Smooth anisotropic Calderón uniqueness from full boundary data

Smooth anisotropic Calderón uniqueness from full boundary data

OpenAI's claim

Recovery of a metric and connection: Zero-frequency measurements with input and observation on one arbitrary open boundary patch are claimed to recover both a smooth metric and a unitary rank-two connection; recovery is up to a diffeomorphism and unitary gauge that fix the measured patch.

One-patch anisotropic uniqueness: The paper claims determination of a smooth Riemannian metric in dimension at least three from its zero-frequency energy form on any one open boundary patch; the only ambiguity is a diffeomorphism equal to the identity on that patch.

Mathematical statement of the claimed result

Mathematical claim 1.

Fix $n\ge3$ and a smooth connected manifold $M$ of that dimension, assumed compact with $\partial M$ smooth and nonempty. Choose a relatively open $\Gamma\subset\partial M$ which is nonempty and proper. On the trivial Hermitian rank-two bundle, set $d_A=d+A$, $A\in C^\infty(T^*M\otimes\mathfrak u(2))$, and define the local Dirichlet energy form using solutions of $d_A^{*g}d_Au=0$ with boundary data supported in $\Gamma$. If two smooth metrics and connections give the same form for every $f,h\in C_c^\infty(\Gamma;\mathbb C^2)$, the claim supplies smooth $\Phi:M\to M$ and $U:M\to U(2)$ with \[\Phi|_\Gamma=\operatorname{id},\quad U|_\Gamma=I,\quad g_2=\Phi^*g_1,\quad A_2=U^{-1}(\Phi^*A_1)U+U^{-1}dU.\] Thus the ambiguity in recovering the two geometric unknowns consists of the displayed diffeomorphism and gauge, each trivial on the observation patch.

Mathematical claim 2.

Fix a dimension $n\ge3$. Assume the smooth $n$-manifold $M$ is compact and connected, with smooth nonempty boundary. For a proper relatively open observation patch $\Gamma\ne\varnothing$, the claim asserts \[\Lambda_{g_1,\Gamma}=\Lambda_{g_2,\Gamma}\quad\Longrightarrow\quad g_2=\Phi^*g_1,\qquad\Phi|_\Gamma=\operatorname{id},\] for some diffeomorphism smooth through the boundary. Both metrics are smooth, and the local maps are the harmonic Dirichlet energy forms for data supported in $\Gamma$. Equality for all full-boundary data gives a diffeomorphism fixing the entire boundary. This concerns smooth anisotropic metrics, not arbitrary bounded measurable scalar conductivities.

Outline of the argument

Recovery of a metric and connection: Boundary factorization identifies normal-gauge jets, and a common exterior cap starts equality of Green matrices. Source solutions provide separating points, harmonic jets, and bundle frames. Mixed-kernel continuation in the product, followed by moving-sphere transfer and angular coercivity, extends the maximal local metric-and-bundle identification throughout the manifold.

One-patch anisotropic uniqueness: Boundary jets and a common exterior cap identify actual Green kernels in an initial region. Remote source solutions produce harmonic coordinates and enough second-order jets. A mixed Green-kernel continuation theorem extends metric equality across a smooth frontier, so the maximal paired coordinate region cannot stop in the interior.

Formal verification

Family 365 describes Lean coverage for the measurable scalar-conductivity obstruction; smooth anisotropic and coupled connection recovery are separate.

Recovery of a metric and connection: No manuscript-specific Comparator is listed in the pinned inventory.

The documented selection does not cover the entire manuscript assertion relevant here.

One-patch anisotropic uniqueness: No manuscript-specific Comparator is listed in the pinned inventory.

The documented selection does not cover the entire manuscript assertion relevant here.

This site has not run the Lean development. These links describe OpenAI’s selected formal statements and should be read with their stated scope.

Status caveat

OpenAI reports that an internal model produced the manuscripts. Their arguments have not been independently verified as proofs by this site. Any portion outside the documented formal scope may contain mistakes, and the catalogue label reports the announced mathematical claim.

Sources

• OpenAI, Determination of a metric and a unitary connection from one boundary patch, 2026-10-05; citation key OAI:Determination-of-a-metric-and-a-unitary-connection-from-one-boundary-patch-October-5-2026. PDF (GitHub); PDF (direct); Manuscript page and citation; Latest version; Statement source: thm:main.
• OpenAI, Smooth anisotropic uniqueness in the Calderón problem from one boundary patch, 2026-09-24; citation key OAI:Smooth-Anisotropic-Uniqueness-in-the-Calderon-Problem-from-One-Boundary-Patch-September-24-2026. PDF (GitHub); PDF (direct); Manuscript page and citation; Latest version; Statement source: main:full; Statement source: main:patch.
• Family 365: Lean scope.
• OpenAI catalogue entry.

