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ASD EQUATION REGISTRY

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Theory · FOUNDATION
ASD-WEB-MET-001
r=ct,c=299 792 458  m s−1r=ct,\quad c=299\,792\,458\;\mathrm{m\,s^{-1}}

SI fixes the numerical value of c; r is the specified vacuum propagation path, not automatically an arbitrary coordinate distance.
Chapter reference: Final consolidated manuscript · Chapter 1, Eq. (1.1); BIPM metre definition · ASD-BOOK-ddbade997b45

Theory · DERIVED
ASD-WEB-LVE-001
L⊥′=γL⊥,γ=11−β2L^{\prime}_{\perp}=\gamma L_{\perp},\quad\gamma=\frac{1}{\sqrt{1-\beta^2}}

L⊥ is the reference transverse length; L′⊥ is the true length associated with motion as defined by ASD. The compared scale is perpendicular to motion; 0 ≤ v < c.
Chapter reference: Final consolidated manuscript · Chapter 1, Eqs. (1.17)–(1.19) · ASD-BOOK-ddbade997b45

Low-speed expansion

计量与运动学
Theory · DERIVED
ASD-WEB-LVE-002
γ=1+β22+O(β4)\gamma=1+\frac{\beta^2}{2}+O(\beta^4)

β ≪ 1. An algebraic supplement on this website obtained by expanding Chapter 1, Eq. (1.10); the manuscript assigns no separate number to this expansion.
Chapter reference: Final consolidated manuscript · Website low-speed expansion of Chapter 1, Eq. (1.10); no new manuscript number · ASD-BOOK-ddbade997b45

Momentum and total energy

计量与运动学
Theory · DERIVED
ASD-WEB-DYN-001
p=γm0v,E=γm0c2p=\gamma m_0v,\quad E=\gamma m_0c^2

Momentum uses inertial constitutive closure; total energy additionally uses E₀=m₀c². Detailed derivations retain the manuscript equation numbers.
Chapter reference: Final collected manuscript · Chapter 2 §III (2.13)—(2.14), (2.24) · ASD-BOOK-ddbade997b45

Reference Coulomb magnitude

电磁相互作用
Theory · FOUNDATION
ASD-WEB-EM-001
F0=∣q1q2∣4πε0r2F_0=\frac{|q_1q_2|}{4\pi\varepsilon_0r^2}

Here F₀ is a magnitude; the signed vector is registered separately. The legacy ID is retained.
Chapter reference: Final collected manuscript · Chapter 3, equations (3.3) · ASD-BOOK-ddbade997b45

Theory · DERIVED
ASD-WEB-EM-003
FT=F0γ2=(1−β2)F0F_T=\frac{F_0}{\gamma^2}=(1-\beta^2)F_0

Magnitude under the true-distance inverse-square law; distinct from the coordinate generalized force. No low-speed truncation.
Chapter reference: Final collected manuscript · Chapter 3, equations (3.4) · ASD-BOOK-ddbade997b45

Frequency and wavelength ratio

引力与空间尺度
Theory · DERIVED
ASD-WEB-GRV-001
ν2ν1=λ1λ2\frac{\nu_2}{\nu_1}=\frac{\lambda_1}{\lambda_2}

Original ID and formula retained; compare frequencies and wavelengths in a common reference.
Chapter reference: Final collected manuscript · Chapter 2 §I (2.1)—(2.2a); wavelength and frequency ratios in one common reference · ASD-BOOK-ddbade997b45

Theory · EFFECTIVE MODEL
ASD-WEB-WAV-001
∂t2ψL−c2∇2ψL=0,ω2=c2∣k∣2\partial_t^2\psi_L-c^2\nabla^2\psi_L=0,\quad\omega^2=c^2|\mathbf k|^2

The compatibility symbol ψL here denotes the source-free linear density channel σ=-∇·uL. Homogeneous long-wave regime and compatible boundaries; retain its 1/k² kinetic kernel rather than introducing a new scalar constitutive law.
Chapter reference: Ch. 5 (5.4.6), source-free density channel; (5.4.8)–(5.4.10) · ASD-BOOK-ddbade997b45

Theory · EFFECTIVE MODEL
ASD-WEB-WAV-002
∂t2uT−c2∇2uT=0,∇⋅uT=0\partial_t^2\mathbf u_T-c^2\nabla^2\mathbf u_T=0,\quad\nabla\cdot\mathbf u_T=0

Compatibility formula for transverse displacement at constant coefficients, long wavelength and zero source after common-speed matching; retain the divergence-free constraint. It does not replace nonlinear matter or tensor branches.
Chapter reference: Ch. 4 (4.8)–(4.9); Ch. 5 (5.1.9), (5.4.1)–(5.4.2) · ASD-BOOK-ddbade997b45

Theory · CANDIDATE MODEL
ASD-WEB-FCC-001
∑ndn,idn,j=2a02δij\sum_n d_{n,i}d_{n,j}=2a_0^2\delta_{ij}

Geometric second moment of twelve neighbours; a₀ is the conventional cell edge and a₀/√2 the neighbour spacing. Equal vector-sector speeds additionally require link dynamics and constitutive matching.
Chapter reference: Ch. 4 (4.3)–(4.4); Ch. 5 (5.1.3)–(5.1.4) · ASD-BOOK-ddbade997b45

Theory · FOUNDATION
ASD-WEB-LVE-003
r=ct,r=c tr=ct,\qquad \mathbf r=\mathbf c\,t

r denotes the effective path in the specified propagation direction. Vector notation follows the manuscript; r is not automatically an arbitrary three-dimensional Euclidean norm.
Chapter reference: Final consolidated manuscript · Chapter 1, Section II, Eqs. (1.1)–(1.2) · ASD-BOOK-ddbade997b45

Theory · FOUNDATION
ASD-WEB-LVE-004
r=xi+yj+zk,r=ctr′=x′i+y′j+z′k,r′=ct′\begin{aligned}\mathbf r&=x\mathbf i+y\mathbf j+z\mathbf k,&r&=ct\\\mathbf r^{\prime}&=x^{\prime}\mathbf i+y^{\prime}\mathbf j+z^{\prime}\mathbf k,&r^{\prime}&=ct^{\prime}\end{aligned}

The same c is used in each measurement. The primed object, endpoints and propagation direction must be specified together with its configuration.
Chapter reference: Final consolidated manuscript · Chapter 1, Section II, Eqs. (1.3)–(1.6) · ASD-BOOK-ddbade997b45

Theory · DERIVED
ASD-WEB-LVE-005
r′=γ(r−vt)v=vxi+vyj+vzkr=γ(r′+vt′)\begin{aligned}\mathbf r^{\prime}&=\gamma(\mathbf r-\mathbf v t)\\\mathbf v&=v_x\mathbf i+v_y\mathbf j+v_z\mathbf k\\\mathbf r&=\gamma(\mathbf r^{\prime}+\mathbf v t^{\prime})\end{aligned}

Both measurements use the same scale factor. Event pairing and inverse-mapping conditions are addressed in the consistency review. This expression is not assumed to be a Lorentz transformation.
Chapter reference: Final consolidated manuscript · Chapter 1, Section II, Eqs. (1.7)–(1.9) · ASD-BOOK-ddbade997b45

Motion scale factor

计量与运动学
Theory · DERIVED
ASD-WEB-LVE-006
γ(v)=11−v2/c2=11−β2,β=vc\gamma(v)=\frac{1}{\sqrt{1-v^2/c^2}}=\frac{1}{\sqrt{1-\beta^2}},\qquad\beta=\frac vc

The positive root is taken under the manuscript’s forward/reverse pairing and normalization conditions; 0 ≤ v < c. Geometric sufficiency of the normalization remains under formal review.
Chapter reference: Final consolidated manuscript · Chapter 1, Section II, Eq. (1.10) · ASD-BOOK-ddbade997b45

Theory · DERIVED
ASD-WEB-LVE-007
t′=t γ(1−βcos⁡ϕ)t^{\prime}=t\,\gamma(1-\beta\cos\phi)

φ is the angle between observer relative velocity and the specified propagation or measurement direction. This projection result is not a general coordinate-time transformation without a synchronization rule.
Chapter reference: Final consolidated manuscript · Chapter 1, Section II, Eqs. (1.11)–(1.12) · ASD-BOOK-ddbade997b45

Theory · DERIVED
ASD-WEB-LVE-009
x′=γ(x−vt)y′=γyz′=γz\begin{aligned}x^{\prime}&=\gamma(x-vt)\\y^{\prime}&=\gamma y\\z^{\prime}&=\gamma z\end{aligned}

Set v = (v,0,0) and expand the manuscript’s vector construction. The transverse components retain γ and must not be replaced by standard Lorentz transverse-coordinate relations.
Chapter reference: Final consolidated manuscript · Chapter 1, Section II, Eqs. (1.15)–(1.18) · ASD-BOOK-ddbade997b45

Theory · DERIVED
ASD-WEB-LVE-010
λ=r,λ′=λ γ(1−βcos⁡ϕ)\lambda=r,\qquad\lambda^{\prime}=\lambda\,\gamma(1-\beta\cos\phi)

r is the propagation length over one period. The definition of φ follows the manuscript’s measurement configuration.
Chapter reference: Final consolidated manuscript · Chapter 1, Section 3.1, Eqs. (1.21)–(1.22) · ASD-BOOK-ddbade997b45

Theory · DERIVED
ASD-WEB-LVE-011
λ′f′=λf=c,f′=fγ(1−βcos⁡ϕ)\lambda^{\prime}f^{\prime}=\lambda f=c,\qquad f^{\prime}=\frac{f}{\gamma(1-\beta\cos\phi)}

Frequency follows from wavelength change at the same c. Match emission/reception frames and angle definitions before comparison with another theory.
Chapter reference: Final consolidated manuscript · Chapter 1, Section 3.1, Eqs. (1.23)–(1.25) · ASD-BOOK-ddbade997b45

Theory · DERIVED
ASD-WEB-LVE-012
t1′=Lcγ(1−β),t2′=Lcγ(1+β)T∥′=t1′+t2′=2γLc\begin{aligned}t_1^{\prime}&=\frac Lc\gamma(1-\beta),&t_2^{\prime}&=\frac Lc\gamma(1+\beta)\\T_{\parallel}^{\prime}&=t_1^{\prime}+t_2^{\prime}=\frac{2\gamma L}{c}\end{aligned}

Ideal equal-arm configuration with apparatus imperfections omitted. The compared observable is the arm-to-arm difference.
Chapter reference: Final consolidated manuscript · Chapter 1, Section 3.2, Eqs. (1.26)–(1.28) · ASD-BOOK-ddbade997b45

Theory · DERIVED
ASD-WEB-LVE-013
t⊥′=γLc,T⊥′=2t⊥′=2γLc=T∥′t_{\perp}^{\prime}=\frac{\gamma L}{c},\qquad T_{\perp}^{\prime}=2t_{\perp}^{\prime}=\frac{2\gamma L}{c}=T_{\parallel}^{\prime}

The model gives zero differential delay. Agreement in this difference does not mean the common factor γ has been measured independently.
Chapter reference: Final consolidated manuscript · Chapter 1, Section 3.2, Eqs. (1.29)–(1.30) · ASD-BOOK-ddbade997b45

Theory · DERIVED
ASD-WEB-LVE-014
Lopt=nL,Lopt′=nLγ(1−βcos⁡ϕ)t1′=nLcγ(1−β),t2′=nLcγ(1+β)T′=2nLγc\begin{aligned}L_{\mathrm{opt}}&=nL,\quad L_{\mathrm{opt}}^{\prime}=nL\gamma(1-\beta\cos\phi)\\t_1^{\prime}&=\frac{nL}{c}\gamma(1-\beta),\quad t_2^{\prime}=\frac{nL}{c}\gamma(1+\beta)\\T^{\prime}&=\frac{2nL\gamma}{c}\end{aligned}

This is the manuscript’s optical-path model. Its correspondence to individual paths and phases in Hoek’s original apparatus requires review.
Chapter reference: Final consolidated manuscript · Chapter 1, Section 3.3, Eqs. (1.31)–(1.35) · ASD-BOOK-ddbade997b45

Theory · DERIVED
ASD-WEB-LVE-015
trev′=tγ(1−β),tco′=tγ(1+β)Δt=tco′−trev′=2tβγ\begin{aligned}t_{\mathrm{rev}}^{\prime}&=t\gamma(1-\beta),&t_{\mathrm{co}}^{\prime}&=t\gamma(1+\beta)\\\Delta t&=t_{\mathrm{co}}^{\prime}-t_{\mathrm{rev}}^{\prime}=2t\beta\gamma\end{aligned}

This section describes motion of the object and uses the corresponding angle convention. Signs from the observer-motion description cannot be transferred automatically.
Chapter reference: Final consolidated manuscript · Chapter 1, Section 3.5, Eqs. (1.36)–(1.38) · ASD-BOOK-ddbade997b45

Theory · DERIVED
ASD-WEB-LVE-016
t=2πRc,v=R′ω,R′=γRΔt=4πωR′2c2=4ωS′c2,S′=πR′2\begin{aligned}t&=\frac{2\pi R}{c},&v&=R^{\prime}\omega,&R^{\prime}&=\gamma R\\\Delta t&=\frac{4\pi\omega R^{\prime2}}{c^2}=\frac{4\omega S^{\prime}}{c^2},&S^{\prime}&=\pi R^{\prime2}\end{aligned}

Uses the manuscript’s rest radius R and motion-associated radius R′. Comparing with standard Sagnac expressions requires matched area and timing definitions; the area form alone does not establish all-orders agreement.
Chapter reference: Final consolidated manuscript · Chapter 1, Section 3.5, Eqs. (1.39)–(1.43) · ASD-BOOK-ddbade997b45

Theory · DERIVED
ASD-WEB-LVE-017
T=Lv,L′=Lγ(1−β)=L1−β1+βT=\frac Lv,\qquad L^{\prime}=L\gamma(1-\beta)=L\sqrt{\frac{1-\beta}{1+\beta}}

T = L/v describes a process at constant speed in a specified direction. Decay statistics and the correspondence between path and observational frame must still be supplied.
Chapter reference: Final consolidated manuscript · Chapter 1, Section 3.6, Eqs. (1.44)–(1.45) · ASD-BOOK-ddbade997b45

Theory · DERIVED
ASD-WEB-LVE-018
L=L′1+β1−β=6431.9980.002 m=643999 m=1929111 m≈20323.276089 m≈20323.3 m\begin{aligned}L&=L^{\prime}\sqrt{\frac{1+\beta}{1-\beta}}=643\sqrt{\frac{1.998}{0.002}}\,\mathrm m\\&=643\sqrt{999}\,\mathrm m=1929\sqrt{111}\,\mathrm m\approx20323.276089\,\mathrm m\approx20323.3\,\mathrm m\end{aligned}

Website algebraic expansion of Eqs. (1.45)–(1.46); the final manuscript gives approximately 20323 m. Inputs are illustrative approximations; extra digits check arithmetic, not experimental precision or a survival fit.
Chapter reference: Final consolidated manuscript · Chapter 1, Section 3.6, Eq. (1.46) · ASD-BOOK-ddbade997b45

Theory · DERIVED
ASD-WEB-LVE-019
T′T=1−β1+β\frac{T^{\prime}}T=\sqrt{\frac{1-\beta}{1+\beta}}

An idealized manuscript expression omitting gravity and Earth rotation. Actual clock trajectories, synchronization and a total clock-difference calculation have not been supplied.
Chapter reference: Final consolidated manuscript · Chapter 1, Section 3.6, Eq. (1.47) · ASD-BOOK-ddbade997b45

Theory · FOUNDATION
ASD-WEB-MET-003
ΔνCs=9 192 631 770  Hz\Delta\nu_{\mathrm{Cs}}=9\,192\,631\,770\;\mathrm{Hz}

The unperturbed ground-state hyperfine transition of caesium-133; the defining value is exact, while realizations have uncertainty.
Chapter reference: Final consolidated manuscript · Chapter 1, §I, metrological background; BIPM second definition (external definition) · ASD-BOOK-ddbade997b45

