Yau’s nodal bounds: surface growth and smooth counterexamples

Source and verification

This article explains family 350 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.

🏷️ Eigenfunctions, frequency, and nodal measure

Let $(M^d,g)$ be a closed connected smooth Riemannian manifold. With the sign convention ${\Delta }_g={\mathrm{div}}_g{\nabla }_g$, a real Laplace eigenfunction satisfies

$$-{\Delta }_{g}u=\lambda u,\lambda >0.$$

We require $u\not\equiv 0$. Its nodal set is $Z_u=\{u=0\}$, and its frequency is $k=\sqrt{\lambda }$. Where $du\ne 0$, the implicit function theorem makes $Z_u$ a smooth hypersurface. At other zeros it can be singular. The quantity ${\mathcal{H}}_g^{d-1}(Z_u)$ is its Hausdorff measure for the intrinsic Riemannian distance; in dimension two this is length.

On the flat torus $(\mathbb{R}/2\pi \mathbb{Z}{)}^d$, the function $\sin (n{x}_1)$ has eigenvalue $n^2$. Its nodal set consists of $2n$ parallel subtori, so

$${\mathcal{H}}^{d-1}(Z_u)=2n(2\pi {)}^{d-1}.$$

This illustrates the wavelength $k^{-1}$ and the expected total nodal measure of order $k$. Yau’s conjectured two-sided estimate is

$$c_g\sqrt{\lambda }\le {\mathcal{H}}_g^{d-1}(Z_u)\le C_g\sqrt{\lambda }.$$

The constants must work for every eigenfunction of one fixed metric, including every vector in a multiple eigenspace. Allowing a new metric at each frequency would be a different statement. The three manuscripts concern the upper inequality; their counterexamples do not refute the lower inequality. They also distinguish arbitrary smooth metrics from the real-analytic setting, where the sharp two-sided bound is classical. (OpenAI, 2026) (OpenAI, 2026b) (OpenAI, 2026a)

Results reported by this family

On every fixed closed connected smooth surface, one constant $C(M,g)$ bounds nodal length by $C(M,g)\sqrt{\lambda }$ for all nonzero real positive-eigenvalue eigenfunctions. No orientability assumption is imposed. (OpenAI, 2026b)

Every smooth neighborhood of the round metric on $S^3$ contains one smooth metric with eigenfunctions $u_j$ and ${\lambda }_j\to \infty$ for which ${\mathcal{H}}^2(Z_{u_j})/\sqrt{{\lambda }_j}\to \infty$. A separate construction gives the analogous conclusion on $S^2\times {\mathbb{T}}^2$. (OpenAI, 2026c)

On $S^4\times S^1$, one fixed smooth metric and one fixed ${\epsilon }_0>0$ admit a sequence with ${\lambda }_j\to \infty$ and

$$\frac{{\mathcal{H}}_g^4(Z_{u_j})}{{\lambda }_j^{1/2+{\epsilon }_0}}⟶\infty .$$

Products extend this power-law conclusion to all dimensions above five. (OpenAI, 2026a)

Dimension	Reported conclusion	Scope of the metric
$2$	Sharp upper bound	Every fixed smooth closed surface
$3$	Unbounded ratio to $\sqrt{\lambda }$	One metric on $S^3$, arbitrarily close to round
$4$	Unbounded ratio to $\sqrt{\lambda }$	One metric on $S^2\times {\mathbb{T}}^2$
$d\ge 5$	Unbounded ratio to ${\lambda }^{1/2+{\epsilon }_0}$	One fixed metric in each dimension

The first two manuscripts together give the universal dimension threshold among $d\ge 2$: the upper bound holds for every smooth metric precisely in dimension two. Products of the three-dimensional example give qualitative failures above dimension three. A fixed positive power gain in dimensions three or four is not asserted.

Prerequisites. The arguments use elliptic equations, Sobolev estimates, basic spectral theory, subharmonic functions, and Gaussian vectors. We introduce the growth quantities, stopping rules, and weighted operators below. The proofs divide into an averaging argument on surfaces and constructions that preserve large nodal measure while changing the metric.

