OP-SD-003 · Stable Nonlinear Topology
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.
Based on the final manuscript · 第四、五章
NEEDS THEORY · Needs theory
Discuss this task01Established 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.
Constitutive input
A2,A4>0 and norm squared=Tr(X†X)/2; existence and the full stable spectrum remain open.
Chapter reference: Ch. 4 (4.21) · ASD-BOOK-ddbade997b45
Mathematical result
Retains Chapter 4’s convention that increasing λ enlarges the profile; R is a physical length. A₂,A₄>0, A₀≥0. The 6A₀ denominator requires A₀>0; the rationalized expression includes A₀=0. Units: A₂ J/m, A₄ J·m, A₀ J/m³. The analytic radial minimum is not a complete three-dimensional solution or full perturbation stability.
Chapter reference: Ch. 4 (4.20)–(4.22); Ch. 5 (5.6.23)–(5.6.25) · ASD-BOOK-ddbade997b45
02Research 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.
03The 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.
04Scope 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 · master equation · electron solution · master equation · space dynamics · electron solution
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Adopted source, conditions and previous versions
第四章式（4.20）—（4.22）；第五章式（5.6.21）—（5.6.25）
已得结果、继续研究任务与下一份交付；保留原问题 ID、优先级及状态。
Conditions retained in this adoption
- 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.
Source locators: p1374 · p1375 · p2275 · p2281 · p2282
Manuscript SHA-256: ddbade997b45913248e588953b1d27ccb9c7c206c3c8d992d3aef2364a64836e
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Previous source, retained as history
SD-51779505a476 · Ch. 4 §4.1; Ch. 5 §§4, 6
CH4-9587f3c44d39 · 第四章-ASD空间本体论-出版整理稿.docx9587f3c44d3974f44e387ed03d92691fa5473a90d7b1c9c989155adf0622849a
CH5-ca737ec4e327 · 第五章-出版整理稿.docxca737ec4e327d72c5507bbc8e00295b6e56029c7583ebad8e505cdb9c68c6262
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These links index inputs and research relationships. The completion status of each derivation is specified in its text. Untranslated supporting pages open in Chinese.
Revision history
第四、五章 · 第四章式（4.20）—（4.22）；第五章式（5.6.21）—（5.6.25）
This research task is based on the cited final Chinese manuscript. It is not a new canonical result. Both languages use the same source snapshot; original equation numbers and website IDs remain distinct.
Knowledge scopeASD: OP-SD-003 · Stable Nonlinear Topology. OP-SD-003-EN. Version 1.0.0; 2026-10-24. https://asd-physics.org/en/open-problems/space-dynamics/stable-topology. 第四章式（4.20）—（4.22）；第五章式（5.6.21）—（5.6.25）. Source ASD-BOOK-ddbade997b45.
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