---
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  "summary": "Competition between positive second- and fourth-gradient energies and a nonnegative volume potential gives a unique positive stationary R_* with positive radial second derivative. The fourth-order mechanism resisting collapse is analytic. The scale parameters of Chapters 4 and 5 are reciprocal, so powers must be compared using their definitions.",
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---
## Established results

Competition between positive second- and fourth-gradient energies and a nonnegative volume potential gives a unique positive stationary R_* with positive radial second derivative. The fourth-order mechanism resisting collapse is analytic. The scale parameters of Chapters 4 and 5 are reciprocal, so powers must be compared using their definitions.

<Equation name="sd-topology-energy" />

<Equation name="sd-derrick" />

## Research still required

With one target space, vacuum and boundary, construct the full three-dimensional finite-energy solution and compute a Hessian or periodic-background Floquet spectrum. Check fission, radiation, zero modes and amplitude-zero paths before mapping topology to particle properties.

## The next reproducible deliverable

First reproduce the analytic radius and A_0→0 limit, then deliver a field solution and nonradial spectrum with equation residuals and boundary/grid convergence; state exactly which perturbations beyond radial curvature were tested.

## Scope and research links

The one-parameter scale model solves its radial extremum problem: this must neither be called an absence of an analytic scale nor enlarged into a complete electron/proton stability proof.

[stability](/en/theory/matter/stability) · [master equation](/en/theory/space-dynamics/master-equation) · [electron solution](/en/open-problems/matter/electron-solution) · [master equation](/en/theory/space-dynamics/master-equation) · [space dynamics](/en/theory/space-dynamics) · [electron solution](/en/open-problems/matter/electron-solution)

<CanonicalSource />