🏷️ Surface reduction to average local growth

In a smooth isothermal chart, $g=p(x)(d{x}_1^2+d{x}_2^2)$ with $p>0$, so the eigenfunction equation becomes

$$\Delta u+k^{2}p(x)u=0.$$

For a square $Q$ of side $s$, let $D_r$ denote the Euclidean disk of radius $r$ centered at its center. Fix a large ratio $R$, and define the logarithmic growth

$$E(Q,0)=\log \frac{\Vert u{\Vert }_{L^2(D_{Rs})}}{\Vert u{\Vert }_{L^2(D_s)}}.$$

These are unnormalized norms. A quantitative unique-continuation argument gives nonzero local norms and growth $O(k)$ on each fixed initial chart. The manuscript’s main new target is much stronger after averaging: subdivide to a common side $s$ comparable to $k^{-1}$, and prove

$${\mathrm{average}}_{Q\text{ terminal}}E(Q,0)\le C.$$

At a sufficiently small multiple of the wavelength, the rescaled potential is small. Roy-Fortin’s planar nodal estimate then gives

$${\mathcal{H}}^1(Q\cap Z_u)\le Cs(1+E(Q,0)).$$

Interior elliptic estimates convert its supremum-norm growth to the displayed $L^2$ growth. There are $O(s^{-2})$ terminal squares in a fixed region. Summing therefore gives $O(s^{-1})=O(k)$ length. Grid-edge zeros cause no difficulty because only subadditivity is used. The constant is uniform over the eigenspaces of the fixed metric. (OpenAI, 2026b)

🏷️ Logarithmic profiles and the two-dimensional estimate

Pointwise $\log |u|$ is singular at zeros, so the paper extracts exponential growth from measures. If $S_j\to \infty$ and ${\mu }_j$ have bounded mass, a subsequence has an upper-semicontinuous profile $V$ satisfying compact upper and open lower bounds:

$$\begin{array}{rl}\underset{j}{\mathrm{limsup}}\frac{\log {\mu }_j(F)}{2S_j}& \le \underset{F}{\sup }V,\\ \underset{j}{\mathrm{liminf}}\frac{\log {\mu }_j(G)}{2S_j}& \ge \underset{G}{\sup }V.\end{array}$$

Here $F$ is compact and $G$ is open. Continuous exponential weights replace $V$ by $V+f$. The same subsequence works for all these comparisons. The identically minus-infinite profile is initially allowed; mass in a compact inner disk later rules it out when needed.

For rescaled solutions $\Delta U_j+K_j^2{p}_j{U}_j=0$, consider tilted measures $c_j{e}^{-2B_j\cdot x}|U_j{|}^{2}dx$. Put $D_j=S_j+|B_j|$. The crucial assertion is that their profiles are subharmonic when

$$\begin{array}{c}{K}_j/D_j\to 0,\\ \frac{K_j^2\Vert \nabla p_j{\Vert }_{\infty }}{S_j{D}_j}\to 0,\end{array}$$

with uniformly bounded potentials. The ratio $|B_j|/S_j$ can be arbitrarily large.

The proof tests a putative strictly superharmonic upper contact $\chi$ using the weight $T_j=-B_j\cdot x-S_j\chi$. A Carleman estimate, meaning a coercive estimate after exponential conjugation, supplies

$$\begin{array}{rl}& \Vert e^{T_j}(\Delta +K_j^2{p}_j)w{\Vert }_2\\ & \ge c\sqrt{S_j}{D}_j\Vert e^{T_j}w{\Vert }_2\end{array}$$

for compactly supported $w$ near a nonexceptional contact. The symmetric and skew-symmetric parts of the conjugated operator yield a commutator involving the Hessian of $T_j$.