Theory · FOUNDATION
ASD-WEB-TIME-001
t=∫SaSbdSv(S)t=\int_{S_a}^{S_b}\frac{\mathrm dS}{v(S)}

Integrate along increasing arc length at positive speed; a stationary interval needs a reference process. This explicitly expands the manuscript’s variable-speed instruction.
Chapter reference: Final consolidated manuscript · Chapter 2, §I.1; explicit website expansion for variable speed, without a manuscript number · ASD-BOOK-ddbade997b45

Theory · FOUNDATION
ASD-WEB-TIME-003
N=∫uaubf(u) du,Δϑ=2πN,tdisp=Nfref.\begin{aligned}N&=\int_{u_a}^{u_b}f(u)\,\mathrm du,\\ \Delta\vartheta&=2\pi N,\qquad t_{\mathrm{disp}}=\frac{N}{f_{\mathrm{ref}}}.\end{aligned}

u is an agreed reference counting parameter; f(u) is actual oscillator frequency relative to it. f_ref is the fixed calibration frequency used for the display. Accumulated phase is ϑ, distinct from the motion angle φ and long-range stretch potential varphi.
Chapter reference: Final consolidated manuscript · Chapter 2, §I.1, counting and phase interpretation; website operational elaboration, no manuscript number · ASD-BOOK-ddbade997b45

Theory · FOUNDATION
ASD-WEB-TIME-004
τ=Scycc,Scyc=cf\tau=\frac{S_{\mathrm{cyc}}}{c},\qquad S_{\mathrm{cyc}}=\frac cf

For an atomic standard, S_cyc is a phase-length scale, not an electron orbit or an internal photon round trip.
Chapter reference: Final consolidated manuscript · Chapter 2, §I.1, phase-scale discussion and Eq. (2.1) · ASD-BOOK-ddbade997b45

Theory · FOUNDATION
ASD-WEB-TIME-005
L(r)L0=Ω(r),f(r)f0=Ω(r)−1\frac{L(r)}{L_0}=\Omega(r),\qquad\frac{f(r)}{f_0}=\Omega(r)^{-1}

The scale–oscillation coupling adopted in Chapter 2; each clock technology still needs microscopic matching. All quantities use one reference calibration.
Chapter reference: Final consolidated manuscript · Chapter 2, §I.2, Eq. (2.2) · ASD-BOOK-ddbade997b45

Theory · FOUNDATION
ASD-WEB-TIME-006
f(r2)f(r1)−1=Ψ(r2)−Ψ(r1)c2+O ⁣(Ψ2c4)\frac{f(r_2)}{f(r_1)}-1=\frac{\Psi(r_2)-\Psi(r_1)}{c^2}+O\!\left(\frac{\Psi^2}{c^4}\right)

Final Chapter 2 Eq. (2.2a) is an exact ratio in the specified static constitutive model. This registry entry retains its first-order weak-field expansion, requiring |Ψ|/c²≪1 at both positions and a common reference calibration; it is not the full Eq. (2.2a).
Chapter reference: Final consolidated manuscript · First-order weak-field expansion of Chapter 2, Eq. (2.2a), and its following unnumbered discussion · ASD-BOOK-ddbade997b45

Theory · DERIVED
ASD-WEB-DYN-002
mdef=ρ0∫D[1−qρ(x)] dV0m_{\mathrm{def}}=\rho_0\int_{\mathcal D}[1-q_\rho(\mathbf x)]\,\mathrm dV_0

q_ρ is a near-core volume-density ratio and D an effective core domain; the far-field line-density coefficient Φ is a different quantity.
Chapter reference: Final collected manuscript · Chapter 2 §III.1 (2.8) · ASD-BOOK-ddbade997b45

Theory · DERIVED
ASD-WEB-DYN-003
Veff=∫D(1−qρ) dV0=4πReff33,m0=Cmmdef,Cm=1.\begin{aligned}V_{\mathrm{eff}}&=\int_{\mathcal D}(1-q_\rho)\,\mathrm dV_0=\frac{4\pi R_{\mathrm{eff}}^3}{3},\\m_0&=C_m m_{\mathrm{def}},\qquad C_m=1.\end{aligned}

R_eff is the equal-deficit-volume radius; C_m=1 is the mass normalization adopted in this chapter.
Chapter reference: Final collected manuscript · Chapter 2 §III.1 (2.9) and normalization · ASD-BOOK-ddbade997b45

Theory · DERIVED
ASD-WEB-DYN-004
L′=LK(ϕ),K(ϕ)=γ(1−βcos⁡ϕ),β=vc,γ=(1−β2)−1/2L^\prime=LK(\phi),\quad K(\phi)=\gamma(1-\beta\cos\phi),\quad\beta=\frac vc,\quad\gamma=(1-\beta^2)^{-1/2}

Retain Chapter 1’s directional true-length interpretation and mathematical review conditions; χ is the angle to velocity.
Chapter reference: Final collected manuscript · Chapter 2 §III.2 (2.10) · ASD-BOOK-ddbade997b45

Theory · DERIVED
ASD-WEB-DYN-005
K+=γ(1−β),K−=γ(1+β),K+K−=1,K+−1+K−−12=γK_+=\gamma(1-\beta),\quad K_-=\gamma(1+\beta),\quad K_+K_-=1,\quad\frac{K_+^{-1}+K_-^{-1}}2=\gamma

This is an identity for two axial factors; a full inertial angular distribution requires an additional constitutive input.
Chapter reference: Final collected manuscript · Chapter 2 §III.2 (2.10a) · ASD-BOOK-ddbade997b45

Theory · DERIVED
ASD-WEB-DYN-006
ρI(ϕ)=ρ0γ(1+βcos⁡ϕ),14π∫ρI dΩang=ρ0γ\rho_I(\phi)=\rho_0\gamma(1+\beta\cos\phi),\qquad\frac1{4\pi}\int\rho_I\,\mathrm d\Omega_{\mathrm{ang}}=\rho_0\gamma

The lowest angular bias and mean gain are additional inputs; ρ_I is not generally ρ₀/K(χ).
Chapter reference: Final collected manuscript · Chapter 2 §III.2 (2.11) · ASD-BOOK-ddbade997b45

Integral of total momentum

计量与运动学
Theory · DERIVED
ASD-WEB-DYN-007
p=v∫0Reffr2 dr∫02πdα∫0πρI(ϕ)sin⁡ϕ dϕ\mathbf p=\mathbf v\int_0^{R_{\mathrm{eff}}}r^2\,dr\int_0^{2\pi}d\alpha\int_0^\pi\rho_I(\phi)\sin\phi\,d\phi

The integral uses the chapter’s effective core and common translational velocity; microscopic nonuniform structure still needs matching.
Chapter reference: Final collected manuscript · Chapter 2 §III.2 (2.12) · ASD-BOOK-ddbade997b45

Total momentum

计量与运动学
Theory · DERIVED
ASD-WEB-DYN-008
p=4π3ρ0Reff3γv=m0γv,p=m0γv\mathbf p=\frac{4\pi}{3}\rho_0R_{\mathrm{eff}}^3\gamma\mathbf v=m_0\gamma\mathbf v,\qquad p=m_0\gamma v

The odd angular term integrates to zero; the result uses the stated mass normalization.
Chapter reference: Final collected manuscript · Chapter 2 §III.2 (2.13)—(2.14) · ASD-BOOK-ddbade997b45

Theory · DERIVED
ASD-WEB-DYN-012
F=m0[γI+γ3c2vvT]a=m0γ3a∥+m0γa⊥.\begin{aligned}\mathbf F&=m_0\left[\gamma\mathbf I+\frac{\gamma^3}{c^2}\mathbf v\mathbf v^{\mathsf T}\right]\mathbf a\\&=m_0\gamma^3\mathbf a_\parallel+m_0\gamma\mathbf a_\perp.\end{aligned}

Resolve acceleration parallel and perpendicular to velocity; this does not introduce two rest masses.
Chapter reference: Final collected manuscript · Chapter 2 §III.3 (2.18) · ASD-BOOK-ddbade997b45

Low-speed Newtonian limit

计量与运动学
Theory · DERIVED
ASD-WEB-DYN-014
F=m0a+O ⁣(m0∣a∣v2c2)\mathbf F=m_0\mathbf a+O\!\left(\frac{m_0|\mathbf a|v^2}{c^2}\right)

Website low-speed expansion of (2.18). Final equation (2.20) is the inverse force response, registered as ASD-WEB-DYN-025.
Chapter reference: Final collected manuscript · Chapter 2 §III.3 (2.18) website low-speed expansion; not final equation (2.20) · ASD-BOOK-ddbade997b45

Kinetic energy from work

计量与运动学
Theory · DERIVED
ASD-WEB-DYN-017
dEk=v⋅dp,Ek=m0∫0vw dw(1−w2/c2)3/2\mathrm dE_k=\mathbf v\cdot\mathrm d\mathbf p,\qquad E_k=m_0\int_0^v\frac{w\,\mathrm dw}{(1-w^2/c^2)^{3/2}}

Set kinetic energy to zero at v=0; w is the integration speed.
Chapter reference: Final collected manuscript · Chapter 2 §III.4 (2.22) · ASD-BOOK-ddbade997b45

Theory · DERIVED
ASD-WEB-DYN-018
Ek=m0c2(γ−1)=m0v22+3m0v48c2+O ⁣(m0v6c4)E_k=m_0c^2(\gamma-1)=\frac{m_0v^2}{2}+\frac{3m_0v^4}{8c^2}+O\!\left(\frac{m_0v^6}{c^4}\right)

Follows from the momentum–work integral without first specifying rest energy.
Chapter reference: Final collected manuscript · Chapter 2 §III.4 (2.23); low-speed series expanded on the website · ASD-BOOK-ddbade997b45

Free-motion Lagrangian

计量与运动学
Theory · DERIVED
ASD-WEB-DYN-022
L0=−m0c21−v2/c2,p=∂L0∂v,E=p⋅v−L0\mathcal L_0=-m_0c^2\sqrt{1-v^2/c^2},\qquad\mathbf p=\frac{\partial\mathcal L_0}{\partial\mathbf v},\qquad E=\mathbf p\cdot\mathbf v-\mathcal L_0

The integration constant is fixed by L₀(0)=−E₀; 𝓛₀ denotes a Lagrangian, distinct from length L.
Chapter reference: Final collected manuscript · Chapter 2 §III.4 (2.24b) · ASD-BOOK-ddbade997b45

Theory · DERIVED
ASD-WEB-DYN-024
Uext=m0Ψext=12m0c2(Xext−1),Fg=−∇Uext=m0gXU_{\mathrm{ext}}=m_0\Psi_{\mathrm{ext}}=\frac12m_0c^2(X_{\mathrm{ext}}-1),\qquad\mathbf F_g=-\nabla U_{\mathrm{ext}}=m_0\mathbf g_X

An input interface for the later Gravity module; background-field and orbital equations are outside this module.
Chapter reference: Final collected manuscript · Chapter 2 §III.5 (2.24d) · ASD-BOOK-ddbade997b45

Spherical exterior scale law

引力与空间尺度
Theory · EFFECTIVE MODEL
ASD-WEB-GRV-002
Ω=(1−2G0Mc2r)−1/2,Φ=Ω−1\Omega=\left(1-\frac{2G_0M}{c^2r}\right)^{-1/2},\quad\Phi=\Omega^{-1}

r is areal radius with r > 2G₀M/c². The full strong-field function is a constitutive input, not uniquely fixed by weak redshift.
Chapter reference: Final collected manuscript · Chapter 2 §I.2 (2.3)—(2.4) · ASD-BOOK-ddbade997b45

Spatial state and potential

引力与空间尺度
Theory · DERIVED
ASD-WEB-GRV-003
X=Φ2=1−2G0Mc2r,Ψ=c22(X−1)=−G0MrX=\Phi^2=1-\frac{2G_0M}{c^2r},\quad\Psi=\frac{c^2}{2}(X-1)=-\frac{G_0M}{r}

Conditional on the stated inputs and domain. Manuscript numbering is separate from the website ID.
Chapter reference: Final collected manuscript · Chapter 2 §II.2 (2.6) · ASD-BOOK-ddbade997b45

Theory · DERIVED
ASD-WEB-GRV-004
gX=−∇Ψ=−c22∇X=−G0Mr2r^\mathbf g_X=-\nabla\Psi=-\frac{c^2}{2}\nabla X=-\frac{G_0M}{r^2}\hat{\mathbf r}

Conditional on the stated inputs and domain. Manuscript numbering is separate from the website ID.
Chapter reference: Final collected manuscript · Chapter 2 §II.2 (2.7) · ASD-BOOK-ddbade997b45

Theory · EFFECTIVE MODEL
ASD-WEB-GRV-005
∇2Ψ=4πG0ρg,∇2X=8πG0c2ρg,M=c28πG0∮∇X⋅dS\nabla^2\Psi=4\pi G_0\rho_g,\quad\nabla^2X=\frac{8\pi G_0}{c^2}\rho_g,\quad M=\frac{c^2}{8\pi G_0}\oint\nabla X\cdot d\mathbf S

Weak static source equations in reference coordinates, not a closed dynamical strong-field master equation.
Chapter reference: Final collected manuscript · Chapter 2 §II.2 (2.7a)—(2.7b) · ASD-BOOK-ddbade997b45

Spatial length law

引力与空间尺度
Theory · EFFECTIVE MODEL
ASD-WEB-GRV-006
dℓs2=X−1dr2+r2dφ2d\ell_s^2=X^{-1}dr^2+r^2d\varphi^2

The radial law is constitutive; areal radius already determines the tangential arc without an extra Ω.
Chapter reference: Final collected manuscript · Chapter 2 §IV.1 (2.25) · ASD-BOOK-ddbade997b45

Local cycles and speed

引力与空间尺度
Theory · DERIVED
ASD-WEB-GRV-007
dN=f0Φ dt,v∗=f0dℓsdN=Φ−1dℓsdtdN=f_0\Phi\,dt,\quad v_* =\frac{f_0d\ell_s}{dN}=\Phi^{-1}\frac{d\ell_s}{dt}

Conditional on the stated inputs and domain. Manuscript numbering is separate from the website ID.
Chapter reference: Final collected manuscript · Chapter 2 §IV.1, equation (2.25) and the following unnumbered relation · ASD-BOOK-ddbade997b45

Theory · DERIVED
ASD-WEB-GRV-008
dℓopt2=Φ−2dℓs2=X−2dr2+X−1r2dφ2,dt=dℓoptcd\ell_{\mathrm{opt}}^2=\Phi^{-2}d\ell_s^2=X^{-2}dr^2+X^{-1}r^2d\varphi^2,\quad dt=\frac{d\ell_{\mathrm{opt}}}{c}

Conditional on the stated inputs and domain. Manuscript numbering is separate from the website ID.
Chapter reference: Final collected manuscript · Chapter 2 §IV.1 (2.26) · ASD-BOOK-ddbade997b45

Theory · DERIVED
ASD-WEB-GRV-009
f(r2)f(r1)=Φ(r2)Φ(r1)=1−2G0M/(c2r2)1−2G0M/(c2r1)\frac{f(r_2)}{f(r_1)}=\frac{\Phi(r_2)}{\Phi(r_1)}=\sqrt{\frac{1-2G_0M/(c^2r_2)}{1-2G_0M/(c^2r_1)}}

Conditional on the stated inputs and domain. Manuscript numbering is separate from the website ID.
Chapter reference: Final collected manuscript · Chapter 2 §I.2 (2.2a) in the spherically symmetric exterior special case · ASD-BOOK-ddbade997b45

Stationary optical path

引力与空间尺度
Theory · DERIVED
ASD-WEB-GRV-010
T[Γ]=1c∫Γdℓopt,δT=0T[\Gamma]=\frac1c\int_\Gamma d\ell_{\mathrm{opt}},\quad\delta T=0

Conditional on the stated inputs and domain. Manuscript numbering is separate from the website ID.
Chapter reference: Final collected manuscript · Chapter 2 §IV.4 (2.43) · ASD-BOOK-ddbade997b45

