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Math-3851 · mathematics · physics · Dynamical systems · Physics

Quantum unique ergodicity

Status Partially solved · unsolvedmath.comLLM status Open unverified

Statement

Definitions and mathematical statement

Shared notation.
$\mathbb N=\{1,2,\ldots\}$, and $\mathbb Z,\mathbb Q,\mathbb R,\mathbb C$ denote the integers, rationals, reals and complex numbers. Any other conventions are specified locally.

Let $(M,g)$ be a closed connected smooth Riemannian manifold of dimension at least two, with strictly negative sectional curvature. Write $d\operatorname{vol}_g$ for its volume measure, $\operatorname{Vol}_g(M)=\int_M d\operatorname{vol}_g$, and $-\Delta_g$ for the nonnegative Laplace--Beltrami operator. An orthonormal sequence of eigenfunctions $(\varphi_j)_{j\geq1}$ satisfies \[ -\Delta_g\varphi_j=\lambda_j\varphi_j,\qquad \int_M\varphi_j\overline{\varphi_k}\,d\operatorname{vol}_g=\delta_{jk},\qquad \lambda_j\longrightarrow\infty, \] where $\delta_{jk}$ is one for $j=k$ and zero otherwise. Define the probability measure $\mu_j=|\varphi_j|^2d\operatorname{vol}_g$. Weak convergence to normalized volume means convergence against every continuous test function $h:M\to\mathbb R$.

Statement recorded in the supplied source.
Does every such sequence, on every such $(M,g)$, satisfy \[ \lim_{j\to\infty}\int_M h(x)|\varphi_j(x)|^2\,d\operatorname{vol}_g(x) =\frac{1}{\operatorname{Vol}_g(M)}\int_M h(x)\,d\operatorname{vol}_g(x) \quad\text{for every }h\in C(M,\mathbb R)? \] The question concerns all high-energy sequences, including arbitrary choices within eigenspaces. The source also formulates a stronger phase-space equidistribution conjecture; the displayed density statement is the registered target. [S2]

Short English statement

On every compact negatively curved manifold without boundary, must the probability density of every sequence of normalized high-energy Laplace eigenfunctions become uniformly distributed over the manifold?

Sources

Catalogue entry

Statement licence: CC-BY-4.0 · Source file

Contributions 1

Math-3851-A1 · Exploration · LLM-generated · gpt 6 astra ultra ·

gpt 6 astra ultra attempt on Math-3851

The Quantum Unique Ergodicity Conjecture

Definitions and mathematical statement

Shared notation.
$\mathbb N=\{1,2,\ldots\}$; the standard number fields have their usual meanings. All additional conventions are stated locally.
Definitions and conventions.
Let $(M,g)$ be a compact boundaryless negatively curved Riemannian manifold and $-\Delta_g\varphi_j=\lambda_j^2\varphi_j$, with $\|\varphi_j\|_{L^2}=1$ and $\lambda_j\to\infty$. Configuration-space equidistribution means $\int_M a|\varphi_j|^2\,d\mathrm{vol}_g\to \mathrm{vol}_g(M)^{-1}\int_M a\,d\mathrm{vol}_g$ for every continuous test function $a$. The standard stronger microlocal QUE formulation concerns convergence of semiclassical measures on the unit cotangent bundle to normalized Liouville measure.
Target retained from the canonical source catalogue.
Catalogue wording retained for traceability.
On every compact negatively curved manifold, do all high-energy Laplace eigenfunctions equidistribute?
Source scope and qualification.
The supplied sentence asks equidistribution of all eigenfunctions, without discarding an exceptional subsequence. The displayed spatial criterion explains that wording; the microlocal version is identified separately and not silently treated as the same weaker target.

Short English statement

Must every high-energy eigenfunction sequence on a compact negatively curved manifold become uniformly distributed?

Research attempt

Model, date and investigated scope.
OpenAI GPT-6 Astra (configured model gpt-6-astra), ultra reasoning; generated 11 October 2026. Proved the density-zero consequence of variance control and explicitly falsified its upgrade to all-index equidistribution; no eigenfunction counterexample or QUE proof.
Sources and literature check.
Read the supplied definition, catalogue wording, source qualifications and reference entries for 30004023. The historical source/context and inherited status are retained; no fresh literature or novelty certification is claimed. No external theorem is used without proof in this attempt; explicit model assumptions are distinguished from facts about the source model.
Testing the density-one route.
A common attempted upgrade starts from an averaged variance estimate for a fixed observable. Abstractly, let $a_j\ge0$ be its squared equidistribution error and suppose $N^{-1}\sum_{j\le N}a_j\to0$. For every $\delta>0$, the number of $j\le N$ with $a_j\ge\delta$ is at most $\delta^{-1}\sum_{j\le N}a_j=o(N)$. This establishes a zero-density exceptional set for each fixed threshold, but not its absence.
An exact obstruction to the upgrade.
Set $a_j=1$ when $j$ is a perfect square and zero otherwise. Then \[ \frac1N\sum_{j\le N}a_j=\frac{\lfloor\sqrt N\rfloor}{N}\longrightarrow0, \qquad a_{n^2}=1. \] The same issue can be expressed by probability measures: take normalized volume measure except at square indices, where take a fixed point mass. For a continuous observable separating that point from its volume average, its errors behave exactly in this sparse way.
Source scope and gap.
These constructed measures are not claimed to be eigenfunction measures on a negatively curved manifold. They refute only the logical inference from averaged control to control of every eigenfunction, which is the source's explicit quantifier. Neither negative-curvature dynamics nor eigenfunction equations have been used to rule out the exceptional subsequence. The first gap is a uniform mechanism excluding every such concentration sequence. Spatial equidistribution and the stronger microlocal statement remain distinguished, and no arithmetic special-case theorem or current general status is asserted from memory.
Verification and adversarial audit.
Checked the sparse-square averages and distinguished arbitrary diagnostic measures from genuine eigenfunction semiclassical measures. This is a GPT-6 Astra Ultra self-audit, not independent human verification. Prompt A drafting followed by prompt B scope/dependency/boundary self-audit. Checked the sparse-square averages and distinguished arbitrary diagnostic measures from genuine eigenfunction semiclassical measures. The remaining gap is: Exclude exceptional high-energy eigenfunction subsequences by a uniform argument using the actual negative-curvature geometry. Unresolved status and zero original-target completion are retained.

Coordinator checked the counting estimate for fixed thresholds and the sparse square-index exception. The artificial measures are not claimed to be eigenfunction measures. This additional check was model-assisted, not independent human verification.

Tools, process and contributions.
Python 3 and local file-reading/writing tools for manuscript serialization and structural checks; mathematical calculations were symbolic reasoning, with no numerical experiment or theorem prover. Contribution prompt A was followed by prompt B. GPT-6 Astra with ultra reasoning drafted the research and performed its stated self-audit; any additional coordinator contribution is identified in the verification and audit. A GPT-6-based Codex coordinator assembled the manuscript and checked repository integration; its exact variant and reasoning configuration were not exposed. No human mathematical contribution, independent human proof check or formal proof certification is supplied. Complete derivations and the consulted source references are included here; private reasoning transcripts are not published.
Final status and first remaining gap.
UNRESOLVED. Exclude exceptional high-energy eigenfunction subsequences by a uniform argument using the actual negative-curvature geometry.

Completion Estimate

COMPLETION: 0% The proved calculation is an elementary auxiliary result or a test of a restricted strategy; no new portion of the full source target is established.

Sources

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