The dimension enters at a precise point. In a hypothetical failure, normalized derivative energy becomes perpendicular to the limiting weight gradient $a\ne 0$. In two dimensions there is only one perpendicular unit direction $e$. If $H=-{\chi }^{\prime \prime }$, the limiting Hessian contribution is

$$\begin{array}{rl}& 4(|a{|}^{2}H(e,e)+H(a,a))\\ & =4|a{|}^2\mathrm{tr}H>0.\end{array}$$

This contradicts failure of coercivity. In higher dimensions, energy perpendicular to $a$ need not recover the full transverse trace.

A cutoff turns this estimate into an impossible comparison between an inner region where $V-\chi$ attains zero and an outer annulus where it is strictly negative. The factor $D_j$ cancels before comparing exponential rates, so no hidden bound on $\log D_j/S_j$ is required. An elementary linear perturbation of a quadratic upper test removes the exceptional gradient. This proves subharmonicity without assuming local integrability beforehand. (OpenAI, 2026b)

🏷️ Affine correction, stopped descent, and packing

A profile can be almost affine. Subdivision then reduces its oscillation and the local frequency $ks$ by the same factor, preventing improvement of normalized growth. For a square $Q$ and tilt $b$, set $u_b(x)=e^{-kb\cdot (x-x_Q)}u(x)$ and write

$$\begin{array}{c}E(Q,b)=\log \frac{\Vert u_b{\Vert }_{L^2(D_{Rs})}}{\Vert u_b{\Vert }_{L^2(D_s)}},\\ m(Q,b)=\frac{E(Q,b)}{ks},\\ N(Q)=m(Q,0).\end{array}$$

Removing an affine part lowers the remaining excess, but its slope must be tracked: $N(Q)\le m(Q,b)+(R+1)|b|$.

A normalized subharmonic profile has a local decomposition into a harmonic function and a logarithmic potential of a positive measure of bounded mass. Subtract the harmonic gradient and the gradient of the far part of that potential. At child side ratio $\rho$, the remaining error is bounded by a constant times

$${\rho }^2+\int \min \left\{1,\frac{{\rho }^2}{|z-y{|}^2}\right\}d\nu (z).$$

Averaging over the planar grid gives $O({\rho }^2(1+|\log \rho |))$, while the mean slope correction stays bounded. The logarithm comes from dyadic annuli: each annulus contributes the same $O({\rho }^2)$ amount. Near atoms, disk averaging of the logarithmic kernel replaces an invalid pointwise Taylor expansion.

Choose the subdivision factor $A={\rho }^{-1}$ large enough. Transferring the profile estimate to eigenfunctions gives child tilts with

$$\begin{array}{rl}\mathbb{E}[m^{\prime }\mid Q,b]& \le \frac{1}{2}(m+f_Q),\\ \mathbb{E}[|b^{\prime }-b|\mid Q,b]& \le C(m+f_Q),\\ f_Q& =\frac{C_0}{ks(Q)}+s(Q).\end{array}$$

This applies when the untilted growth or tilt is sufficiently large. A random path chooses each child uniformly. The floors $f_Q$ have a bounded sum down to wavelength scale: the side lengths form a decreasing geometric series, and inverse local frequencies form a reversed geometric series.

For any permitted stopping rule, summing conditional expectations yields

$$\begin{array}{rl}& \mathbb{E}\left[\sum _{i=0}^T{m}_i+\sum _{i<T}|b_{i+1}-b_i|\right]\\ & \le C(m_0+1).\end{array}$$

The last value and the increment that triggers stopping are included. A large initial slope with small relative excess consequently survives on a positive fraction of paths; Markov’s inequality bounds the probability of losing too much slope or acquiring too much excess. A separate profile argument converts that surviving slope into high untilted growth. (OpenAI, 2026b)

There can still be many separate high-growth excursions. An excursion enters above a fixed high threshold and ends below a lower one. Each entry square $Q$ produces a nearby set $Y_Q$ of area at least $\delta |Q|$ on which the number of finer entries is uniformly bounded. Analytically, a nonconstant limiting profile has a point with nonzero weak gradient; its small affine excess near that point supplies the persistent slope.

