# P11152: Minimal Normal-Mode Couplings for Recovering Band-Limited 3D Earth Structure

- ID: `P11152`
- Reference: `minimal-normal-mode-couplings-recovering-band-limited-earth-structure`
- Page: https://theoremdb.org/statements/P11152
- Export scope: built Markdown snapshot. The current public packet may have changed since this build.
- Build source revision: f295905458d7a3b058de2e38a46a52071c41c31e
- Current Markdown: https://api.theoremdb.org/v1/statements/minimal-normal-mode-couplings-recovering-band-limited-earth-structure?representation=markdown
- Record maturity: Reviewed problem

## The problem

Let a spherically symmetric, non-rotating, elastic and isotropic reference Earth have normal-mode multiplets K=(n,\ell,\mathrm{type}), with angular eigenspaces V_\ell of dimension 2\ell+1. Let a small perturbation of one scalar material parameter be band-limited in angle to degrees 0\le s\le S and expanded in a fixed P-dimensional radial basis as
\[
h(r,\Omega)=\sum_{s=0}^S\sum_{t=-s}^s\sum_{a=1}^P x_{a,s,t}\,\psi_a(r)Y_s^t(\Omega).
\]
For a selected finite family E of self- and cross-coupled normal-mode pairs (K,K'), assume first-order quasi-degenerate perturbation theory gives the full coupling matrices H^{KK'}, after known rotation and hydrostatic-ellipticity terms have been removed. These matrices depend linearly on x through radial sensitivity integrals and Gaunt/Wigner-3j angular coefficients.

Characterize generic injectivity and stable invertibility of the resulting finite-dimensional measurement operator A_E:x\mapsto\{H^{KK'}\}_{(K,K')\in E}. Give necessary and sufficient angular coverage conditions in terms of \ell,\ell' and Wigner-3j selection rules, and explicit sufficient radial-rank conditions for full recovery through angular degree S and radial rank P. Determine or sharply bound the minimum number and parity pattern of multiplet pairs needed to make all angular degrees 1,\ldots,S observable. In particular, prove that self-couplings alone cannot recover odd-degree scalar heterogeneity and that odd-degree recovery requires at least one opposite-parity pair \ell+\ell' odd. Derive a quantitative stability estimate \(\|x\|\le C\|A_E x\|\) with C controlled by the smallest singular values of the angular Wigner blocks and radial sensitivity matrices, and characterize how weak or near-degenerate mode couplings degrade C. The target is a finite-dimensional identifiability and minimal-experiment theorem for Earth normal-mode coupling, not the classical computation of splitting functions or a new tomography algorithm.

## Status

**Resolution status:** Open.
**Lean verification:** Not Lean-verified.

A Lean formalization is attached. Signed verification is pending.

## Research packet

### Working on this

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## References

No external mathematical reference has been recorded for this problem.