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ASDThree-dimensional scaling and analytic stationary radius
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Three-dimensional scaling and analytic stationary radius

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.

Theory · CANDIDATE MODELEquation version 1.0.0

Based on the final manuscript · Chapters 4 and 5

ASD-WEB-SD-032
Qλ(x)=Q(x/λ),E(λ)=λE2+λ−1E4+λ3EV,E2−E4+3EV=0(λ=1),E(R)=A2R+A4/R+A0R3,E′(R)=A2−A4/R2+3A0R2,R∗2=A22+12A0A4−A26A0(A0>0),=2A4A2+A22+12A0A4(A0≥0),R∗2=A4/A2(A0=0),E′′(R∗)=2A4R∗3+6A0R∗>0.\begin{aligned}Q_\lambda(\mathbf x)&=Q(\mathbf x/\lambda),\\E(\lambda)&=\lambda E_2+\lambda^{-1}E_4+\lambda^3E_V,\\E_2-E_4+3E_V&=0\quad(\lambda=1),\\E(R)&=A_2R+A_4/R+A_0R^3,\\E'(R)&=A_2-A_4/R^2+3A_0R^2,\\R_*^2&=\frac{\sqrt{A_2^2+12A_0A_4}-A_2}{6A_0}\quad(A_0>0),\\&=\frac{2A_4}{A_2+\sqrt{A_2^2+12A_0A_4}}\quad(A_0\ge0),\\R_*^2&=A_4/A_2\quad(A_0=0),\\E''(R_*)&=\frac{2A_4}{R_*^3}+6A_0R_*>0.\end{aligned}

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

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Ch. 4 (4.20)–(4.22); Ch. 5 (5.6.23)–(5.6.25)

Registered formula checked individually against the final manuscript; analytic radius, rationalization, zero-potential limit and positive radial second derivative; source-grounded scope note synchronized

Conditions retained in this adoption

  • 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.

Source locators: p1368–1375 · p2275–2282

Manuscript SHA-256: ddbade997b45913248e588953b1d27ccb9c7c206c3c8d992d3aef2364a64836e

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SD-51779505a476 · Ch. 4 (4.20)–(4.22)

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algebraOrientation stabilizer candidateASD-WEB-SD-031
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