Smooth anisotropic Calderón uniqueness from full boundary data

Definitions and mathematical statement

For a smooth metric $g$ on a compact connected manifold $M$ with boundary, solve the Dirichlet problem $\Delta_g u=0$, $u|_{\partial M}=f$, and define the Dirichlet-to-Neumann operator $\Lambda_g f=\partial_{\nu_g}u|_{\partial M}$, with outward metric-unit normal $\nu_g$. In dimension $n\ge3$, the target is \[ \Lambda_{g_1}=\Lambda_{g_2}\quad\Longrightarrow\quad g_2=\Phi^*g_1\text{ for some }\Phi|_{\partial M}=\mathrm{id}. \] Both metrics live on the same smooth manifold and the operators are compared on the entire boundary using its identity marking. Pullback by a boundary-fixing diffeomorphism is the unavoidable gauge ambiguity. No real-analyticity, product structure, or partial-boundary restriction is included.

Formulation and concept references: [S1].

Short English statement

Do all boundary voltage-to-flux measurements determine a smooth interior metric, apart from a change of coordinates that fixes the boundary?

Research attempt

Current frontier.

The 2026 formulation paper explicitly distinguishes the general smooth problem from its quasianalytic, symmetric, and ordered settings [S1], Introduction and Section 1.1. Its new results are manuscript results with additional hypotheses, not a proof of the present universal assertion. Boundary determination fixes jets in boundary-normal coordinates; it does not identify arbitrary smooth metrics on an interior collar merely from equality of those jets. We use the standard energy and unique continuation mechanisms discussed in that source, but no unverified global rigidity theorem from it.

Attack route.

A boundary-jet route requires a continuation principle unavailable for arbitrary smooth coefficients. A monotonicity route works if the metric comparison produces a sign-definite energy defect. A linearized route can use products of harmonic functions, but must distinguish injectivity from an invertible nonlinear reconstruction. We pursue the latter two routes: prove an ordered conformal case directly, compute the Euclidean linearized kernel in a gauge, and test the attempted passage to the full problem.

Attempt: an ordered conformal case on the stated manifold.

Suppose, as additional hypotheses, that $g_2=e^{2\varphi}g_1$, $\varphi\ge0$, and $\varphi$ vanishes on a neighborhood of the boundary. Put $a=e^{(n-2)\varphi}$. The Dirichlet energy of $g_2$, written using $g_1$, is \[ E_2(u)=\int_M a|\nabla u|_{g_1}^2\,dV_{g_1}, \qquad E_1(u)=\int_M|\nabla u|_{g_1}^2\,dV_{g_1}. \] For boundary data $f$, let $u_j$ minimize its respective energy. The metrics coincide near the boundary, so the surface measures and outward normals agree there. Equality of the normal-derivative DN maps in the question therefore gives equality of the boundary energy pairings, and hence $E_2(u_2)=E_1(u_1)$. Minimality of $u_1$ yields \begin{equation}\label{wsref-0-65-71-31-37-36-2d-6d-6f-6e-6f-74-6f-6e-65} 0\le\int_M(a-1)|\nabla u_2|_{g_1}^2\,dV_{g_1} =E_2(u_2)-E_1(u_2) \le E_2(u_2)-E_1(u_1)=0. \end{equation} If $\varphi$ is positive somewhere, it is bounded below by a positive number on an interior ball. Thus $u_2$ is constant on that ball. Interior unique continuation for smooth uniformly elliptic scalar equations then makes $u_2$ constant on connected $M$. Choose nonconstant smooth boundary data to get a contradiction. The unique continuation input, including its applicability to smooth divergence-form coefficients, is the one recalled in [S1], Section 5.1. Hence $\varphi=0$. The case $\varphi\le0$ follows by interchanging the metrics.

This proves a genuine special-case uniqueness statement using the exact boundary convention. It does not assume that all tensor-valued metric differences can be made conformal or ordered by a boundary-fixing coordinate change. Removing that additional structure removes the sign in (eq176-monotone).

Attempt: the Euclidean linearized gauge calculation.

Let $\Omega\subset{\mathbb{R}}^n$, $n\ge3$, be a smooth bounded connected domain, and let $h\in C_c^\infty(\Omega;\operatorname{Sym}_n)$. For $g_\varepsilon=I+\varepsilon h$, the conductivity density is \begin{equation}\label{wsref-0-65-71-31-37-36-2d-64-65-6e-73-69-74-79} \sqrt{\det g_\varepsilon}\,g_\varepsilon^{-1} =I+\varepsilon B+O(\varepsilon^2),\qquad B=\tfrac12(\operatorname{tr}h)I-h. \end{equation} For harmonic $u,v$, differentiation of the energy pairing gives the linearized DN form \[ \int_\Omega B\nabla u\cdot\nabla v\,dx. \] The derivatives of the harmonic extensions have zero boundary trace and pair to zero with harmonic functions by integration by parts. Compact support of $h$ also prevents a derivative of the boundary surface measure. Thus vanishing of the linearized data implies that this integral is zero for every pair of smooth harmonic functions.