The witness $Y_Q$ may lie outside $Q$, so disjoint-descendant reasoning alone does not pack the entries. The manuscript groups squares by their number of entry ancestors and selects disjoint dilates at each rank. For an exterior witness, spatial membership in a larger square can disagree with ancestry only near that larger square’s boundary. At fine side $t$ and coarse side $s$, this collar has area $O(st+t^2)$. Summing $t/s=A^{-j}$ bounds the total mismatch cost. Removing witnesses with too many mismatches leaves bounded overlap across ranks, giving

$$\sum _{Q\text{ entry}}|Q|\le C|Q_0|.$$

Each terminal path has at most one final active excursion. Apply stopped descent separately at its entry and sum using this packing estimate. The result is bounded mean $N$, hence bounded mean $E$ at wavelength scale. This completes the surface mechanism. (OpenAI, 2026b)

🏷️ Amplified envelopes and approximate waves

The counterexamples start with explicit background modes. On $S^3$ they use $\mathrm{Re}(x_1+i{x}_2{)}^n$, with eigenvalue $n(n+2)$. On $S^2\times {\mathbb{T}}^2$, in spherical coordinates $(p,\theta )$ and torus coordinates $(z,z^{\prime })$, they use

$$\begin{array}{c}{u}_k^0={\sin }^{k}p\cos (k(\theta +z)),\\ {\Lambda }_k=2k^2+k.\end{array}$$

The sphere contributes $k(k+1)$ and the first circle contributes $k^2$. A region of this explicit solution remains unchanged throughout each local construction.

An envelope $e^{kh}$ controls the amplitude of complex oscillatory packets. In the three-dimensional branch, write $a=\nabla h$ and $H={\nabla }^{2}h$. The required strict condition is the existence of $b$ with

$$\begin{array}{c}a\perp b,|b{|}^2=1+|a{|}^2,\\ H(a,a)+H(b,b)>0.\end{array}$$

For the four-dimensional background, the corresponding constant is $2$ instead of $1$, reflecting ${\Lambda }_k/k^2\to 2$. The condition permits complex phase Hessians whose real parts lie strictly below the envelope Hessian while satisfying the differentiated eikonal equation. Finite Taylor solutions of the phase and transport equations then give packets $e^{kS}V$ with arbitrarily small residual relative to the envelope.

Corrugations increase $\int |\nabla h|$ while preserving this strict inequality and changing $h$ arbitrarily little in supremum norm. In dimension three, preparation first makes all transverse tests positive on a definite fraction of gradient mass; radial corrugation in two transverse variables then gives a definite gain. The four-dimensional construction reserves a favorable direction and corrugates in two further directions. These qualitative constructions allow thresholds and derivative bounds to depend on the entire amplified envelope. The envelope is fixed before frequency is sent to infinity. (OpenAI, 2026c)

A Gaussian superposition of packets must satisfy two different requirements: many sign changes and a quantitatively nonvanishing first jet, the vector consisting of the value and first derivatives. The latter prevents correction formulas from dividing by zero. In the three-dimensional branch, the normalized real jet has four components. The small-ball probability at one point is bounded by $C{n}^{28-4B}$; a net has $O(n^{3(B+10)})$ points. The union bound is therefore $C{n}^{58-B}$, which tends to zero for the fixed choice $B=110$. Derivative estimates extend the net bound to all points. The four-dimensional branch uses five real jet components and its own construction. Increasing residual accuracy does not worsen the fixed noncancellation exponent. (OpenAI, 2026c)

🏷️ Stable signs as lower bounds for nodal measure

Large gradients alone do not prove large nodal measure. The qualitative constructions use a sign score. In a cube where the local oscillation scale is $\ell$, count starting points for which $u(x)$ and $u(x+\ell e_i)$ have opposite signs, integrate over the interior where the segment remains in the cube, and divide by $\ell$. Sum over coordinate directions and disjoint cubes.