Isotropic radial coordinate

引力与空间尺度
Theory · DERIVED
ASD-WEB-GRV-011
r=ρ(1+q)2,q=ℓg2ρ,X=(1−q)2(1+q)2,drdρ=1−q2r=\rho(1+q)^2,\quad q=\frac{\ell_g}{2\rho},\quad X=\frac{(1-q)^2}{(1+q)^2},\quad\frac{dr}{d\rho}=1-q^2

Conditional on the stated inputs and domain. Manuscript numbering is separate from the website ID.
Chapter reference: Final collected manuscript · Chapter 2 §IV.4 (2.44) and adjacent substitutions · ASD-BOOK-ddbade997b45

Isotropic optical coefficient

引力与空间尺度
Theory · DERIVED
ASD-WEB-GRV-012
dℓopt=nisods0,niso=(1+q)31−q=1+2ℓgρ+O(ℓg2/ρ2)d\ell_{\mathrm{opt}}=n_{\mathrm{iso}}ds_0,\quad n_{\mathrm{iso}}=\frac{(1+q)^3}{1-q}=1+\frac{2\ell_g}{\rho}+O(\ell_g^2/\rho^2)

Conditional on the stated inputs and domain. Manuscript numbering is separate from the website ID.
Chapter reference: Final collected manuscript · Chapter 2 §IV.4 (2.45)—(2.46) · ASD-BOOK-ddbade997b45

Fermat ray equation

引力与空间尺度
Theory · DERIVED
ASD-WEB-GRV-013
dds0(nisoe)=∇niso,e=dxds0\frac{d}{ds_0}(n_{\mathrm{iso}}\mathbf e)=\nabla n_{\mathrm{iso}},\quad\mathbf e=\frac{d\mathbf x}{ds_0}

Conditional on the stated inputs and domain. Manuscript numbering is separate from the website ID.
Chapter reference: Final collected manuscript · Chapter 2 §IV.5 (2.47) · ASD-BOOK-ddbade997b45

Theory · DERIVED
ASD-WEB-GRV-014
dθdx≃∂niso∂y∣y=b=−2ℓgb(x2+b2)3/2\frac{d\theta}{dx}\simeq\left.\frac{\partial n_{\mathrm{iso}}}{\partial y}\right|_{y=b}=-\frac{2\ell_gb}{(x^2+b^2)^{3/2}}

Conditional on the stated inputs and domain. Manuscript numbering is separate from the website ID.
Chapter reference: Final collected manuscript · Chapter 2 §IV.5 (2.48); retain linear small-deflection accuracy · ASD-BOOK-ddbade997b45

Deflection integral

引力与空间尺度
Theory · DERIVED
ASD-WEB-GRV-015
αdef=2ℓg∫−∞+∞b dx(x2+b2)3/2\alpha_{\mathrm{def}}=2\ell_g\int_{-\infty}^{+\infty}\frac{b\,dx}{(x^2+b^2)^{3/2}}

Conditional on the stated inputs and domain. Manuscript numbering is separate from the website ID.
Chapter reference: Final collected manuscript · Chapter 2 §IV.5 (2.49) · ASD-BOOK-ddbade997b45

Weak-field light deflection

引力与空间尺度
Theory · DERIVED
ASD-WEB-GRV-016
αdef=4G0Mc2b+O(ℓg2/b2)\alpha_{\mathrm{def}}=\frac{4G_0M}{c^2b}+O(\ell_g^2/b^2)

Conditional on the stated inputs and domain. Manuscript numbering is separate from the website ID.
Chapter reference: Final collected manuscript · Chapter 2 §IV.5 (2.50); higher-order remainder stated explicitly on the website · ASD-BOOK-ddbade997b45

Fixed-endpoint travel time

引力与空间尺度
Theory · DERIVED
ASD-WEB-GRV-017
T[Γ]=1c∫Γniso ds0T[\Gamma]=\frac1c\int_\Gamma n_{\mathrm{iso}}\,ds_0

Conditional on the stated inputs and domain. Manuscript numbering is separate from the website ID.
Chapter reference: Final collected manuscript · Chapter 2 §IV.6 (2.51) · ASD-BOOK-ddbade997b45

Excess optical-length integral

引力与空间尺度
Theory · DERIVED
ASD-WEB-GRV-019
ΔT≃2G0Mc3∫−x1x2dxx2+b2\Delta T\simeq\frac{2G_0M}{c^3}\int_{-x_1}^{x_2}\frac{dx}{\sqrt{x^2+b^2}}

Conditional on the stated inputs and domain. Manuscript numbering is separate from the website ID.
Chapter reference: Final collected manuscript · Chapter 2 §IV.6 (2.53)—(2.54); first-order weak-field integral · ASD-BOOK-ddbade997b45

Theory · DERIVED
ASD-WEB-GRV-020
ΔT=2G0Mc3ln⁡ ⁣[(ρ1+x1)(ρ2+x2)b2]\Delta T=\frac{2G_0M}{c^3}\ln\!\left[\frac{(\rho_1+x_1)(\rho_2+x_2)}{b^2}\right]

Equality is at first-order accuracy, with endpoints across closest approach.
Chapter reference: Final collected manuscript · Chapter 2 §IV.6 (2.55) · ASD-BOOK-ddbade997b45

Theory · DERIVED
ASD-WEB-GRV-021
ΔT=2G0Mc3ln⁡ ⁣(ρ1+ρ2+R12ρ1+ρ2−R12)\Delta T=\frac{2G_0M}{c^3}\ln\!\left(\frac{\rho_1+\rho_2+R_{12}}{\rho_1+\rho_2-R_{12}}\right)

Equality is at first-order accuracy; all endpoint distances use one coordinate convention.
Chapter reference: Final collected manuscript · Chapter 2 §IV.6 (2.56)—(2.57) · ASD-BOOK-ddbade997b45

Distant-endpoint approximation

引力与空间尺度
Theory · DERIVED
ASD-WEB-GRV-022
ΔT≃2G0Mc3ln⁡ ⁣(4ρ1ρ2b2)\Delta T\simeq\frac{2G_0M}{c^3}\ln\!\left(\frac{4\rho_1\rho_2}{b^2}\right)

Distant-endpoint asymptote of the first-order weak-field result, ρ₁,ρ₂≫b. Final (2.57) is the exact identity between endpoint logarithm arguments.
Chapter reference: Final collected manuscript · Chapter 2 §IV.6 (2.58) and the following distant-endpoint asymptote; not final equation (2.57) · ASD-BOOK-ddbade997b45

Static round-trip delay

引力与空间尺度
Theory · DERIVED
ASD-WEB-GRV-023
ΔTround=2ΔT=4G0Mc3ln⁡ ⁣(ρ1+ρ2+R12ρ1+ρ2−R12)\Delta T_{\mathrm{round}}=2\Delta T=\frac{4G_0M}{c^3}\ln\!\left(\frac{\rho_1+\rho_2+R_{12}}{\rho_1+\rho_2-R_{12}}\right)

Full endpoint formula at first weak-field order for fixed endpoints and equal outgoing/returning geometry; no additional distant-endpoint assumption.
Chapter reference: Final collected manuscript · Chapter 2 §IV.6 (2.58) · ASD-BOOK-ddbade997b45

Theory · EFFECTIVE MODEL
ASD-WEB-GRV-024
L=ΦL0(v∗)=−m0c2X−r˙2c2X−r2φ˙2c2\mathcal L=\Phi\mathcal L_0(v_*)=-m_0c^2\sqrt{X-\frac{\dot r^2}{c^2X}-\frac{r^2\dot\varphi^2}{c^2}}

In addition to the free action, adopt universal static energy conversion L dt = L₀(v*) Φ dt.
Chapter reference: Final collected manuscript · Chapter 2 §IV.2 (2.27) · ASD-BOOK-ddbade997b45

Theory · DERIVED
ASD-WEB-GRV-025
Lm0=−c2+v22+U+1c2(v48+Uv22+Uvr2+U22)+O(c−4)\frac{\mathcal L}{m_0}=-c^2+\frac{v^2}2+U+\frac1{c^2}\left(\frac{v^4}8+\frac{Uv^2}2+Uv_r^2+\frac{U^2}2\right)+O(c^{-4})

Website 1PN expansion of final (2.27) for U/c² and v²/c²≪1; final (2.28) instead defines canonical momenta.
Chapter reference: Final collected manuscript · Chapter 2 §IV.2 (2.27) website 1PN expansion; not final equation (2.28) · ASD-BOOK-ddbade997b45

Orbital velocity domain

引力与空间尺度
Theory · DERIVED
ASD-WEB-GRV-026
D=X−r˙2c2X−r2φ˙2c2>0D=X-\frac{\dot r^2}{c^2X}-\frac{r^2\dot\varphi^2}{c^2}>0

Conditional on the stated inputs and domain. Manuscript numbering is separate from the website ID.
Chapter reference: Final collected manuscript · Chapter 2 §IV.2 (2.29) · ASD-BOOK-ddbade997b45

Theory · DERIVED
ASD-WEB-GRV-027
j=1m0∂L∂φ˙=r2φ˙Dj=\frac1{m_0}\frac{\partial\mathcal L}{\partial\dot\varphi}=\frac{r^2\dot\varphi}{\sqrt D}

Conditional on the stated inputs and domain. Manuscript numbering is separate from the website ID.
Chapter reference: Final collected manuscript · Chapter 2 §IV.2 (2.28), (2.30) · ASD-BOOK-ddbade997b45

Conserved reference energy

引力与空间尺度
Theory · DERIVED
ASD-WEB-GRV-028
pr=m0r˙XD,pφ=m0r2φ˙D,E=E∞m0c2=prr˙+pφφ˙−Lm0c2=XD.\begin{aligned}p_r&=\frac{m_0\dot r}{X\sqrt D},\qquad p_\varphi=\frac{m_0r^2\dot\varphi}{\sqrt D},\\\mathcal E&=\frac{E_\infty}{m_0c^2}=\frac{p_r\dot r+p_\varphi\dot\varphi-\mathcal L}{m_0c^2}=\frac X{\sqrt D}.\end{aligned}

Differentiate final (2.27) with respect to both generalized velocities to obtain (2.28), then apply the Legendre transform; D>0.
Chapter reference: Final collected manuscript · Chapter 2 §IV.2 (2.28)—(2.31) · ASD-BOOK-ddbade997b45

Theory · DERIVED
ASD-WEB-GRV-029
u=1r,u′=dudφ,φ˙=jD u2,r˙=−jD u′u=\frac1r,\quad u^\prime=\frac{du}{d\varphi},\quad\dot\varphi=j\sqrt D\,u^2,\quad\dot r=-j\sqrt D\,u^\prime

Conditional on the stated inputs and domain. Manuscript numbering is separate from the website ID.
Chapter reference: Final collected manuscript · Chapter 2 §IV.3 (2.32)—(2.33) · ASD-BOOK-ddbade997b45

First integral of the orbit

引力与空间尺度
Theory · DERIVED
ASD-WEB-GRV-030
(u′)2=c2j2(E2−X)−Xu2(u^\prime)^2=\frac{c^2}{j^2}(\mathcal E^2-X)-Xu^2

Conditional on the stated inputs and domain. Manuscript numbering is separate from the website ID.
Chapter reference: Final collected manuscript · Chapter 2 §IV.3 (2.34) · ASD-BOOK-ddbade997b45

Shape equation for general X

引力与空间尺度
Theory · DERIVED
ASD-WEB-GRV-031
u′′=−c22j2dXdu−Xu−12u2dXduu^{\prime\prime}=-\frac{c^2}{2j^2}\frac{dX}{du}-Xu-\frac12u^2\frac{dX}{du}

Conditional on the stated inputs and domain. Manuscript numbering is separate from the website ID.
Chapter reference: Final collected manuscript · Chapter 2 §IV.3 (2.35) · ASD-BOOK-ddbade997b45

Spherical orbital equation

引力与空间尺度
Theory · DERIVED
ASD-WEB-GRV-032
u′′+u=μj2+3ℓgu2,μ=G0M,ℓg=μ/c2u^{\prime\prime}+u=\frac{\mu}{j^2}+3\ell_gu^2,\quad\mu=G_0M,\quad\ell_g=\mu/c^2

Conditional on the stated inputs and domain. Manuscript numbering is separate from the website ID.
Chapter reference: Final collected manuscript · Chapter 2 §IV.3 (2.36) · ASD-BOOK-ddbade997b45

Zeroth-order ellipse

引力与空间尺度
Theory · DERIVED
ASD-WEB-GRV-033
u0=1+ecos⁡φporb,porb=j2μ=a(1−e2)u_0=\frac{1+e\cos\varphi}{p_{\mathrm{orb}}},\quad p_{\mathrm{orb}}=\frac{j^2}{\mu}=a(1-e^2)

Conditional on the stated inputs and domain. Manuscript numbering is separate from the website ID.
Chapter reference: Final collected manuscript · Chapter 2 §IV.3 (2.37) · ASD-BOOK-ddbade997b45

Theory · DERIVED
ASD-WEB-GRV-034
δu′′+δu=3ℓgporb2(1+2ecos⁡φ+e2cos⁡2φ)\delta u^{\prime\prime}+\delta u=\frac{3\ell_g}{p_{\mathrm{orb}}^2}(1+2e\cos\varphi+e^2\cos^2\varphi)

Conditional on the stated inputs and domain. Manuscript numbering is separate from the website ID.
Chapter reference: Final collected manuscript · Chapter 2 §IV.3 (2.38) · ASD-BOOK-ddbade997b45

Theory · DERIVED
ASD-WEB-GRV-035
δusec=3ℓgeporb2φsin⁡φ,u≃1+ecos⁡[(1−κ)φ]porb,κ=3ℓgporb\delta u_{\mathrm{sec}}=\frac{3\ell_ge}{p_{\mathrm{orb}}^2}\varphi\sin\varphi,\quad u\simeq\frac{1+e\cos[(1-\kappa)\varphi]}{p_{\mathrm{orb}}},\quad\kappa=\frac{3\ell_g}{p_{\mathrm{orb}}}

Conditional on the stated inputs and domain. Manuscript numbering is separate from the website ID.
Chapter reference: Final collected manuscript · Chapter 2 §IV.3 (2.39)—(2.41); rotated ellipse matched at first order · ASD-BOOK-ddbade997b45

Periapsis advance per orbit

引力与空间尺度
Theory · DERIVED
ASD-WEB-GRV-036
Δϖ=6πG0Ma(1−e2)c2+O ⁣(ℓg2porb2)\Delta\varpi=\frac{6\pi G_0M}{a(1-e^2)c^2}+O\!\left(\frac{\ell_g^2}{p_{\mathrm{orb}}^2}\right)

κ is known only at first order; re-expanding 2π/(1−κ) is not a complete second-order derivation.
Chapter reference: Final collected manuscript · Chapter 2 §IV.3 (2.42) · ASD-BOOK-ddbade997b45

Near-boundary lengths

引力与空间尺度
Theory · EFFECTIVE MODEL
ASD-WEB-GRV-038
ℓs(rh,r)=r(r−rh)+rhln⁡r+r−rhrh,ℓopt(r1,r2)=(r2−r1)+rhln⁡r2−rhr1−rh.\begin{aligned}\ell_s(r_h,r)&=\sqrt{r(r-r_h)}+r_h\ln\frac{\sqrt r+\sqrt{r-r_h}}{\sqrt{r_h}},\\\ell_{\mathrm{opt}}(r_1,r_2)&=(r_2-r_1)+r_h\ln\frac{r_2-r_h}{r_1-r_h}.\end{aligned}

Exact analytic integrals for the adopted X=1−r_h/r, with r>r_h and r₂≥r₁>r_h. Actual length remains finite while the reference optical path diverges logarithmically as its inner endpoint approaches the boundary.
Chapter reference: Final collected manuscript · Chapter 2 §V.1 (2.60)—(2.62) · ASD-BOOK-ddbade997b45

Theory · EFFECTIVE MODEL
ASD-WEB-GRV-039
κX=c22lim⁡r→rh+X′(r)=G0Mrh2=c44G0M\kappa_X=\frac{c^2}{2}\lim_{r\to r_h^+}X^\prime(r)=\frac{G_0M}{r_h^2}=\frac{c^4}{4G_0M}

