Global counterexamples to uniqueness for a Calderón problem with $C^k$ conductivities
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Bibliographic record
Abstract
Let $Ω\subset R^n$, $n \geq 3$, be a fixed smooth bounded domain, and let $γ$ be a smooth conductivity in $\overlineΩ$. Consider a non-zero frequency $λ_0$ which does not belong to the Dirichlet spectrum of $L_γ= -{\rm div} (γ\nabla \cdot)$. Then, for all $k \geq 1$, there exists an infinite number of pairs of non-isometric $C^k$ conductivities $(γ_1, γ_2)$ on $\overlineΩ$, which are close to $γ$ such that the associated DN maps at frequency $λ_0$ satisfy \begin{equation*} Λ_{γ_1,λ_0} = Λ_{γ_2,λ_0}. \end{equation*}
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Full frame distilled prediction
Teacher imitationNot calibrated prevalence, not ground truth. Human validation pending. Learned from the 10,348 direct Codex labels and 10,348 direct Gemma labels. Candidate is the union of thresholded teacher heads; consensus is their intersection. These outputs are machine_predicted_unvalidated and are not human labels or direct frontier model labels.
Codex and Gemma teacher scores by category
| Category | Codex | Gemma |
|---|---|---|
| Metaresearch | 0.001 | 0.000 |
| Meta-epidemiology (narrow) | 0.001 | 0.001 |
| Meta-epidemiology (broad) | 0.001 | 0.000 |
| Bibliometrics | 0.000 | 0.001 |
| Science and technology studies | 0.000 | 0.000 |
| Scholarly communication | 0.000 | 0.000 |
| Open science | 0.001 | 0.002 |
| Research integrity | 0.000 | 0.001 |
| Insufficient payload (model declined to judge) | 0.000 | 0.000 |
Machine scores (provisional)
The two teacher heads of the student model, read on this work. A score orders the frame for review; it never asserts a category, and the validation status ships verbatim with every row.
Baseline scores from an immature model (maturity gate not passed, 7 training rounds). Scores rank; they never assert a category.
score_only:v0-immature-baseline · verbatim from the scoring run: score_only means the number may rank works, and no category label ships from it