Every successful segment contains a zero. Cover the zeros by small balls of radius $r_j$ and sweep those balls backward along the segment direction. The swept $d$-volume is at most

$$C\sum _j(\ell r_j^{d-1}+r_j^d).$$

Taking arbitrarily fine covers proves that the score is bounded above by a fixed multiple of ${\mathcal{H}}^{d-1}(Z_u)$. This requires no smoothness or transversality of the zero set. Coordinate functions extended with cutoffs supply the global Lipschitz comparison to intrinsic Hausdorff measure.

Strict endpoint inequalities also make the score lower semicontinuous under uniform convergence, by Fatou’s lemma. A positive multiplier preserves it exactly. Thus it survives both the correction of the equation and later spectral perturbations.

The packet phases oscillate on scale $(k\sqrt{1+|\nabla h{|}^2}{)}^{-1}$, with the analogous $2+|\nabla h{|}^2$ expression in dimension four. Comparing Gaussian endpoint correlations shows that at least one coordinate direction has a uniformly positive opposite-sign probability. The imaginary phase gradients give this correlation gap even if one carrier direction dominates. Consequently the expected score is at least a fixed multiple of $k\int \sqrt{1+|\nabla h{|}^2}$. Its deterministic upper bound has the same scale. Subtracting the small probability of first-jet failure selects a wave with both properties; independence of the two events is unnecessary. (OpenAI, 2026c)

🏷️ Exact equations and the metric determinant condition

An approximate eigenfunction is insufficient. First allow the principal tensor and volume density to vary independently:

$$Lf={\rho }^{-1}{\mathrm{div}}_{\mu }(Adf).$$

Here $A$ is positive symmetric, $\rho >0$, and $\mu$ is a fixed reference density. The operator is symmetric in $L^2(\rho d\mu )$. It need not yet be a Laplace–Beltrami operator for a metric in the same dimension.

The power-law manuscript supplies a useful exact identity. Let $I$ be the background inverse metric and let

$$\begin{array}{rl}\mathfrak{r}& ={\mathrm{div}}_{\mu }(Adu)+k^2\rho u,\\ D& =u^2+I(du,du),\\ F& =-\mathfrak{r}/D,\\ \delta A& =FuI,\\ \delta \rho & =k^{-2}Fu\\ & -k^{-2}{\mathrm{div}}_{\mu }(FIdu).\end{array}$$

The first-jet lower bound gives $D>0$. The product rule cancels the differentiated flux terms and gives

$${\mathrm{div}}_{\mu }(\delta Adu)+k^2\delta \rho u=FD=-\mathfrak{r}.$$

Thus the corrected equation is exact. Weighted derivative estimates make both corrections small in any prescribed finite norm: exponentially small amplitudes cancel between numerator and denominator. The requested residual order is chosen before the large frequency. (OpenAI, 2026a)

For an ordinary metric relative to a background $g$, a conductivity endomorphism $A$ and relative volume density $\rho$ must satisfy

$$\det A={\rho }^{d-2}.$$

Indeed, if the metric endomorphism is $C$, then $\rho =\sqrt{\det C}$ and $A=\rho C^{-1}$. The qualitative constructions enforce this constraint in two distinct ways. (OpenAI, 2026c)

Dimension three. A scalar correction first gives ${\mathrm{div}}_g(b\nabla U)+\Lambda sU=0$ with $b,s$ close to one. Divide by a positive factor $v$. For $u=U/v$, the new scalar conductivity and density are

$$\begin{array}{rl}q& =b{v}^2,\\ d& =s{v}^2+{\Lambda }^{-1}v{\mathrm{div}}_g(b\nabla v).\end{array}$$

The needed identity $d=q^3$ becomes

$$-{\mathrm{div}}_g(b\nabla v)+\Lambda b^3{v}^5-\Lambda sv=0.$$

At $b=s=v=1$, its linearization is $-{\Delta }_g+4\Lambda$, a coercive operator. A contraction argument finds $v$ close to one. Cutting $v$ off to one in the protected exterior creates a determinant defect only in a clean annulus where $\nabla u\ne 0$. Let $P_{\perp }$ project onto the two-plane perpendicular to that gradient. The repaired conductivity

$$A=q\left[I+\left(\sqrt{d/q^3}-1\right)P_{\perp }\right]$$

has $A\nabla u=q\nabla u$ and $\det A=d$. Thus it preserves the equation and realizes an ordinary metric. Division by a positive factor preserves every sign.