Conditional on the stated inputs and domain. Manuscript numbering is separate from the website ID.
Chapter reference: Final collected manuscript · Chapter 2 §V.2 (2.63) · ASD-BOOK-ddbade997b45

Local hovering force

引力与空间尺度
Theory · EFFECTIVE MODEL
ASD-WEB-GRV-040
dW∞=m0G0Mr2X dr,Fhold,locm0=G0Mr2XdW_\infty=\frac{m_0G_0M}{r^2\sqrt X}\,dr,\quad\frac{F_{\mathrm{hold,loc}}}{m_0}=\frac{G_0M}{r^2\sqrt X}

Conditional on the stated inputs and domain. Manuscript numbering is separate from the website ID.
Chapter reference: Final collected manuscript · Chapter 2 §V.2 (2.64)—(2.65) · ASD-BOOK-ddbade997b45

Strong-field optical functions

引力与空间尺度
Theory · EFFECTIVE MODEL
ASD-WEB-GRV-041
dℓopt2=A2dr2+B2dφ2,A=X−1,B=rX−1/2d\ell_{\mathrm{opt}}^2=\mathcal A^2dr^2+\mathcal B^2d\varphi^2,\quad\mathcal A=X^{-1},\quad\mathcal B=rX^{-1/2}

Conditional on the stated inputs and domain. Manuscript numbering is separate from the website ID.
Chapter reference: Final collected manuscript · Chapter 2 §V.3 (2.66) · ASD-BOOK-ddbade997b45

Theory · EFFECTIVE MODEL
ASD-WEB-GRV-042
b=B2dφdsopt=Bsin⁡ψoptb=\mathcal B^2\frac{d\varphi}{ds_{\mathrm{opt}}}=\mathcal B\sin\psi_{\mathrm{opt}}

Conditional on the stated inputs and domain. Manuscript numbering is separate from the website ID.
Chapter reference: Final collected manuscript · Chapter 2 §V.3 (2.67) · ASD-BOOK-ddbade997b45

Optical radial equation

引力与空间尺度
Theory · EFFECTIVE MODEL
ASD-WEB-GRV-043
A2(drdsopt)2=1−b2B2\mathcal A^2\left(\frac{dr}{ds_{\mathrm{opt}}}\right)^2=1-\frac{b^2}{\mathcal B^2}

Conditional on the stated inputs and domain. Manuscript numbering is separate from the website ID.
Chapter reference: Final collected manuscript · Chapter 2 §V.3 (2.68) · ASD-BOOK-ddbade997b45

Circular critical-ray condition

引力与空间尺度
Theory · EFFECTIVE MODEL
ASD-WEB-GRV-044
B=r1−2ℓg/r,B′=1−3ℓg/r(1−2ℓg/r)3/2=0\mathcal B=\frac{r}{\sqrt{1-2\ell_g/r}},\quad\mathcal B^\prime=\frac{1-3\ell_g/r}{(1-2\ell_g/r)^{3/2}}=0

Conditional on the stated inputs and domain. Manuscript numbering is separate from the website ID.
Chapter reference: Final collected manuscript · Chapter 2 §V.3 (2.69) · ASD-BOOK-ddbade997b45

Static photon-sphere radius

引力与空间尺度
Theory · EFFECTIVE MODEL
ASD-WEB-GRV-045
rph=3ℓg=3G0Mc2=32rhr_{\mathrm{ph}}=3\ell_g=\frac{3G_0M}{c^2}=\frac32r_h

Conditional on the stated inputs and domain. Manuscript numbering is separate from the website ID.
Chapter reference: Final collected manuscript · Chapter 2 §V.3 (2.70) · ASD-BOOK-ddbade997b45

Theory · EFFECTIVE MODEL
ASD-WEB-GRV-046
bc=B(rph)=33G0Mc2=332rhb_c=\mathcal B(r_{\mathrm{ph}})=3\sqrt3\frac{G_0M}{c^2}=\frac{3\sqrt3}{2}r_h

Capture requires an absorbing boundary; b_c/Dobs is not a complete observed emission-ring model.
Chapter reference: Final collected manuscript · Chapter 2 §V.3 (2.71) · ASD-BOOK-ddbade997b45

Coulomb law in true space

电磁相互作用
Theory · FOUNDATION
ASD-WEB-EM-005
FT=q1q24πε0rT2r^,F0=q1q24πε0r2r^\mathbf F_T=\frac{q_1q_2}{4\pi\varepsilon_0r_T^2}\hat{\mathbf r},\qquad\mathbf F_0=\frac{q_1q_2}{4\pi\varepsilon_0r^2}\hat{\mathbf r}

Using true length as the interaction distance is a constitutive input, not a deduction from metrology alone.
Chapter reference: Final collected manuscript · Chapter 3, equations (3.3) · ASD-BOOK-ddbade997b45

Vector magnetic correction

电磁相互作用
Theory · DERIVED
ASD-WEB-EM-006
ΔFm≡FT−F0=−β2F0\Delta\mathbf F_m\equiv\mathbf F_T-\mathbf F_0=-\beta^2\mathbf F_0

The correction points inward for like charges and outward for unlike charges; either interaction decreases in magnitude.
Chapter reference: Final collected manuscript · Chapter 3, equations (3.5) · ASD-BOOK-ddbade997b45

Theory · DERIVED
ASD-WEB-EM-007
ε0μ0=c−2,ΔFm=−μ0q1q2v24πr2r^\varepsilon_0\mu_0=c^{-2},\qquad\Delta\mathbf F_m=-\frac{\mu_0q_1q_2v^2}{4\pi r^2}\hat{\mathbf r}

The product relation is exact; the separate values of ε₀ and μ₀ are experimentally determined in the present SI.
Chapter reference: Final collected manuscript · Chapter 3, equations (3.6) · ASD-BOOK-ddbade997b45

Theory · DERIVED
ASD-WEB-EM-010
FESR=γF0,FmSR,red=γβ2F0,F⊥SR=F0γF_E^{\rm SR}=\gamma F_0,\quad F_m^{\rm SR,red}=\gamma\beta^2F_0,\quad F_\perp^{\rm SR}=\frac{F_0}{\gamma}

Standard-theory comparison only; like-charge magnitudes with laboratory definitions. Not an ASD derivation input.
Chapter reference: Final collected manuscript · Chapter 3, equations (3.9) · ASD-BOOK-ddbade997b45

Theory · DERIVED
ASD-WEB-EM-011
FTF0=1−β2,F⊥SRF0=1−β22+O(β4),γβ2=β2+O(β4)\frac{F_T}{F_0}=1-\beta^2,\quad\frac{F_\perp^{\rm SR}}{F_0}=1-\frac{\beta^2}{2}+O(\beta^4),\quad\gamma\beta^2=\beta^2+O(\beta^4)

The same leading magnetic coefficient does not imply the same total force; observables must first be aligned.
Chapter reference: Final collected manuscript · Chapter 3, equations (3.10), (3.9) · ASD-BOOK-ddbade997b45

Theory · CANDIDATE MODEL
ASD-WEB-EM-012
Ecand(R,ϕ)=q4πε0R21−β2(1−βcos⁡ϕ)2R^\mathbf E_{\rm cand}(R,\phi)=\frac{q}{4\pi\varepsilon_0R^2}\frac{1-\beta^2}{(1-\beta\cos\phi)^2}\hat{\mathbf R}

Assumes a reference-radial field and ordinary spherical area elements; not a closed moving-field solution. Web motion angle φ corresponds to the manuscript’s ϕ in this section, without changing its meaning.
Chapter reference: Final collected manuscript · Chapter 3, equations (3.17) · ASD-BOOK-ddbade997b45

Theory · CANDIDATE MODEL
ASD-WEB-EM-013
Φcand=q4πε0RK(ϕ),−∇Φcand=q4πε0R2(R^K+K′K2ϕ^)\Phi_{\rm cand}=\frac{q}{4\pi\varepsilon_0RK(\phi)},\quad-\nabla\Phi_{\rm cand}=\frac{q}{4\pi\varepsilon_0R^2}\left(\frac{\hat{\mathbf R}}K+\frac{K^{\prime}}{K^2}\hat{\boldsymbol\phi}\right)

Uses the ordinary reference spherical gradient; demonstrates nonuniqueness of a field reconstruction from the length factor. Web motion angle φ corresponds to the manuscript’s ϕ in this section, without changing its meaning.
Chapter reference: Final collected manuscript · Chapter 3, equations (3.18) · ASD-BOOK-ddbade997b45

Theory · DERIVED
ASD-WEB-EM-014
2π∫−111−β2(1−βz)2 dz=4π,ε0∮Ecand⋅dS=q2\pi\int_{-1}^{1}\frac{1-\beta^2}{(1-\beta z)^2}\,dz=4\pi,\qquad\varepsilon_0\oint\mathbf E_{\rm cand}\cdot d\mathbf S=q

Total flux checks charge normalization, not local source closure or detector response.
Chapter reference: Final collected manuscript · Chapter 3, equations (3.19) · ASD-BOOK-ddbade997b45

Theory · FOUNDATION
ASD-WEB-EM-015
Φτ=q4πε0r,Estatic=−∇Φτ=q4πε0r2r^\Phi_\tau=\frac{q}{4\pi\varepsilon_0r},\qquad\mathbf E_{\rm static}=-\nabla\Phi_\tau=\frac{q}{4\pi\varepsilon_0r^2}\hat{\mathbf r}

Exterior, static homogeneous baseline; electromagnetic normalization is separately specified.
Chapter reference: Final collected manuscript · Chapter 3, equations (3.11) · ASD-BOOK-ddbade997b45

Theory · EFFECTIVE MODEL
ASD-WEB-EM-016
E=−∇Φτ−∂tAτ,B=∇×Aτ\mathbf E=-\nabla\Phi_\tau-\partial_t\mathbf A_\tau,\qquad\mathbf B=\nabla\times\mathbf A_\tau

Effective continuum definitions introduced after the LVE magnetic-force derivation.
Chapter reference: Final collected manuscript · Chapter 3, equations (3.12) · ASD-BOOK-ddbade997b45

Theory · DERIVED
ASD-WEB-EM-017
Aτ↦Aτ+∇χ,Φτ↦Φτ−∂tχ\mathbf A_\tau\mapsto\mathbf A_\tau+\nabla\chi,\qquad\Phi_\tau\mapsto\Phi_\tau-\partial_t\chi

For smooth functions and compatible boundaries E and B are unchanged; arbitrary-gauge A cannot be uniquely equated to physical displacement.
Chapter reference: Final collected manuscript · Chapter 3, equations (3.13) · ASD-BOOK-ddbade997b45

Theory · EFFECTIVE MODEL
ASD-WEB-EM-018
Aτ≃Φstaticc2u=μ0q4πru\mathbf A_\tau\simeq\frac{\Phi_{\rm static}}{c^2}\mathbf u=\frac{\mu_0q}{4\pi r}\mathbf u

Leading slow-source, near-zone quasistatic representation; u is the source velocity.
Chapter reference: Final collected manuscript · Chapter 3, equations (3.15) · ASD-BOOK-ddbade997b45

Theory · EFFECTIVE MODEL
ASD-WEB-EM-019
B≃1c2∇Φstatic×u=1c2u×Estatic\mathbf B\simeq\frac1{c^2}\nabla\Phi_{\rm static}\times\mathbf u=\frac1{c^2}\mathbf u\times\mathbf E_{\rm static}

u is constant under spatial differentiation; this is not a general all-speed moving-charge field.
Chapter reference: Final collected manuscript · Chapter 3, equations (3.16) · ASD-BOOK-ddbade997b45

Theory · DERIVED
ASD-WEB-EM-020
L0=p⋅v−Ef=−m0c2γ,∂L0∂vi=m0γviL_0=\mathbf p\cdot\mathbf v-E_f=-\frac{m_0c^2}{\gamma},\quad\frac{\partial L_0}{\partial v_i}=m_0\gamma v_i

Uses the same physical variables and dEf=v·dp as Chapter 2, rather than assuming a relativistic action.
Chapter reference: Final collected manuscript · Chapter 3, equations (3.20a), (3.20c) · ASD-BOOK-ddbade997b45

Theory · DERIVED
ASD-WEB-EM-022
p˙i=q[−∂iΦτ−∂tAi+vj(∂iAj−∂jAi)],p˙=q(E+v×B)\dot p_i=q[-\partial_i\Phi_\tau-\partial_t A_i+v_j(\partial_i A_j-\partial_j A_i)],\quad\dot{\mathbf p}=q(\mathbf E+\mathbf v\times\mathbf B)

Follows from the chosen minimal coupling; it is not used to prove the origin of the LVE magnetic correction.
Chapter reference: Final collected manuscript · Chapter 3, equations (3.21), (3.23) · ASD-BOOK-ddbade997b45

Theory · EFFECTIVE MODEL
ASD-WEB-EM-023
Fm≃qc2[u(v⋅Estatic)−Estatic(v⋅u)]\mathbf F_m\simeq\frac q{c^2}[\mathbf u(\mathbf v\cdot\mathbf E_{\rm static})-\mathbf E_{\rm static}(\mathbf v\cdot\mathbf u)]

u is source velocity and v probe velocity; the common transverse configuration checks only the O(β²) magnetic correspondence.
Chapter reference: Final collected manuscript · Chapter 3, equations (3.24) · ASD-BOOK-ddbade997b45

Theory · DERIVED
ASD-WEB-EM-024
dEfdt=qv⋅E,v⋅(v×B)=0\frac{dE_f}{dt}=q\mathbf v\cdot\mathbf E,\qquad\mathbf v\cdot(\mathbf v\times\mathbf B)=0

Energy exchange within the same effective coupling; radiation and external sources are accounted for separately.
Chapter reference: Final collected manuscript · Chapter 3, equations (3.25) · ASD-BOOK-ddbade997b45

Theory · EFFECTIVE MODEL
ASD-WEB-EM-025
ρe=∑aρa,J=∑aρaua\rho_e=\sum_a\rho_a,\qquad\mathbf J=\sum_a\rho_a\mathbf u_a

Electrical neutrality does not imply zero current; both signs and cross-interactions must be retained.
Chapter reference: Final collected manuscript · Chapter 3, equations (3.26) · ASD-BOOK-ddbade997b45

Theory · EFFECTIVE MODEL
ASD-WEB-EM-026
Aτ(r)=μ04π∫J(r′)∣r−r′∣ d3r′,∇⋅Aτ=0\mathbf A_\tau(\mathbf r)=\frac{\mu_0}{4\pi}\int\frac{\mathbf J(\mathbf r')}{|\mathbf r-\mathbf r'|}\,d^3r',\quad\nabla\cdot\mathbf A_\tau=0

Localized steady divergence-free current and decaying boundaries; a linear static effective kernel.
Chapter reference: Final collected manuscript · Chapter 3, equations (3.27) · ASD-BOOK-ddbade997b45

Theory · EFFECTIVE MODEL
ASD-WEB-EM-027
B(r)=μ04π∫J(r′)×RR3 d3r′,R=r−r′\mathbf B(\mathbf r)=\frac{\mu_0}{4\pi}\int\frac{\mathbf J(\mathbf r')\times\mathbf R}{R^3}\,d^3r',\quad\mathbf R=\mathbf r-\mathbf r'

Obtained by taking the curl of the effective current potential; the microscopic wire force does not follow from the common-velocity pair law alone.
Chapter reference: Final collected manuscript · Chapter 3, equations (3.28), (3.29) · ASD-BOOK-ddbade997b45

Theory · EFFECTIVE MODEL
ASD-WEB-EM-028
B1(d)=μ0I12πd,FL=μ0I1I22πdB_1(d)=\frac{\mu_0I_1}{2\pi d},\qquad\frac FL=\frac{\mu_0 I_1I_2}{2\pi d}

Long straight steady macroscopic wires with negligible end effects; parallel currents attract, antiparallel currents repel.
Chapter reference: Final collected manuscript · Chapter 3, equations (3.30), (3.31) · ASD-BOOK-ddbade997b45

