or height constraints on short arcs. These events may have very small probabilities.

Choose all pins and insertion neighborhoods strictly on the same side of the reflection row, preserving positive separations. Let $P_1,\dots ,P_m$ be pins with disjoint filled neighborhoods, let $p_i=\mathbb{P}(P_i)$, and let $A$ be a bounded observable, possibly multiplied by one zero-total height insertion supported away from those neighborhoods. For reflection $\vartheta$, set

$$V=A\prod _i\frac{1_{P_i}}{p_i},\Vert V{\Vert }_{\vartheta }^2=\mathbb{E}[\overline{\vartheta V}V].$$

The squared norm has denominator ${\prod }_i{p}_i^2$. An absolute mixing error would be useless if that denominator vanished. The required lemma is a relative probability estimate:

$$\mathbb{P}\left(\bigcap _i{P}_i\cap \bigcap _i\vartheta P_i\right)\le C\prod _i{p}_i^2.$$

Uniform conditional second moments of the height insertion, followed by Cauchy–Schwarz, then give $\Vert V{\Vert }_{\vartheta }\le C$. This supplies a bounded analytic family when pins are translated across a separating line.

There is a second bookkeeping issue. Disconnected height lifts initially have separate additive offsets. The proof inserts unit-modulus phase fields and uses Fourier projection to impose equality of those offsets before closing the gaps between pins. The estimates with separated neighborhoods do not have constants uniform as the gaps shrink; common offsets permit a different, anchored height comparison at that stage. The order of these operations is part of the argument. (OpenAI, 2026b)

Harmonic moments become Wick’s rule

After tightness and boundary estimates, the limiting $j$-point absolute-height moment $T_j$ is harmonic in each variable away from collisions. Bounds near the collision diagonals also exclude distributions concentrated on those diagonals. Fix all variables except $x$, and approach another point $y$. A punctured-disk harmonic expansion gives

$$T_j(x,y,\mathbf{z})=L(y,\mathbf{z})\log \frac{1}{|x-y|}+R(x;y,\mathbf{z}),$$

where $R$ extends harmonically through $y$. Indeed, the negative radial Fourier modes $r^{-n}$ grow faster than the available polylogarithmic bound, so all their coefficients vanish. The logarithmic coefficient can be read stably from a circle:

$$L(y,\mathbf{z})=-\frac{1}{2\pi }{\int }_{|w-y|=a}{\partial }_{n_w}{T}_j(w,y,\mathbf{z})d{s}_w.$$

To identify $L$, smear the colliding variables with two small zero-total tests. Explore separating circuits from outside inward, stopping before their interior is examined. The resulting coupling compares that interior to the plane while keeping the exterior height observables as spectators. The mesh tends to zero first, then the test scale tends to zero. Uniform moments control the unsuccessful coupling event.

If $c$ is the plane covariance multiplier, this yields

$$L(y,\mathbf{z})=\frac{c}{2\pi }{T}_{j-2}(\mathbf{z}).$$

Subtract the Green-kernel terms with these coefficients. Their singularities cancel; the remainder is harmonic with the prescribed boundary values, hence is fixed by uniqueness. Induction gives the Gaussian moment recursion, including the harmonic mean for two-arc data. Centering by the actual discrete mean is justified by convergence of that mean; it is not a substitute for proving the boundary estimates.

A tangent cell problem identifies the real model

For real Lipschitz heights, the finite-volume law is uniform on a polytope. A reflected diffusion preserves this law. Couple two such diffusions with nearby forcing and study their difference. Local slack estimates control exceptional clusters, while projection updates dissipate a square norm. The resulting stationary tangent problem has a mean flux $j(q)$ for an imposed macroscopic slope $q$.

The essential uniqueness lemma says that a controlled tangent solution of slope zero has zero relative heights. Join two solutions over the same environment and apply this lemma to their difference: their fluxes agree. Joining and adding solutions makes $j$ linear. Rotation and reflection symmetry then force

$$j(q)=Aq.$$

Testing stationarity against a smooth function gives the weak response equation

$$A{\int }_D\nabla m\cdot \nabla f={\int }_{D}bf.$$

A nonzero nonnegative source rules out both the absence of nonzero slopes and $A=0$. Monotonicity of the response rules out $A<0$. Thus $0<A<\infty$, and the zero-boundary response is $m=A^{-1}(-{\Delta }_D{)}^{-1}b$. The boundary condition itself requires a zero-extension estimate across the common pinned boundary.

To get Gaussianity, let $v=2/\sqrt{3}$ be the site density and let $X_n(f)$ be the centered height pairing normalized by $1/(v{n}^2)$. Tilting by $e^{t{X}_n(f)}$ adds the small drift $n^{-2}tf/v$. Therefore, for sufficiently small $|t|$, its log moment-generating function satisfies

$$\frac{d}{dt}\log \mathbb{E}{e}^{t{X}_n(f)}⟶\frac{t}{vA}{\int }_{D}f(-{\Delta }_D{)}^{-1}f.$$

Uniform exponential moments justify integration in $t$ and identify a Gaussian law. Applying this to every linear combination of tests identifies the joint law; wavelet estimates supply distributional tightness. Consequently ${\sigma }^2=(vA{)}^{-1}$. (OpenAI, 2026c)

The interface height gap is fixed by variance conservation

For the real model, the remaining interface analysis establishes a stopped harmonic observable and conditional Green covariance in the slit domain. Uniform estimates for deterministic strings are needed before they can be applied to the random explored interface.

Once the driving process is $2B_t$, write $Z_t(z)=g_t(z)-2B_t$. Itô’s formula gives $d\log Z_t=-2Z_t^{-1}d{B}_t$. A boundary height gap $\pm l$ therefore produces conditional-mean cross variation with coefficient $16l^2/{\pi }^2$. Meanwhile the Dirichlet Green kernel loses covariance with coefficient $2/\pi$, multiplied by ${\sigma }^2$. Pair these formulas with smooth tests; the law of total variance equates gained mean variance and lost conditional variance. Cancelling their common positive integral gives

$$\frac{16l^2}{{\pi }^2}=\frac{2{\sigma }^2}{\pi },l^2=\frac{\pi {\sigma }^2}{8}.$$

This calculation uses field pairings, not nonexistent pointwise GFF values. It explains both the tuned boundary height and the normalization of the SLE conclusion, which belongs to the real-valued manuscript in this family.

See Also

• Family 225 gives a height limit with an explicitly known variance multiplier.
• Family 233 couples a height field to entire current-cluster collections.
• Family 216 explains a related distinction between height averages and microscopic insertions.

🏷️ Manuscripts and verification scope

• Gaussian free-field limits of weighted integer Lipschitz heights — pinned manuscript and companion source.
• The Gaussian free field limit of integer Lipschitz heights with two-arc boundary data — pinned manuscript and companion source.
• Uniform real Lipschitz surfaces on the triangular lattice — pinned manuscript and companion source.

No family scope page was present at lean/docs/232.md in the pinned snapshot. This does not establish that all related formalization code is absent. 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 2026a. Gaussian free-field limits of weighted integer Lipschitz heights.

🐻  OpenAI 2026b. The Gaussian free field limit of integer Lipschitz heights with two-arc boundary data.

🐻  OpenAI 2026c. Uniform real Lipschitz surfaces on the triangular lattice.