Dimension four. After a weighted correction, conjugation by a positive factor produces ${\Delta }_{h_1}v=(V_0-\Lambda )v$. Seek $g=Z^2{h}_1$ and $t=v/Z$. The required equation is

$${\Delta }_{h_1}Z=V_{0}Z+\Lambda (Z^3-Z).$$

Its linearization near the background is ${\Delta }_h-2\Lambda$. The inverse has an $O({\Lambda }^{-1})$ bound at equal Sobolev order and a uniform bound gaining two derivatives, so no high-energy nonresonance hypothesis is needed. After localization, change the conductivity on the rank-three bundle $\ker dt$. Scaling its three tangent eigenvalues enforces $\det A={\rho }^2$ while leaving $Adt$ unchanged. This produces an exact ordinary metric eigenfunction that is a positive multiple of the original wave. (OpenAI, 2026c)

🏷️ Preserving witnesses in one smooth metric

Even exact eigenpairs for a sequence of different metrics would not settle the problem. Each construction therefore makes the selected eigenvalue simple while preserving its eigenfunction.

In a protected background patch, a tensor variation annihilating $du$ leaves its flux unchanged. For ordinary metrics, a trace-free variation also preserves volume through the matrix exponential. Other eigenfunctions in the same eigenspace have nonzero transverse gradient somewhere: otherwise they locally equal $F(u)$, and the background gradient identity and eigenfunction equation force $F$ to be linear. Unique continuation then makes them multiples of $u$. Suitable transverse variations split the unwanted multiplicity. The weighted construction can instead use a nonnegative tensor variation, since density and tensor are independent. (OpenAI, 2026c) (OpenAI, 2026a)

Near a simple eigenvalue, the min–max principle and elliptic compactness give a continuous normalized eigenfunction, up to sign. A fixed strict sign-score threshold or finite collection of strict disk signs therefore persists in a neighborhood. No uniform spectral gap across all stages is required.

For the qualitative examples, choose nested closed balls in a complete metric for the smooth tensor topology. Each new center lies inside every previous witness neighborhood, has a larger target nodal ratio, and retains an exterior for the next construction. Shrinking radii give a smooth positive limiting metric in every witness neighborhood. Each neighborhood supplies its own exact eigenfunction of that same limit. The final eigenvalues may move from their prescribed intermediate values; the persistence intervals force them to infinity. (OpenAI, 2026c)

🏷️ A fixed power gain in dimension five

The quantitative companion works first on a four-dimensional weighted base $S^4$. Its improvement is a repeatable gain on the region prepared for the next generation, not merely on the total current region. For a parent cube $P$ and its prepared children ${\mathcal{R}}_P$, the corrugation gives

$${\int }_{{\mathcal{R}}_P}|\nabla h_{i+1}|\ge (1+\zeta ){\int }_P|\nabla h_i|.$$

Radial wells in three transverse variables have two favorable tangential Hessian directions. A finite packing of these wells and smaller cores leaves enough gradient mass for the next generation, even after cutoff and boundary losses. With a fixed scale ratio, the derivative increments add, and disjoint supports prevent a factor counting cubes in supremum norms. The estimates are

$$\begin{array}{rl}{I}_m& \ge I_0(1+\zeta {)}^m,\\ \Vert h_m{\Vert }_{C^r}& \le C_r{e}^{C_{\ast }(r+1)m}.\end{array}$$