Theory · EFFECTIVE MODEL
ASD-WEB-EM-030
∂tρe+∇⋅J=0,∇⋅E=ρe/ε0 ⇒ ∇⋅(J+ε0∂tE)=0\partial_t\rho_e+\nabla\cdot\mathbf J=0,\quad\nabla\cdot\mathbf E=\rho_e/\varepsilon_0\ \Rightarrow\ \nabla\cdot(\mathbf J+\varepsilon_0\partial_t\mathbf E)=0

Constant ε₀; the divergence constraint permits additional solenoidal terms and does not uniquely determine the field equations.
Chapter reference: Final collected manuscript · Chapter 3, equations (3.37), (3.38) · ASD-BOOK-ddbade997b45

Theory · EFFECTIVE MODEL
ASD-WEB-EM-031
LT(2)=ρs2∣∂tuT∣2−KT2∣∇×uT∣2,∇⋅uT=0\mathcal L_T^{(2)}=\frac{\rho_s}{2}|\partial_t\mathbf u_T|^2-\frac{K_T}{2}|\nabla\times\mathbf u_T|^2,\quad\nabla\cdot\mathbf u_T=0

Homogeneous long-wavelength weak-perturbation sector; positive coefficients and compatible boundaries are inputs.
Chapter reference: Final collected manuscript · Chapter 3, equations (3.44) · ASD-BOOK-ddbade997b45

Theory · DERIVED
ASD-WEB-EM-032
ρs∂t2uT+KT∇×(∇×uT)=0,∂t2uT−KTρs∇2uT=0\rho_s\partial_t^2\mathbf u_T+K_T\nabla\times(\nabla\times\mathbf u_T)=0,\quad\partial_t^2\mathbf u_T-\frac{K_T}{\rho_s}\nabla^2\mathbf u_T=0

From transverse variation and the curl–curl identity with constant coefficients.
Chapter reference: Final collected manuscript · Chapter 3, equations (3.45) · ASD-BOOK-ddbade997b45

Theory · EFFECTIVE MODEL
ASD-WEB-EM-033
AT=NTuT,ΩT=∇×uT,B=NTΩT,ET=−NT∂tuT\mathbf A_T=N_T\mathbf u_T,\quad\boldsymbol\Omega_T=\nabla\times\mathbf u_T,\quad\mathbf B=N_T\boldsymbol\Omega_T,\quad\mathbf E_T=-N_T\partial_t\mathbf u_T

Source-free transverse sector with specified gauge and boundaries; ΩT is dimensionless, not temporal angular velocity or a topological number.
Chapter reference: Final collected manuscript · Chapter 3, equations (3.46) · ASD-BOOK-ddbade997b45

Theory · EFFECTIVE MODEL
ASD-WEB-EM-034
ρs=ε0NT2,KT=NT2μ0,cT2=KTρs=1ε0μ0=c2\rho_s=\varepsilon_0N_T^2,\quad K_T=\frac{N_T^2}{\mu_0},\quad c_T^2=\frac{K_T}{\rho_s}=\frac1{\varepsilon_0\mu_0}=c^2

Coefficient matching is a condition; LVE alone has not determined the microscopic constants.
Chapter reference: Final collected manuscript · Chapter 3, equations (3.47) · ASD-BOOK-ddbade997b45

Theory · DERIVED
ASD-WEB-EM-035
∇⋅ET=∇⋅B=0,∇×ET=−∂tB,∇×B=c−2∂tET\nabla\cdot\mathbf E_T=\nabla\cdot\mathbf B=0,\quad\nabla\times\mathbf E_T=-\partial_t\mathbf B,\quad\nabla\times\mathbf B=c^{-2}\partial_t\mathbf E_T

Derived under the stated transverse action, normalization and coefficient matching; source extension is recorded separately.
Chapter reference: Final collected manuscript · Chapter 3, equations (3.47a) · ASD-BOOK-ddbade997b45

Theory · DERIVED
ASD-WEB-EM-036
uem=ε0E22+B22μ0,S=1μ0E×B,∂tuem+∇⋅S=−J⋅Eu_{\rm em}=\frac{\varepsilon_0E^2}{2}+\frac{B^2}{2\mu_0},\quad\mathbf S=\frac1{\mu_0}\mathbf E\times\mathbf B,\quad\partial_tu_{\rm em}+\nabla\cdot\mathbf S=-\mathbf J\cdot\mathbf E

J=0 in the free region; the sourced exchange law uses the same effective coupling.
Chapter reference: Final collected manuscript · Chapter 3, equations (3.48) · ASD-BOOK-ddbade997b45

Theory · DERIVED
ASD-WEB-EM-037
ω2=c2k2,k⋅E=k⋅B=0,B=1ck^×E\omega^2=c^2k^2,\quad\mathbf k\cdot\mathbf E=\mathbf k\cdot\mathbf B=0,\quad\mathbf B=\frac1c\hat{\mathbf k}\times\mathbf E

Homogeneous isotropic nondispersive free sector; the material γ factor is not evaluated at a free photon’s propagation speed c.
Chapter reference: Final collected manuscript · Chapter 3, equations (3.54), (3.56) · ASD-BOOK-ddbade997b45

Theory · EFFECTIVE MODEL
ASD-WEB-EM-038
Lem=ε02∣E∣2−12μ0∣B∣2−ρeΦτ+J⋅Aτ\mathcal L_{\rm em}=\frac{\varepsilon_0}{2}|\mathbf E|^2-\frac1{2\mu_0}|\mathbf B|^2-\rho_e\Phi_\tau+\mathbf J\cdot\mathbf A_\tau

The free part has a transverse realization; source coupling remains a macroscopic slow-source effective choice.
Chapter reference: Final collected manuscript · Chapter 3, equations (3.39) · ASD-BOOK-ddbade997b45

Theory · DERIVED
ASD-WEB-EM-039
∇⋅E=ρeε0,∇×B=μ0J+c−2∂tE\nabla\cdot\mathbf E=\frac{\rho_e}{\varepsilon_0},\quad\nabla\times\mathbf B=\mu_0\mathbf J+c^{-2}\partial_t\mathbf E

From variation of the chosen effective action; this does not close the general high-speed topological source.
Chapter reference: Final collected manuscript · Chapter 3, equations (3.40), (3.42) · ASD-BOOK-ddbade997b45

Theory · EFFECTIVE MODEL
ASD-WEB-EM-040
Φτ=14πε0∫ρe(r′,t−R/c)R d3r′,Aτ=μ04π∫J(r′,t−R/c)R d3r′\Phi_\tau=\frac1{4\pi\varepsilon_0}\int\frac{\rho_e(\mathbf r',t-R/c)}R\,d^3r',\quad\mathbf A_\tau=\frac{\mu_0}{4\pi}\int\frac{\mathbf J(\mathbf r',t-R/c)}R\,d^3r'

Lorenz gauge, localized conserved sources, no incident free field; inherits the slow-source effective domain.
Chapter reference: Final collected manuscript · Chapter 3, equations (3.51), (3.53) · ASD-BOOK-ddbade997b45

Theory · CANDIDATE MODEL
ASD-WEB-EM-041
η=−∇⋅uL,ΩT=∇×uT,LLT=−gLTη∣ΩT∣2\eta=-\nabla\cdot\mathbf u_L,\quad\boldsymbol\Omega_T=\nabla\times\mathbf u_T,\quad\mathcal L_{LT}=-g_{LT}\eta|\boldsymbol\Omega_T|^2

A lowest-order rotational scalar candidate, not uniquely derived from FCC or LVE.
Chapter reference: Final collected manuscript · Chapter 3, equations (3.74), (3.75) · ASD-BOOK-ddbade997b45

Theory · CANDIDATE MODEL
ASD-WEB-EM-042
fLnl=−gLT∇∣ΩT∣2,fTnl=−2gLT∇×(ηΩT)\mathbf f_L^{\rm nl}=-g_{LT}\nabla|\boldsymbol\Omega_T|^2,\quad\mathbf f_T^{\rm nl}=-2g_{LT}\nabla\times(\eta\boldsymbol\Omega_T)

Generalized force densities in N/m³, not electric current densities.
Chapter reference: Final collected manuscript · Chapter 3, equations (3.76) · ASD-BOOK-ddbade997b45

Theory · CANDIDATE MODEL
ASD-WEB-EM-043
KL2η2+gLTη∣ΩT∣2+λT4∣ΩT∣4=KL2(η+gLTKL∣ΩT∣2)2+(λT4−gLT22KL)∣ΩT∣4\frac{K_L}{2}\eta^2+g_{LT}\eta|\boldsymbol\Omega_T|^2+\frac{\lambda_T}{4}|\boldsymbol\Omega_T|^4=\frac{K_L}{2}\left(\eta+\frac{g_{LT}}{K_L}|\boldsymbol\Omega_T|^2\right)^2+\left(\frac{\lambda_T}{4}-\frac{g_{LT}^2}{2K_L}\right)|\boldsymbol\Omega_T|^4

K_L>0 and λ_T≥2g_LT²/K_L bound this local energy below. At fixed transverse intensity η_eq=−g_LT|Ω_T|²/K_L, giving expansion for g_LT>0. This is not a complete stable localized particle solution.
Chapter reference: Final collected manuscript · Chapter 3, equations (3.77) · ASD-BOOK-ddbade997b45

Theory · FOUNDATION
ASD-WEB-SD-001
u=uL+uT,∇×uL=0,∇⋅uT=0\mathbf u=\mathbf u_L+\mathbf u_T,\quad\nabla\times\mathbf u_L=0,\quad\nabla\cdot\mathbf u_T=0

Decaying fields on all space or compatible boundaries; harmonic components require separate conditions.
Chapter reference: Ch. 4 (4.5) · ASD-BOOK-ddbade997b45

Fourier projectors

空间动力学
Theory · FOUNDATION
ASD-WEB-SD-002
PL,ij=kikjk2,PT,ij=δij−PL,ij,k≠0P_{L,ij}=\frac{k_ik_j}{k^2},\quad P_{T,ij}=\delta_{ij}-P_{L,ij},\quad\mathbf k\ne0

The zero mode is not fixed here.
Chapter reference: Ch. 5 (5.1.8) · ASD-BOOK-ddbade997b45

Theory · FOUNDATION
ASD-WEB-SD-003
s=∇⋅u,σ=−∇⋅uL=−s,ns−ns,0ns,0=σ+O(∣∇u∣2)s=\nabla\cdot\mathbf u,\quad\sigma=-\nabla\cdot\mathbf u_L=-s,\quad\frac{n_s-n_{s,0}}{n_{s,0}}=\sigma+O(|\nabla\mathbf u|^2)

Chapter 4 η corresponds to σ here; neither is the inertial coefficient ρs.
Chapter reference: Ch. 4 (4.14); Ch. 5 (5.4.5) · ASD-BOOK-ddbade997b45

Theory · FOUNDATION
ASD-WEB-SD-004
ΩT=∇×uT,ω=12ΩT,eijdev=12(∂iuj+∂jui)−13δij∇⋅u\boldsymbol\Omega_T=\nabla\times\mathbf u_T,\quad\boldsymbol\omega=\tfrac12\boldsymbol\Omega_T,\quad e^{\rm dev}_{ij}=\tfrac12(\partial_i u_j+\partial_j u_i)-\tfrac13\delta_{ij}\nabla\cdot\mathbf u

Curl is not the full symmetric shear strain; independent nonlinear orientation requires its own definition.
Chapter reference: Ch. 5 (5.4.19), (5.6.1)–(5.6.2), (5.6.22); kinematic expansion · ASD-BOOK-ddbade997b45

Theory · EFFECTIVE MODEL
ASD-WEB-SD-005
L2=12ρs∣∂tu∣2−12KL(∇⋅u)2−12KT∣∇×u∣2\mathcal L_2=\tfrac12\rho_s|\partial_t\mathbf u|^2-\tfrac12K_L(\nabla\cdot\mathbf u)^2-\tfrac12K_T|\nabla\times\mathbf u|^2

Homogeneous, small-amplitude model with constant positive ρs,KL,KT.
Chapter reference: Ch. 4 (4.6) · ASD-BOOK-ddbade997b45

Theory · DERIVED
ASD-WEB-SD-007
(∂t2−cA2∇2)uA=0,cL2=KL/ρs,cT2=KT/ρs,A=L,T(\partial_t^2-c_A^2\nabla^2)\mathbf u_A=0,\quad c_L^2=K_L/\rho_s,\quad c_T^2=K_T/\rho_s,\quad A=L,T

Projection of the complete constant-coefficient equation.
Chapter reference: Ch. 4 (4.8) · ASD-BOOK-ddbade997b45

Theory · EFFECTIVE MODEL
ASD-WEB-SD-008
KLeff=KTeff=ρsc2⟹cL=cT=cK_L^{\rm eff}=K_T^{\rm eff}=\rho_sc^2\quad\Longrightarrow\quad c_L=c_T=c

Effective constitutive matching in the homogeneous matter-free ASD vacuum; speed follows from coefficient ratios, and nonzero order-parameter feedback requires rematching the full spectrum.
Chapter reference: Ch. 4 (4.9); Ch. 5 (5.1.9) · ASD-BOOK-ddbade997b45

Theory · DERIVED
ASD-WEB-SD-009
E2=12ρs∣∂tu∣2+12KL(∇⋅u)2+12KT∣∇×u∣2≥0\mathcal E_2=\tfrac12\rho_s|\partial_t\mathbf u|^2+\tfrac12K_L(\nabla\cdot\mathbf u)^2+\tfrac12K_T|\nabla\times\mathbf u|^2\ge0

Positive coefficients give nonnegative quadratic energy; zero modes and boundaries still need specification.
Chapter reference: Ch. 4 (4.10) · ASD-BOOK-ddbade997b45

Theory · EFFECTIVE MODEL
ASD-WEB-SD-010
L0=∑A=L,T[12ZA∣∂tuA∣2−12ZAcA2∣∇uA∣2+JA⋅uA]\mathcal L_0=\sum_{A=L,T}\left[\tfrac12 Z_A|\partial_t\mathbf u_A|^2-\tfrac12Z_Ac_A^2|\nabla\mathbf u_A|^2+\mathbf J_A\cdot\mathbf u_A\right]

ZA>0 with projection constraints; matching Chapter 4’s normalization uses ZA=ρs.
Chapter reference: Ch. 5 (5.4.1) · ASD-BOOK-ddbade997b45

Theory · DERIVED
ASD-WEB-SD-012
EA=12ZA(∣∂tuA∣2+c2∣∇uA∣2),SA,i=−ZAc2∂tuA,a∂iuA,a,∂tEA+∇⋅SA=JA⋅∂tuA.\begin{aligned}\mathcal E_A&=\tfrac12Z_A(|\partial_t\mathbf u_A|^2+c^2|\nabla\mathbf u_A|^2),\\S_{A,i}&=-Z_Ac^2\partial_tu_{A,a}\partial_i u_{A,a},\\\partial_t\mathcal E_A+\nabla\cdot\mathbf S_A&=\mathbf J_A\cdot\partial_t\mathbf u_A.\end{aligned}

Component indices are summed, cA=c; conservation permits geometric spreading.
Chapter reference: Ch. 5 (5.4.3)–(5.4.4) · ASD-BOOK-ddbade997b45

Theory · DERIVED
ASD-WEB-SD-013
(∂t2−c2∇2)σ=sσ,sσ=−ZL−1∇⋅JL(\partial_t^2-c^2\nabla^2)\sigma=s_\sigma,\quad s_\sigma=-Z_L^{-1}\nabla\cdot\mathbf J_L

Minus the divergence of displacement dynamics; the source is not an arbitrarily specified mass density.
Chapter reference: Ch. 5 (5.4.6) · ASD-BOOK-ddbade997b45

Theory · DERIVED
ASD-WEB-SD-014
uL(k)=ikk2σ,EL=ZL2∫d3k(2π)3[∣∂tσ∣2k2+c2∣σ∣2]\mathbf u_L(\mathbf k)=\frac{i\mathbf k}{k^2}\sigma,\quad E_L=\frac{Z_L}{2}\int\frac{d^3k}{(2\pi)^3}\left[\frac{|\partial_t\sigma|^2}{k^2}+c^2|\sigma|^2\right]

k≠0; real-space kinetic energy contains the inverse Laplacian.
Chapter reference: Ch. 5 (5.4.8)–(5.4.10) · ASD-BOOK-ddbade997b45