Extend $B$ by zero and use complex harmonic exponentials $u=e^{\zeta\cdot x}$, $v=e^{\eta\cdot x}$ with $\zeta\cdot\zeta=\eta\cdot\eta=0$. Complex functions are legitimate by complex-linear extension of the real bilinear identity. Fix a nonzero real frequency $\xi$, choose a real $w\perp\xi$ of length $|\xi|/2$, and set \[ \zeta=i\xi/2+w,\qquad \eta=i\xi/2-w. \] They are null vectors and sum to $i\xi$. Symmetry of $B$ cancels the cross terms, so, writing $\widehat B(\xi)=\int e^{i\xi\cdot x}B(x)\,dx$, \begin{equation}\label{wsref-0-65-71-31-37-36-2d-66-6f-75-72-69-65-72} \tfrac14\xi^t\widehat B(\xi)\xi+ w^t\widehat B(\xi)w=0. \end{equation} Rotate coordinates so $\xi=|\xi|e_1$. Since (eq176-fourier) holds for every $w$ on the transverse sphere, polarization gives \[ \widehat B_{ab}=-\widehat B_{11}\delta_{ab} \qquad(2\le a,b\le n). \] The invertible linear relation in (eq176-density) is \[ \widehat h=\frac{\operatorname{tr}\widehat B}{n-2}I-\widehat B. \] Because $\operatorname{tr}\widehat B=-(n-2)\widehat B_{11}$, it follows that $\widehat h_{ab}=0$ for all transverse indices $a,b\ge2$.

Now impose the explicit linear gauge $\operatorname{div}h=0$. It implies $\xi_i\widehat h_{ij}=0$, so in these coordinates all first-row and first-column entries vanish too. Hence $\widehat h(\xi)=0$ for every $\xi\ne0$, and continuity gives the same at zero. Fourier injectivity proves $h=0$. This is a complete injectivity calculation for compactly supported, divergence-free infinitesimal perturbations, not the nonlinear problem on an arbitrary manifold.

The omitted linear directions are exactly of the expected potential form at each nonzero frequency: a symmetric matrix with only first row and column nonzero can be written as $\xi\otimes v+v\otimes\xi$. Indeed a boundary-fixing infinitesimal diffeomorphism produces $h=\nabla V+(\nabla V)^t$, which has zero linearized DN data. This also follows directly by differentiating the exact pullback invariance. The calculation does not claim a global gauge-fixing theorem with estimates from the frequency description alone.

Attempt: trying to close the nonlinear gap.

A possible local strategy would choose a gauge and prove an estimate controlling $h$ by the linearized data in spaces where the nonlinear remainder is smaller than that control. The injectivity above supplies no bounded inverse in such spaces. Harmonic exponentials grow with frequency; using them in estimates incurs costs absent from a bare zero-data argument. The assertion that an injective derivative automatically gives a local inverse in an infinite-dimensional problem is false without additional range and stability hypotheses.

The boundary-jet route has its own explicit failure. A smooth function $e^{-1/t^2}$ for $t>0$, extended by zero for $t\le0$, has every derivative zero at zero but is not zero on any right neighborhood. Multiplying this by a tensor in a collar produces smooth perturbations invisible to their boundary Taylor series, while not proving anything about their full DN maps. The example refutes inference from equal jets to equal smooth metrics, not uniqueness from the full boundary data.

Stress test.

Dimension two is an important negative check on the algebra: $n-2=0$ makes the tensor inversion in (eq176-density) impossible, and conformal rescaling with factor one at the boundary preserves the harmonic equation and its measured normal derivative. That mechanism is excluded by the stated $n\ge3$ hypothesis; it is not a counterexample here.

The weak boundary flux is a density whereas the catalogue DN map is a function. In both calculations the metrics agree near the boundary, so the distinction has been resolved explicitly rather than ignored. General equality of DN operators would first need the appropriate boundary identification, as discussed in [S1]. Neither positivity of a conformal factor nor a Euclidean Fourier representation is available for two unrestricted smooth metrics in the target.

Outcome.

Outcome: Exploratory approach with unresolved gap.

The attempt supplies proofs of an ordered conformal special case and linearized Euclidean injectivity in a divergence-free compact-support gauge. These are diagnostic derivations of familiar mechanisms, not claims of new general inverse-problem theorems. The unresolved steps are obtaining sufficient harmonic probing and quantitative nonlinear control for arbitrary smooth geometry, or another global rigidity mechanism without order or quasianalyticity. No full uniqueness theorem or counterexample is obtained.

Sources

• S1 Quasianalyticity and geometric rigidity in anisotropic Calderon's problem; arXiv2608.15813v1 August16,2026. https://arxiv.org/pdf/2608.15813v1. Catalogue formulation source.