One constant $C_{\ast }$ works for all finite orders. Taking $m=\lfloor \theta \log k\rfloor$ with fixed small $\theta$ yields

$$\begin{array}{c}{\int }_V\sqrt{1+|\nabla h_k{|}^2}\ge c{k}^{\gamma },\\ \gamma =\theta \log (1+\zeta )>0.\end{array}$$

The derivative growth remains a controlled power of $k$, even when arbitrarily high finite residual accuracy is requested. This uniformity of exponents, not uniformity of constants or frequency thresholds, is what the qualitative construction alone does not supply. The phase-Hessian argument separately handles zero real slope, avoiding division by a vanishing gradient. (OpenAI, 2026a)

At local scale $R_x=(k\sqrt{1+|\nabla h_k(x){|}^2}{)}^{-1}$, select disjoint balls. Packing gives ${\sum }_x{R}_x^3\ge c{k}^{1+\gamma }$. Within each ball, Gaussian conditional covariance leaves a derivative with positive variance after fixing the value. A small value, a definite derivative, and a controlled second derivative produce opposite signs on two parallel three-dimensional disks. Their directions and locations can be fixed before sampling. A bounded weighted-success variable gives a positive probability of total disk area at least $c{k}^{1+\gamma }$, without independence between balls.

The five-component first jet is simultaneously nonvanishing with probability tending to one. In the paper’s explicit bookkeeping, the small-ball and net exponents combine as

$$\begin{array}{rl}& (260-5D)+(4D+192)\\ & =452-D.\end{array}$$

Taking $D=500$ makes the failure probability decay. The exact tensor–density correction above then applies, and a protected round patch supplies simplicity. (OpenAI, 2026a)

At stage $j$, the sign certificate has size $c_j{k}_j^{1+\gamma }$. Its constant and starting frequency may depend badly on previous tolerances, but $\gamma$ is fixed. Choose $k_j$ sufficiently large that $c_j{k}_j^{\gamma /2}>j$, then choose future perturbations summably small in every required derivative norm. Persistence gives a limiting weighted pair with eigenvalues between $k_j^2/2$ and $2k_j^2$ and certificates exceeding $j{k}_j^{1+\gamma /2}$. Each new persistence tolerance is imposed only on later stages, avoiding a circular frequency choice.

Finally let $G^{-1}=A/\rho$ for the limiting pair and put

$$\begin{array}{c}q=\frac{\rho d\mu }{d{\mathrm{vol}}_G},\\ g=G+q^{2}d{\phi }^2\text{on }{S}^4\times S^1.\end{array}$$

For functions independent of the circle coordinate, the divergence formula gives ${\Delta }_g=L$. Thus the weighted eigenfunctions become ordinary five-dimensional eigenfunctions. Each sign cylinder, multiplied by a fixed circle interval, projects its nodal subset onto a transverse disk times that interval. The Lipschitz projection inequality converts the certificates into intrinsic four-dimensional nodal measure, without assuming the zero set is regular.

Set ${\epsilon }_0=\gamma /4$. Since ${\lambda }_j\asymp k_j^2$, the retained bound $j{k}_j^{1+\gamma /2}$ gives precisely the divergent ratio to ${\lambda }_j^{1/2+{\epsilon }_0}$. Taking further fixed products preserves the eigenvalues and extends the projection certificates to higher dimensions. This last conversion is why the quantitative theorem is stated in dimension five although its weighted construction occurs on a four-dimensional base. (OpenAI, 2026a)

🏷️ Related analytic and geometric arguments

• The Solomon–Yau least-volume problem also uses upper tests and a compactness argument, but its minimal-hypersurface area problem is separate from nodal measure.
• Einstein four-manifold rigidity provides another setting where elliptic estimates and geometric limiting arguments must preserve the exact intended hypotheses.
• The collection guide places this family among the differential geometry and elliptic-equation articles and records prerequisites.

🏷️ Key steps and proof outlines

In two dimensions the affine correction has a summable average cost

A near-affine logarithmic growth profile may retain its normalized growth under subdivision. Subtracting an affine function removes this obstruction, but the changing slope must remain part of the bookkeeping. The harmonic part and the far logarithmic potential supply that slope; the remaining local potential has error controlled by $\min \{1,{\rho }^2/|z-y{|}^2\}$ at child scale $\rho$.