Theory · EFFECTIVE MODEL
ASD-WEB-SD-015
AT=NTuT,ET=−∂tAT,Bem=∇×AT\mathbf A_T=N_T\mathbf u_T,\quad\mathbf E_T=-\partial_t\mathbf A_T,\quad\mathbf B_{\rm em}=\nabla\times\mathbf A_T

Source-free radiation zone with specified gauge and normalization; sources also require a scalar potential and coupling.
Chapter reference: Ch. 5 §4.3, (5.4.11) · ASD-BOOK-ddbade997b45

Theory · CANDIDATE MODEL
ASD-WEB-SD-017
JLnl=−gLT∇∣ΩT∣2,JTnl=−2gLT∇×(σΩT)\mathbf J_L^{\rm nl}=-g_{LT}\nabla|\boldsymbol\Omega_T|^2,\quad\mathbf J_T^{\rm nl}=-2g_{LT}\nabla\times(\sigma\boldsymbol\Omega_T)

J is generalized force density, not electric current; the variation is conditional on the candidate.
Chapter reference: Ch. 5 (5.4.20) · ASD-BOOK-ddbade997b45

Theory · CANDIDATE MODEL
ASD-WEB-SD-018
12Ksσ2+gLTσ∣ΩT∣2+14λT∣ΩT∣4=12Ks(σ+gLTKs∣ΩT∣2)2+(λT4−gLT22Ks)∣ΩT∣4.\begin{aligned}\tfrac12K_s\sigma^2+g_{LT}\sigma|\boldsymbol\Omega_T|^2+\tfrac14\lambda_T|\boldsymbol\Omega_T|^4&=\tfrac12K_s(\sigma+\tfrac{g_{LT}}{K_s}|\boldsymbol\Omega_T|^2)^2\\&\quad+(\tfrac{\lambda_T}{4}-\tfrac{g_{LT}^2}{2K_s})|\boldsymbol\Omega_T|^4.\end{aligned}

Ks>0 and λT≥2gLT²/Ks suffice for the local bound, not full stability.
Chapter reference: Ch. 5 (5.4.21) · ASD-BOOK-ddbade997b45

Theory · EFFECTIVE MODEL
ASD-WEB-SD-019
∂t(Z∂tq)−∂i(Kij∂jq)=J,Kij=Kji\partial_t(Z\partial_tq)-\partial_i(K_{ij}\partial_jq)=J,\quad K_{ij}=K_{ji}

Divergence form for a scalar/channel model; multiple fields require the full constitutive tensor.
Chapter reference: Ch. 5 (5.1.10) and following text · ASD-BOOK-ddbade997b45

FCC neighbour moments

空间动力学
Theory · CANDIDATE MODEL
ASD-WEB-SD-020
dn∈a02{(±1,±1,0),(±1,0,±1),(0,±1,±1)},∑ndn,idn,j=2a02δij\mathbf d_n\in\tfrac{a_0}{2}\{(\pm1,\pm1,0),(\pm1,0,\pm1),(0,\pm1,\pm1)\},\quad\sum_nd_{n,i}d_{n,j}=2a_0^2\delta_{ij}

Twelve neighbours, four sign combinations per group; a geometric result.
Chapter reference: Ch. 4 (4.3)–(4.4); Ch. 5 (5.1.3)–(5.1.4) · ASD-BOOK-ddbade997b45

Theory · CANDIDATE MODEL
ASD-WEB-SD-021
ω2=4Γ0[3−cos⁡a0kx2cos⁡a0ky2−cos⁡a0kx2cos⁡a0kz2−cos⁡a0ky2cos⁡a0kz2]\omega^2=4\Gamma_0\left[3-\cos\tfrac{a_0k_x}{2}\cos\tfrac{a_0k_y}{2}-\cos\tfrac{a_0k_x}{2}\cos\tfrac{a_0k_z}{2}-\cos\tfrac{a_0k_y}{2}\cos\tfrac{a_0k_z}{2}\right]

Equal-link scalar/selected-polarization channel, Γ₀>0 and c²=Γ₀a₀². Uses the full cosine spectrum (5.1.11)/(4.12); the printed half-angle coefficient preceding (5.1.12) is 2 rather than the equivalent 4, recorded locally.
Chapter reference: Ch. 4 (4.12)–(4.13); Ch. 5 (5.1.11); coefficient discrepancy preceding (5.1.12) · ASD-BOOK-ddbade997b45

Theory · CANDIDATE MODEL
ASD-WEB-SD-022
Ψ=χQ,Q∈SU(2),Q†Q=I,det⁡Q=1,χ2=12Tr⁡(Ψ†Ψ)\Psi=\chi Q,\quad Q\in SU(2),\quad Q^\dagger Q=I,\quad\det Q=1,\quad\chi^2=\tfrac12\operatorname{Tr}(\Psi^\dagger\Psi)

χ is a real amplitude and Q compact orientation; this is not a solved particle or gauge field.
Chapter reference: Ch. 4 (4.16) · ASD-BOOK-ddbade997b45

Theory · CANDIDATE MODEL
ASD-WEB-SD-023
LuΘ=12Zu∣∂tu∣2+12IΘ∑A(∂tΘA)2−E0+Jext⋅u,E0=12Ku(s2+∣w∣2)+12AΘ∑A∣∇ΘA∣2+W(Θ)+gLsF(Θ)+gTw⋅T+Estab+Enl,F=∑AΘA2−v02,s=∇⋅u,w=∇×u,Ku=Zucu2.\begin{aligned}\mathcal L_{u\Theta}&=\tfrac12Z_u|\partial_t\mathbf u|^2+\tfrac12I_\Theta\sum_A(\partial_t\Theta_A)^2-\mathcal E_0+\mathbf J_{\rm ext}\cdot\mathbf u,\\\mathcal E_0&=\tfrac12K_u(s^2+|\mathbf w|^2)+\tfrac12A_\Theta\sum_A|\nabla\Theta_A|^2+W(\Theta)\\&\quad+g_LsF(\Theta)+g_T\mathbf w\cdot\mathbf T+\mathcal E_{\rm stab}+\mathcal E_{\rm nl},\\F&=\sum_A\Theta_A^2-v_0^2,\quad s=\nabla\cdot\mathbf u,\quad\mathbf w=\nabla\times\mathbf u,\quad K_u=Z_uc_u^2.\end{aligned}

An explicit local dual-field action; Θ has real components and W subtracts vacuum energy. Specify stabilizers, constitutive parameters and target space; complete microscopic unification still needs joint solution.
Chapter reference: Ch. 5 (5.6.3)–(5.6.5) · ASD-BOOK-ddbade997b45

Theory · CANDIDATE MODEL
ASD-WEB-SD-024
Zu(∂t2−cu2∇2)u=gL∇F−gT∇×T+Jadd+JextZ_u(\partial_t^2-c_u^2\nabla^2)\mathbf u=g_L\nabla F-g_T\nabla\times\mathbf T+\mathbf J_{\rm add}+\mathbf J_{\rm ext}

Jadd=−δEadd/δu; sources and feedback come from one functional.
Chapter reference: Ch. 5 (5.6.8) · ASD-BOOK-ddbade997b45

Theory · CANDIDATE MODEL
ASD-WEB-SD-025
IΘ∂t2ΘA−AΘ∇2ΘA+∂W∂ΘA+2gLsΘA+gT[∂C∂ΘA−∂i∂C∂(∂iΘA)]+δEaddδΘA=0,C=w⋅T.\begin{aligned}I_\Theta\partial_t^2\Theta_A-A_\Theta\nabla^2\Theta_A+\frac{\partial W}{\partial\Theta_A}+2g_Ls\Theta_A&+g_T\left[\frac{\partial\mathcal C}{\partial\Theta_A}-\partial_i\frac{\partial\mathcal C}{\partial(\partial_i\Theta_A)}\right]+\frac{\delta E_{\rm add}}{\delta\Theta_A}=0,\\\mathcal C&=\mathbf w\cdot\mathbf T.\end{aligned}

Parity invariance requires axial T; its internal symmetry must be specified.
Chapter reference: Ch. 5 (5.6.9) · ASD-BOOK-ddbade997b45

Theory · CANDIDATE MODEL
ASD-WEB-SD-026
(cu2k2−ω2)(ΩΘ2+cΘ2k2−ω2)−λ2k2ZuIΘ=0,λ=2gLv0(c_u^2k^2-\omega^2)(\Omega_\Theta^2+c_\Theta^2k^2-\omega^2)-\frac{\lambda^2k^2}{Z_uI_\Theta}=0,\quad\lambda=2g_Lv_0

ΩΘ²>0; the full linearized spectrum determines the branches.
Chapter reference: Ch. 5 (5.6.12) · ASD-BOOK-ddbade997b45

Theory · CANDIDATE MODEL
ASD-WEB-SD-027
cL,eff2=cu2−λ2ZuIΘΩΘ2,cT,eff2=cu2c_{L,\rm eff}^2=c_u^2-\frac{\lambda^2}{Z_uI_\Theta\Omega_\Theta^2},\quad c_{T,\rm eff}^2=c_u^2

For this candidate’s low-k gapless branch; equal bare speeds do not ensure equal effective speeds.
Chapter reference: Ch. 5 (5.6.13) · ASD-BOOK-ddbade997b45

Theory · CANDIDATE MODEL
ASD-WEB-SD-028
∇2(Kus+gLF)=0⟹s=−gLKuF,σ=gLKuF\nabla^2(K_us+g_LF)=0\quad\Longrightarrow\quad s=-\frac{g_L}{K_u}F,\quad\sigma=\frac{g_L}{K_u}F

Smooth fields on all space, localized source, no singular harmonic term and zero boundary at infinity.
Chapter reference: Ch. 5 (5.6.14)–(5.6.15) · ASD-BOOK-ddbade997b45

Theory · CANDIDATE MODEL
ASD-WEB-SD-029
Lφ=12Zφ[(∂tφ)2−c2∣∇φ∣2]+φJg[Θ,∇Θ]\mathcal L_\varphi=\tfrac12Z_\varphi[(\partial_t\varphi)^2-c^2|\nabla\varphi|^2]+\varphi\mathcal J_g[\Theta,\nabla\Theta]

Its energy kernel differs from displacement divergence; keeping both requires constraints or scale separation.
Chapter reference: Ch. 5 (5.6.16) · ASD-BOOK-ddbade997b45

Theory · CANDIDATE MODEL
ASD-WEB-SD-030
Zφ(∂t2−c2∇2)φ=Jg,φ(r)=14πZφc2∫Jg(r′)∣r−r′∣ d3r′Z_\varphi(\partial_t^2-c^2\nabla^2)\varphi=\mathcal J_g,\quad\varphi(\mathbf r)=\frac{1}{4\pi Z_\varphi c^2}\int\frac{\mathcal J_g(\mathbf r')}{|\mathbf r-\mathbf r'|}\,d^3r'

Dimensionless varphi; the right side is the static Green integral for a localized source and zero far boundary. Spherical compact support gives exact exterior 1/r; an integrable tail gives an asymptote. Matter fixes the source and feedback.
Chapter reference: Ch. 5 (5.6.17) · ASD-BOOK-ddbade997b45

Theory · CANDIDATE MODEL
ASD-WEB-SD-031
EQ=∫[A22∑i∥Li∥2+A44∑i<j∥[Li,Lj]∥2+W(Q,χ)]d3x,Li=Q−1∂iQE_Q=\int\left[\frac{A_2}{2}\sum_i\|L_i\|^2+\frac{A_4}{4}\sum_{i<j}\|[L_i,L_j]\|^2+W(Q,\chi)\right]d^3x,\quad L_i=Q^{-1}\partial_iQ

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

Theory · CANDIDATE MODEL
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

Theory · CANDIDATE MODEL
ASD-WEB-SD-033
ME=E0/c2,Lcoll(2)=−E0+12MijX˙iX˙j,Mg=c2Glim⁡r→∞rφ(r)=∫Jg d3r4πGZφ,qg=−∂Etest∂φ∣φ=0,Mp=qg/c2,∫Jg d3r=E0, Mg=ME ⟹ Zφ=c24πG,ME=Mi=Mg=Mp(matching requirement).\begin{aligned}M_E&=E_0/c^2,\quad L_{\rm coll}^{(2)}=-E_0+\tfrac12M_{ij}\dot X_i\dot X_j,\\M_g&=\frac{c^2}{G}\lim_{r\to\infty}r\varphi(r)=\frac{\int\mathcal J_g\,d^3r}{4\pi GZ_\varphi},\\q_g&=-\left.\frac{\partial E_{\rm test}}{\partial\varphi}\right|_{\varphi=0},\quad M_p=q_g/c^2,\\\int\mathcal J_g\,d^3r=E_0,\ M_g=M_E&\ \Longrightarrow\ Z_\varphi=\frac{c^2}{4\pi G},\\M_E=M_i=M_g=M_p&\quad\text{(matching requirement).}\end{aligned}

Define all four masses from the same complete core–field system. Equality and the Zφ condition require source–energy matching, not ontology alone. Setting coefficients by observed G is effective normalization.
Chapter reference: Ch. 5 (5.6.26)–(5.6.29) · ASD-BOOK-ddbade997b45

Theory · DERIVED
ASD-WEB-SD-034
nsns,0=1det⁡F,Fij=δij+∂jui,F=eφI ⟹ nsns,0=e−3φ,ns−ns,0ns,0=e−3φ−1=−3φ+O(φ2).\begin{aligned}\frac{n_s}{n_{s,0}}&=\frac{1}{\det F},\quad F_{ij}=\delta_{ij}+\partial_j u_i,\\F=e^\varphi I&\ \Longrightarrow\ \frac{n_s}{n_{s,0}}=e^{-3\varphi},\\\frac{n_s-n_{s,0}}{n_{s,0}}&=e^{-3\varphi}-1=-3\varphi+O(\varphi^2).\end{aligned}

Use detF for finite deformation; sigma is the small-strain density term. The second part is an isotropic frame with linear matching sigma=-3varphi; do not conflate general anisotropy, density energy kernels and independent stretch actions.
Chapter reference: Ch. 4 (4.14)–(4.15); Ch. 5 (5.4.5) and isotropic-frame discussion · ASD-BOOK-ddbade997b45

Theory · FOUNDATION
ASD-WEB-MAT-001
Φ(x)→Φ0 (∣x∣→∞),E[Φ]=∫R3(H[Φ]−H0) d3x<∞\Phi(\mathbf x)\to\Phi_0\ (|\mathbf x|\to\infty),\qquad E[\Phi]=\int_{\mathbb R^3}(\mathcal H[\Phi]-\mathcal H_0)\,d^3x<\infty

Φ denotes all participating fields; approaching the vacuum alone does not ensure convergence.
Chapter reference: Ch. 4 §4.2, (4.18), (4.34); conditions collected for review · ASD-BOOK-ddbade997b45

Theory · DERIVED
ASD-WEB-MAT-003
Q(∞)=Q∞,Q:Sspace3→SU(2)≃S3,π3(S3)=Z,π1(SU(2))=0Q(\infty)=Q_\infty,\quad Q:S^3_{\rm space}\to SU(2)\simeq S^3,\quad\pi_3(S^3)=\mathbb Z,\quad\pi_1(SU(2))=0

The boundary orientation is direction independent; amplitude zeros or a different target require a new classification.
Chapter reference: Ch. 4 (4.18), §4.2.1 · ASD-BOOK-ddbade997b45

Theory · DERIVED
ASD-WEB-MAT-004
Bμ=ϵμνρσ24π2Tr⁡(LνLρLσ),Lμ=Q−1∂μQ,∂μBμ=0,NQ=∫B0 d3x∈ZB^\mu=\frac{\epsilon^{\mu\nu\rho\sigma}}{24\pi^2}\operatorname{Tr}(L_\nu L_\rho L_\sigma),\quad L_\mu=Q^{-1}\partial_\mu Q,\quad\partial_\mu B^\mu=0,\quad N_Q=\int B^0\,d^3x\in\mathbb Z

x⁰=ct and ε⁰¹²³=+1; target orientation follows the manuscript. The identity requires smooth Q and fixed far boundary.
Chapter reference: Ch. 4 (4.25)–(4.26) · ASD-BOOK-ddbade997b45

Theory · CANDIDATE MODEL
ASD-WEB-MAT-005
Nem=NQ,ρem=eB0,jem=ec BtopN_{\rm em}=N_Q,\quad\rho_{\rm em}=eB^0,\quad\mathbf j_{\rm em}=ec\,\mathbf B_{\rm top}

Minimal source identification in (4.27): Nem=NQ is a model input, e fixes the unit-charge normalization, and x0=ct supplies c in the spatial current. Btop is topological current, not magnetic field.
Chapter reference: Ch. 4 (4.27) · ASD-BOOK-ddbade997b45

Theory · CANDIDATE MODEL
ASD-WEB-MAT-006
∂tρem+∇⋅jem=0,Qem=eNem,ϵ0∮∂VE⋅dS=Qem(V)\partial_t\rho_{\rm em}+\nabla\cdot\mathbf j_{\rm em}=0,\qquad Q_{\rm em}=eN_{\rm em},\qquad\epsilon_0\oint_{\partial V}\mathbf E\cdot d\mathbf S=Q_{\rm em}(V)

The first two follow from the candidate identification; the flux law also needs effective EM source equations.
Chapter reference: Ch. 4 (4.28)–(4.29) · ASD-BOOK-ddbade997b45

Theory · DERIVED
ASD-WEB-MAT-007
Qλ(x)=Q(x/λ),E(λ)=λE2+λ3EV,E′(1)=E2+3EV>0Q_\lambda(\mathbf x)=Q(\mathbf x/\lambda),\quad E(\lambda)=\lambda E_2+\lambda^3E_V,\quad E'(1)=E_2+3E_V>0

For a nontrivial static 3D configuration, E2>0, EV≥0 and fixed vacuum boundary on all space; shrinking λ lowers energy.
Chapter reference: Ch. 4 (4.20) · ASD-BOOK-ddbade997b45

Theory · DERIVED
ASD-WEB-MAT-008
3EVλ∗4+E2λ∗2−E4=0,λ∗2=2E4E2+E22+12E4EV,EV=0:λ∗2=E4E2,E′′(λ∗)=2E4λ∗3+6EVλ∗>0\begin{aligned}3E_V\lambda_*^4+E_2\lambda_*^2-E_4&=0,\\ \lambda_*^2&=\frac{2E_4}{E_2+\sqrt{E_2^2+12E_4E_V}},\\ E_V=0:\quad\lambda_*^2&=\frac{E_4}{E_2},\\ E''(\lambda_*)&=\frac{2E_4}{\lambda_*^3}+6E_V\lambda_*>0\end{aligned}

E2,E4>0 and EV≥0 are energy integrals of one reference shape; Qλ(x)=Q(x/λ), with dimensionless λ. The unique positive stationary scale and positive curvature are analytic results for this family, not a full field solution or nonradial stability.
Chapter reference: Final collected manuscript, Chapter 4, Eq. (4.22) and the following analytic root and second derivative (p1374–p1375) · ASD-BOOK-ddbade997b45

Theory · FOUNDATION
ASD-WEB-MAT-009
Erest=∫(H[Q,η,χ,φ]−Hvac) d3x,mE=Erest/c2E_{\rm rest}=\int(\mathcal H[Q,\eta,\chi,\varphi]-\mathcal H_{\rm vac})\,d^3x,\qquad m_E=E_{\rm rest}/c^2

Includes core, accompanying fields and interactions; defining mE does not calculate inertia or gravitational mass.
Chapter reference: Ch. 4 (4.34) · ASD-BOOK-ddbade997b45

Theory · CANDIDATE MODEL
ASD-WEB-MAT-010
Lcoll=−Erest+12MijX˙iX˙j+⋯ ,Mij=mIδij in an isotropic limitL_{\rm coll}=-E_{\rm rest}+\tfrac12M_{ij}\dot X_i\dot X_j+\cdots,\qquad M_{ij}=m_I\delta_{ij}\ \text{in an isotropic limit}

Insert an actual moving family into the same action, including velocity-induced deformation; isotropy is an additional condition.
Chapter reference: Ch. 4 (4.35); Ch. 5 (5.6.26) · ASD-BOOK-ddbade997b45

Theory · CANDIDATE MODEL
ASD-WEB-MAT-011
φ(r)∼GmGc2r,Φgext=−c2φext,Uext=mPΦgext,F=−mP∇Φgext\varphi(r)\sim\frac{Gm_G}{c^2r},\quad\Phi_g^{\rm ext}=-c^2\varphi^{\rm ext},\quad U_{\rm ext}=m_P\Phi_g^{\rm ext},\quad\mathbf F=-m_P\nabla\Phi_g^{\rm ext}

Isolated-source far field and weak slowly varying external field; mP must be obtained from the configuration’s response.
Chapter reference: Ch. 4 (4.36)–(4.38); Ch. 5 (5.6.28) and following text · ASD-BOOK-ddbade997b45

Theory · OPEN
ASD-WEB-MAT-012
mE=mI=mG=mP=m\boxed{m_E=m_I=m_G=m_P=m}

Common matching requirement (4.36): the same solution, parameters and normalization must independently produce four equal masses. Equality signs do not promote the unfinished microscopic calculation to a proof.
Chapter reference: Final collected manuscript, Chapter 4, Eq. (4.36) and its matching condition; Chapter 5, Eq. (5.6.29) · ASD-BOOK-ddbade997b45

Theory · CANDIDATE MODEL
ASD-WEB-MAT-013
QR(x,t)=Q0(R−1(t)x),A∈SU(2),A↦R∈SO(3),Lrot=12ΛabΩaΩbQ_R(\mathbf x,t)=Q_0(R^{-1}(t)\mathbf x),\quad A\in SU(2),\quad A\mapsto R\in SO(3),\quad L_{\rm rot}=\tfrac12\Lambda_{ab}\Omega_a\Omega_b

A stable Q0 is needed before computing inertia; A lifts spatial rotations and must not be identified with the internal Q field.
Chapter reference: Ch. 4 (4.30)–(4.31) · ASD-BOOK-ddbade997b45

Theory · CANDIDATE MODEL
ASD-WEB-MAT-014
ψ(−A)=−ψ(A)⟹j∈{12,32,…}\psi(-A)=-\psi(A)\quad\Longrightarrow\quad j\in\{\tfrac12,\tfrac32,\ldots\}

Requires a nontrivial rotation loop in the full configuration space and this quantization choice; it alone fixes neither j=1/2 nor exchange statistics.
Chapter reference: Ch. 4 (4.32); conditional Finkelstein–Rubinstein construction · ASD-BOOK-ddbade997b45

Theory · HYPOTHESIS
ASD-WEB-MAT-017
n=13(1,1,1)T,PA1=nnT,PE=I−PA1,Q^p=e(qEPE+qAPA1)\mathbf n=\tfrac1{\sqrt3}(1,1,1)^T,\quad P_{A_1}=\mathbf n\mathbf n^T,\quad P_E=I-P_{A_1},\quad\widehat Q_p=e(q_E P_E+q_A P_{A_1})

Assumes an A1⊕E decomposition of an internal three-channel response space; not a spatial projection or a unique FCC consequence.
Chapter reference: Ch. 4 (4.40)–(4.41) · ASD-BOOK-ddbade997b45

Theory · HYPOTHESIS
ASD-WEB-MAT-018
2qE+qA=1,qE−qA=1⏟additional inputs⟹spec⁡Q^p=e{23,23,−13}\underbrace{2q_E+q_A=1,\quad q_E-q_A=1}_{\rm additional\ inputs}\quad\Longrightarrow\quad\operatorname{spec}\widehat Q_p=e\{\tfrac23,\tfrac23,-\tfrac13\}

Symmetry does not fix the eigenvalues; matching the trace to physical charge is also input. Eigenchannels are not three geometric petals.
Chapter reference: Ch. 4 (4.42)–(4.43) · ASD-BOOK-ddbade997b45

Theory · CANDIDATE MODEL
ASD-WEB-MAT-019
Nemin=0,Neme−+Neme+=−1+1=0N_{\rm em}^{\rm in}=0,\quad N_{\rm em}^{e^-}+N_{\rm em}^{e^+}=-1+1=0

Zero total permits neutral final states; it guarantees neither a path or rate nor that any neutral combination is an antiparticle pair.
Chapter reference: Ch. 4 (4.71), (4.78) · ASD-BOOK-ddbade997b45

Theory · DERIVED
ASD-WEB-MAT-020
γ+Z→e−+e++Z,S=M2c4+2Mc2Eγ,Eγ,th=2mec2(1+me/M)\gamma+Z\to e^-+e^++Z,\quad\mathcal S=M^2c^4+2Mc^2E_\gamma,\quad E_{\gamma,\rm th}=2m_ec^2(1+m_e/M)

Z is initially at rest and retains mass M. Assumes E²=m²c⁴+c²p² and Eγ=c|pγ|; the threshold does not determine a rate.
Chapter reference: Ch. 4 (4.72)–(4.74) · ASD-BOOK-ddbade997b45

Theory · DERIVED
ASD-WEB-MAT-022
p1+p2=0,E1+E2=2mec2,Ei=c∣pi∣⟹E1=E2=mec2≃511 keV\mathbf p_1+\mathbf p_2=0,\quad E_1+E_2=2m_ec^2,\quad E_i=c|\mathbf p_i|\quad\Longrightarrow\quad E_1=E_2=m_ec^2\simeq511\,\mathrm{keV}

Zero total momentum, negligible relative kinetic and binding energy, no other participants, and the two-photon channel already selected.
Chapter reference: Ch. 4 (4.79)–(4.80) · ASD-BOOK-ddbade997b45

Theory · FOUNDATION
ASD-WEB-MAT-023
Ein=Eout,γ+Eout,other+Erecoil+ΔEbackgroundE_{\rm in}=E_{{\rm out},\gamma}+E_{{\rm out},{\rm other}}+E_{\rm recoil}+\Delta E_{\rm background}

Use one system boundary without double counting; this does not imply a mandatory undetected longitudinal-energy branch.
Chapter reference: Ch. 4 (4.81) · ASD-BOOK-ddbade997b45

Theory · EFFECTIVE MODEL
ASD-WEB-QM-003
Q(r,t)=Q∗[r−X(t)]+η,[∂t2+H∗(t)]η=S,Str=(a⋅∇)Q∗−(v⋅∇)2Q∗\mathcal Q(\mathbf r,t)=\mathcal Q_*[\mathbf r-\mathbf X(t)]+\eta,\quad [\partial_t^2+\mathcal H_*(t)]\eta=S,\quad S_{\rm tr}=(\mathbf a\cdot\nabla)\mathcal Q_*-(\mathbf v\cdot\nabla)^2\mathcal Q_*

First-order perturbation equation about a specified static configuration with a kinetic-normalized Hessian; full nonlinear source construction is a separate task.
Chapter reference: Ch. 5 (5.1.18)–(5.1.20) · ASD-BOOK-ddbade997b45

Theory · EFFECTIVE MODEL
ASD-WEB-QM-005
ωa2(k)=ω02+c2kiBijkj,ka0≪1\omega_a^2(\mathbf k)=\omega_0^2+c^2k_iB_{ij}k_j,\qquad ka_0\ll1

Final equation 5.1.22 is the dispersion equality of the selected second-order gapped continuum channel. ka₀≪1 limits microscopic applicability; higher lattice terms are separate. B is real symmetric positive definite.
Chapter reference: Ch. 5 (5.1.22) · ASD-BOOK-ddbade997b45

Theory · EFFECTIVE MODEL
ASD-WEB-QM-006
ηa(r,t)=ηa,in(r,t)+∫t0t ⁣dt′ ⁣∫d3r′ Garet(r−r′,t−t′)Sa(r′,t′)\eta_a(\mathbf r,t)=\eta_{a,\mathrm{in}}(\mathbf r,t)+\int_{t_0}^{t}\!dt'\!\int d^3r'\,G_a^{\mathrm{ret}}(\mathbf r-\mathbf r',t-t')S_a(\mathbf r',t')

A solution representation for a specified linear operator, initial data and source; coupled channels require a matrix Green function.
Chapter reference: Ch. 5 (5.1.23)–(5.1.24) · ASD-BOOK-ddbade997b45

Theory · EFFECTIVE MODEL
ASD-WEB-QM-007
ωa(k)=k⋅v+ωs,min⁡k[ω02+c2k2−k⋅v]=ω0/γ\omega_a(\mathbf k)=\mathbf k\cdot\mathbf v+\omega_s,\qquad\min_{\mathbf k}\left[\sqrt{\omega_0^2+c^2k^2}-\mathbf k\cdot\mathbf v\right]=\omega_0/\gamma

Isotropic gapped branch, v<c. ωs<ω₀/γ has no intersection; equality is critical tangency, not strictly subthreshold. Outgoing flux also requires source projection.
Chapter reference: Ch. 5 (5.1.28)–(5.1.29) · ASD-BOOK-ddbade997b45

Theory · EFFECTIVE MODEL
ASD-WEB-QM-008
E=γm0c2,P=γm0v,E2−c2P2=m02c4E=\gamma m_0c^2,\quad\mathbf P=\gamma m_0\mathbf v,\quad E^2-c^2P^2=m_0^2c^4

Motion relations of the complete dressed structure inherited from Chapter 2; the final text connects them to the gapped phase through single-excitation action matching.
Chapter reference: Ch. 5 (5.2.12); upstream ASD dynamics input · ASD-BOOK-ddbade997b45

Theory · EFFECTIVE MODEL
ASD-WEB-QM-009
E=Iω,P=Ik,I=matchingℏ ⟹ P=ℏk,λ=h/PE=\mathcal I\omega,\quad\mathbf P=\mathcal I\mathbf k,\qquad \mathcal I\stackrel{\rm matching}{=}\hbar\ \Longrightarrow\ \mathbf P=\hbar\mathbf k,\quad\lambda=h/P

The final manuscript adopts I=ℏ and total dressed-momentum matching, giving E=ℏω, P=ℏk and λ=h/P. Microscopic locking and universal normalization require a complete moving solution.
Chapter reference: Ch. 5 §2.3, text following (5.2.13) · ASD-BOOK-ddbade997b45

Theory · EFFECTIVE MODEL
ASD-WEB-QM-010
Ekin=piBijpj2m0,pi=ℏki,vg,i=Bijpjm0;B=I: vg=c2kω=c2PE=vE_{\rm kin}=\frac{p_iB_{ij}p_j}{2m_0},\quad p_i=\hbar k_i,\quad v_{g,i}=\frac{B_{ij}p_j}{m_0};\qquad B=I:\ \mathbf v_g=\frac{c^2\mathbf k}{\omega}=\frac{c^2\mathbf P}{E}=\mathbf v

The first set is the low-speed constant-B envelope. The full isotropic gapped branch with E=ℏω and P=ℏk gives vg=c²P/E=v. Momentum-dependent B requires its derivatives.
Chapter reference: Ch. 5 (5.2.14) · ASD-BOOK-ddbade997b45

Theory · EFFECTIVE MODEL
ASD-WEB-QM-011
[∂t2−c2∂i(Bij∂j)+ω02+2ω0ℏV]Φ(+)=S(+)\left[\partial_t^2-c^2\partial_i(B_{ij}\partial_j)+\omega_0^2+\frac{2\omega_0}{\hbar}V\right]\Phi^{(+)}=S^{(+)}

ω₀ and field normalization are constant; V is a weak scalar energy modulation and B is real symmetric positive definite. This is not a closed ASD master equation.
Chapter reference: Ch. 5 (5.1.30) · ASD-BOOK-ddbade997b45