Integrating this kernel over a planar grid costs $O({\rho }^2\log (1/\rho ))$: each dyadic annulus contributes $O({\rho }^2)$ and there are $O(\log (1/\rho ))$ relevant annuli. The near disk contributes $O({\rho }^2)$. Stopped descent uses this average improvement while retaining slope increments. Persistence estimates then assign large-growth squares witnesses in neighboring regions; the packing argument must count those external witnesses rather than pretending they lie inside their parent squares. (OpenAI, 2026b)

An exact divergence equation need not come from a metric

The counterexample constructions first obtain an equation with independently adjustable conductivity and density. In dimension three, a positive endomorphism $A$ with determinant $d$ defines the ordinary metric $g_A=g(d{A}^{-1}\cdot ,\cdot )$. Its relative metric determinant is $d^2$, so its volume density is $d$, and

$${\Delta }_{g_A}f=d^{-1}{\mathrm{div}}_g(A\nabla f).$$

For scalar conductivity $qI$, the required density is therefore $q^3$. An arbitrary corrected density does not automatically satisfy this relation.

Suppose $\mathrm{div}(b\nabla U)+\Lambda sU=0$. Divide by a positive function $v$. The product rule gives a new conductivity $q=b{v}^2$ and density $d=s{v}^2+{\Lambda }^{-1}v\mathrm{div}(b\nabla v)$. Imposing $d=q^3$ is exactly

$$-\mathrm{div}(b\nabla v)+\Lambda b^3{v}^5-\Lambda sv=0.$$

At $b=s=v=1$, its linearization in $v$ is $-\Delta +4\Lambda$, a coercive operator. For $\Lambda \ge 1$, its inverse from $H^{l-2}$ to $H^l$ is uniformly bounded because the spectral multiplier $(1+\mu )/(\mu +4\Lambda )$ is at most one. Sufficiently accurate coefficient corrections and a contraction near $v=1$ solve the determinant condition. Positivity preserves the zero set under $U\mapsto U/v$. Localized annular repair then restores the protected exterior. (OpenAI, 2026c)

Stable sign witnesses survive the passage to one limiting metric

The wave packets provide alternating signs at separated locations with quantitative margins. Geometric slicing converts these signs into a lower bound for nodal measure that persists under sufficiently small function perturbations. Exactification is chosen smaller than these margins. A protected region allows the eigenvalue to be made simple while retaining the eigenfunction construction; spectral continuity then preserves each finite collection of witnesses under later metric changes.

The final metric is obtained by a diagonal sequence: at each stage prescribe a finite list of derivative tolerances small enough to preserve every earlier witness. For every fixed derivative order, the remaining changes are summable, giving one smooth limiting metric. The power-law companion additionally proves one gain exponent that survives arbitrarily high finite-order demands; allowing that exponent to shrink with the requested smoothness would not establish its stated fixed power improvement. (OpenAI, 2026a)

🏷️ Manuscripts and verification scope

• Sharp nodal length on smooth surfaces — pinned manuscript and companion source.
• Smooth counterexamples to Yau’s nodal upper bound in dimensions three and four — pinned manuscript and companion source.
• Power-law violations of Yau’s nodal upper bound — pinned manuscript and companion source.

The family scope page describes selected formalization coverage. It selects the surface upper bound and both qualitative counterexamples. For the manuscript on power-law violations, its documented statement selects only an unbounded ratio to the square root of the eigenvalue on one fixed metric on S⁴ × S¹; it does not select the fixed positive power excess. The surface lower bound is also outside its selected statement. 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. Power-law violations of Yau’s nodal upper bound.

🐻  OpenAI 2026b. Sharp nodal length on smooth surfaces.

🐻  OpenAI 2026c. Smooth counterexamples to Yau’s nodal upper bound in dimensions three and four.