Theory · FOUNDATION
ASD-WEB-QM-012
Φ(+)=Ψe−iω0t,S(+)=S0e−iω0t,Φreal=Re⁡(Ψe−iω0t)\Phi^{(+)}=\Psi e^{-i\omega_0t},\quad S^{(+)}=S_0e^{-i\omega_0t},\quad\Phi_{\rm real}=\operatorname{Re}(\Psi e^{-i\omega_0t})

This page uses the rest carrier k₀=0; the complex number represents two quadratures, not an event probability.
Chapter reference: Ch. 5 (5.1.31) · ASD-BOOK-ddbade997b45

Theory · DERIVED
ASD-WEB-QM-013
∂t2Ψ−2iω0∂tΨ−c2∂i(Bij∂jΨ)+2ω0ℏVΨ=S0\partial_t^2\Psi-2i\omega_0\partial_t\Psi-c^2\partial_i(B_{ij}\partial_j\Psi)+\frac{2\omega_0}{\hbar}V\Psi=S_0

An exact carrier substitution and cancellation of rest terms within the specified second-order channel.
Chapter reference: Ch. 5 (5.1.32)–(5.1.33) · ASD-BOOK-ddbade997b45

Theory · EFFECTIVE MODEL
ASD-WEB-QM-014
ℏω0=matchingm0c2;∣Ω∣ω0≪1,ckω0≪1,ka0≪1,∣V∣ℏω0≪1\hbar\omega_0\stackrel{\rm matching}{=}m_0c^2;\qquad\frac{|\Omega|}{\omega_0}\ll1,\quad\frac{ck}{\omega_0}\ll1,\quad ka_0\ll1,\quad\frac{|V|}{\hbar\omega_0}\ll1

Ω is the slow frequency; use spectral or norm bounds, not division by Ψ at nodes. Energy matching is an input, not an algebraic output.
Chapter reference: Ch. 5 §1.6, (5.1.34) · ASD-BOOK-ddbade997b45

Theory · EFFECTIVE MODEL
ASD-WEB-QM-015
iℏ∂tΨ=−ℏ22m0∂i(Bij∂jΨ)+VΨ+SSch,SSch=−ℏ2ω0S0i\hbar\partial_t\Psi=-\frac{\hbar^2}{2m_0}\partial_i(B_{ij}\partial_j\Psi)+V\Psi+S_{\rm Sch},\qquad S_{\rm Sch}=-\frac{\hbar}{2\omega_0}S_0

Drop the second envelope time derivative and use the matching input; the familiar scalar form additionally takes B=I and no source.
Chapter reference: Ch. 5 (5.1.35) · ASD-BOOK-ddbade997b45

Theory · DERIVED
ASD-WEB-QM-016
z=ck/ω0,Ωexact/ω0=1+z2−1=12z2−18z4+O(z6),ΩSVEA/ω0=12z2z=ck/\omega_0,\quad\Omega_{\rm exact}/\omega_0=\sqrt{1+z^2}-1=\tfrac12z^2-\tfrac18z^4+O(z^6),\quad\Omega_{\rm SVEA}/\omega_0=\tfrac12z^2

A Taylor check of the given branch, not a new complete high-energy ASD prediction.
Chapter reference: Ch. 5 (5.1.22), (5.2.13); website Taylor expansion for B=I, V=0 · ASD-BOOK-ddbade997b45

Theory · EFFECTIVE MODEL
ASD-WEB-QM-017
⟨EΦ⟩=ZΦ4[∣∂tΨ−iω0Ψ∣2+c2Bij∂iΨ∗∂jΨ+ω02∣Ψ∣2]≃ZΦω022∣Ψ∣2\langle\mathcal E_\Phi\rangle=\frac{Z_\Phi}{4}\left[|\partial_t\Psi-i\omega_0\Psi|^2+c^2B_{ij}\partial_i\Psi^*\partial_j\Psi+\omega_0^2|\Psi|^2\right]\simeq\frac{Z_\Phi\omega_0^2}{2}|\Psi|^2

Quadratic field energy without external potential, averaged at leading slow-envelope order; the next order includes cross terms, not only gradients. This is not the Born rule.
Chapter reference: Ch. 5 (5.1.36) · ASD-BOOK-ddbade997b45

Theory · DERIVED
ASD-WEB-QM-018
∂tn+∂iji=2ℏIm⁡(Ψ∗SSch),n=∣Ψ∣2,ji=ℏm0Im⁡(Ψ∗Bij∂jΨ)\partial_t n+\partial_i j_i=\frac2\hbar\operatorname{Im}(\Psi^*S_{\rm Sch}),\quad n=|\Psi|^2,\quad j_i=\frac\hbar{m_0}\operatorname{Im}(\Psi^*B_{ij}\partial_j\Psi)

Real V and real symmetric B; total norm is conserved only without source and net boundary flux. Conserved weight is not automatically core-position probability.
Chapter reference: Ch. 5 (5.2.7)–(5.2.8) · ASD-BOOK-ddbade997b45

Theory · EFFECTIVE MODEL
ASD-WEB-QM-019
ψ(L,Y)=1iλL∫−∞∞T(y)ψin(y)exp⁡ ⁣[ik(Y−y)22L]dy,λ=2π/k\psi(L,Y)=\frac1{\sqrt{i\lambda L}}\int_{-\infty}^{\infty}T(y)\psi_{\rm in}(y)\exp\!\left[\frac{ik(Y-y)^2}{2L}\right]dy,\qquad\lambda=2\pi/k

B=I, nearly monoenergetic, thin aperture and paraxial approximation; common longitudinal and time phases are suppressed.
Chapter reference: Ch. 5 (5.3.1)–(5.3.3) · ASD-BOOK-ddbade997b45

Theory · EFFECTIVE MODEL
ASD-WEB-QM-020
I(ϕ)I0=sinc⁡2 ⁣(πasin⁡ϕλ)cos⁡2 ⁣(πdsin⁡ϕλ),ΔY≃λL/d,sinc⁡z=sin⁡z/z\frac{I(\phi)}{I_0}=\operatorname{sinc}^2\!\left(\frac{\pi a\sin\phi}{\lambda}\right)\cos^2\!\left(\frac{\pi d\sin\phi}{\lambda}\right),\quad\Delta Y\simeq\lambda L/d,\quad\operatorname{sinc}z=\sin z/z

a is slit width and d separation; coherent uniform illumination and D²/(λL)≪1. Web φ replaces manuscript observation angle θ, not wave phases; paraxial spacing remains approximate.
Chapter reference: Ch. 5 (5.3.4)–(5.3.6) · ASD-BOOK-ddbade997b45

Theory · EFFECTIVE MODEL
ASD-WEB-QM-021
ΔθV=−1ℏ∫[V1(t)−V2(t)]dt,∣Δθg∣=MPgAℏv\Delta\theta_V=-\frac1\hbar\int[V_1(t)-V_2(t)]dt,\qquad |\Delta\theta_g|=\frac{M_Pg\mathcal A}{\hbar v}

Equal leading path speeds; the gravitational expression assumes a weak uniform field and specified enclosed area. MP is passive gravitational mass.
Chapter reference: Ch. 5 (5.3.9)–(5.3.10) · ASD-BOOK-ddbade997b45

Theory · FOUNDATION
ASD-WEB-QM-022
ψ~(k)=12π∫ψ(x)e−ikx dx,∫∣ψ∣2dx=∫∣ψ~∣2dk=1\widetilde\psi(k)=\frac1{\sqrt{2\pi}}\int\psi(x)e^{-ikx}\,dx,\quad\int|\psi|^2dx=\int|\widetilde\psi|^2dk=1

Finite second position moment and square-integrable first weak derivative; standard deviations refer to these mathematical weights.
Chapter reference: Ch. 5 (5.3.23) · ASD-BOOK-ddbade997b45

Theory · DERIVED
ASD-WEB-QM-023
F=(x−⟨x⟩)ψ,G=(∂x−i⟨k⟩)ψ,2Re⁡⟨F,G⟩=−1,ΔxΔk≥12;p=ℏk ⟹ ΔxΔp≥ℏ/2F=(x-\langle x\rangle)\psi,\quad G=(\partial_x-i\langle k\rangle)\psi,\quad2\operatorname{Re}\langle F,G\rangle=-1,\quad\Delta x\Delta k\ge\tfrac12;\quad p=\hbar k\ \Longrightarrow\ \Delta x\Delta p\ge\hbar/2

Uses integration by parts, decaying boundaries, Cauchy–Schwarz and Parseval; the final step depends on phase–momentum matching.
Chapter reference: Ch. 5 (5.3.24)–(5.3.25) · ASD-BOOK-ddbade997b45

Theory · DERIVED
ASD-WEB-QM-024
ψ(x)=e−(x−x0)2/(4σx2)+ik0x(2πσx2)1/4,Δx=σx,Δk=1/(2σx)\psi(x)=\frac{e^{-(x-x_0)^2/(4\sigma_x^2)+ik_0x}}{(2\pi\sigma_x^2)^{1/4}},\quad\Delta x=\sigma_x,\quad\Delta k=1/(2\sigma_x)

σx>0; widths are standard deviations, not slit first-zero or half-maximum widths.
Chapter reference: Ch. 5 (5.3.26) · ASD-BOOK-ddbade997b45

Theory · EFFECTIVE MODEL
ASD-WEB-QM-025
MD(q¨D+2γDq˙D+ωD2qD)=gDΦreal+ξD,χD(ω)=1MD(ωD2−ω2−2iγDω)M_D(\ddot q_D+2\gamma_D\dot q_D+\omega_D^2q_D)=g_D\Phi_{\rm real}+\xi_D,\qquad\chi_D(\omega)=\frac1{M_D(\omega_D^2-\omega^2-2i\gamma_D\omega)}

Uses e⁻ⁱωᵗ and γD>0; a linear material-response model, not a universal measurement axiom.
Chapter reference: Ch. 5 (5.3.17)–(5.3.18) · ASD-BOOK-ddbade997b45

Theory · EFFECTIVE MODEL
ASD-WEB-QM-026
P‾D=gD2ω02Im⁡χD(ω0)∣Ψ∣2,RD≃P‾D/εD\overline P_D=\frac{g_D^2\omega_0}{2}\operatorname{Im}\chi_D(\omega_0)|\Psi|^2,\qquad R_D\simeq\overline P_D/\varepsilon_D

The first follows from near-monochromatic average work after subtracting independent noise; the second additionally requires fixed absorbed energy per event and negligible leakage, dead time and saturation.
Chapter reference: Ch. 5 (5.3.19) and following text · ASD-BOOK-ddbade997b45

Theory · EFFECTIVE MODEL
ASD-WEB-QM-027
R(r,t)=ηD(r,t)∣Ψ∣2,RA=∫thicknessR dz,p(Y∣det)=∫TRA(Y,t)dt∫dA∫TRA(Y,t)dt\mathcal R(\mathbf r,t)=\eta_D(\mathbf r,t)|\Psi|^2,\quad\mathcal R_A=\int_{\rm thickness}\mathcal R\,dz,\quad p(Y\mid\mathrm{det})=\frac{\int_{\mathcal T}\mathcal R_A(Y,t)dt}{\int dA\int_{\mathcal T}\mathcal R_A(Y,t)dt}

Nonzero total detection rate with stated efficiency, thickness and time window. For normal-flux sampling jn=vn|Ψ|²; normalized squared modulus also requires constant or cancelling vn across compared positions.
Chapter reference: Ch. 5 (5.3.20)–(5.3.21) · ASD-BOOK-ddbade997b45

Theory · EFFECTIVE MODEL
ASD-WEB-QM-028
R=η1∣Ψ∣2+η2∣Ψ∣4+∫KD(r,r′)∣Ψ(r′)∣2d3r′\mathcal R=\eta_1|\Psi|^2+\eta_2|\Psi|^4+\int K_D(\mathbf r,\mathbf r')|\Psi(\mathbf r')|^2d^3r'

Equation 5.3.22 defines the selected quartic-plus-nonlocal response model; keep event rates nonnegative in its effective range. It is not an all-intensity law for every detector.
Chapter reference: Ch. 5 (5.3.22) · ASD-BOOK-ddbade997b45

Theory · CANDIDATE MODEL
ASD-WEB-QM-029
Ψ=n eiS/ℏ,vi=Bij∂jSm0,∂tn+∂i(nvi)=0;X˙i=candidatevi(X,t)\Psi=\sqrt n\,e^{iS/\hbar},\quad v_i=\frac{B_{ij}\partial_jS}{m_0},\quad\partial_t n+\partial_i(nv_i)=0;\qquad\dot X_i\stackrel{\rm candidate}{=}v_i(\mathbf X,t)

The first three apply to a source-free envelope with n>0; the last is an extra core–field identification. Equivariance preserves an assumed initial distribution.
Chapter reference: Ch. 5 (5.3.12)–(5.3.15) · ASD-BOOK-ddbade997b45

Theory · EFFECTIVE MODEL
ASD-WEB-QM-030
−ℏ22m0ψ′′+Vψ=Eψ,k=2m0E/ℏ,κ=2m0(V0−E)/ℏ,ψII=Aeκx+Be−κx-\frac{\hbar^2}{2m_0}\psi''+V\psi=E\psi,\quad k=\sqrt{2m_0E}/\hbar,\quad\kappa=\sqrt{2m_0(V_0-E)}/\hbar,\quad\psi_{\rm II}=Ae^{\kappa x}+Be^{-\kappa x}

0<E<V₀, equal masses, B=I and barrier width a; ψ and ψ′ are continuous across finite steps.
Chapter reference: Ch. 5 (5.3.28)–(5.3.31) · ASD-BOOK-ddbade997b45

Theory · EFFECTIVE MODEL
ASD-WEB-QM-031
T=[1+V02sinh⁡2(κa)4E(V0−E)]−1,z=e−2κa,D=V02(1−z)2+16E(V0−E)z,T=16E(V0−E)zD,Rrefl=V02(1−z)2D,T+Rrefl=1,jII=2ℏκm0Im⁡(AB∗)\begin{aligned}T&=\left[1+\frac{V_0^2\sinh^2(\kappa a)}{4E(V_0-E)}\right]^{-1},\\ z&=e^{-2\kappa a},\quad D=V_0^2(1-z)^2+16E(V_0-E)z,\\ T&=\frac{16E(V_0-E)z}{D},\quad R_{\rm refl}=\frac{V_0^2(1-z)^2}{D},\quad T+R_{\rm refl}=1,\\ j_{\rm II}&=\frac{2\hbar\kappa}{m_0}\operatorname{Im}(AB^*)\end{aligned}

Exact boundary solution for equal exterior parameters, real potential and 0<E<V₀. z,D abbreviate 5.3.34; Rrefl is reflectance, not core radius. Complex exponential superposition carries flux.
Chapter reference: Ch. 5 (5.3.32)–(5.3.35) · ASD-BOOK-ddbade997b45

Theory · EFFECTIVE MODEL
ASD-WEB-QM-032
E>V0: T=[1+V02sin⁡2(qa)4E(E−V0)]−1, q=2m0(E−V0)/ℏ;E=V0: T=[1+m0V0a22ℏ2]−1E>V_0:\ T=\left[1+\frac{V_0^2\sin^2(qa)}{4E(E-V_0)}\right]^{-1},\ q=\sqrt{2m_0(E-V_0)}/\hbar;\quad E=V_0:\ T=\left[1+\frac{m_0V_0a^2}{2\hbar^2}\right]^{-1}

qa=nπ gives above-barrier perfect transmission; it is not double-barrier quasibound resonance.
Chapter reference: Ch. 5 (5.3.36) and its threshold limit · ASD-BOOK-ddbade997b45

Theory · EFFECTIVE MODEL
ASD-WEB-QM-033
κ(x)=2m0[V(x)−E]ℏ2b(x),A(x)=C0b(x)κ(x),TWKB=Pexp⁡ ⁣[−2∫x1x2κ(x)dx]\kappa(x)=\sqrt{\frac{2m_0[V(x)-E]}{\hbar^2b(x)}},\quad A(x)=\frac{C_0}{\sqrt{b(x)\kappa(x)}},\quad T_{\rm WKB}=\mathcal P\exp\!\left[-2\int_{x_1}^{x_2}\kappa(x)dx\right]

TWKB names the first-order approximation, not exact T for any barrier; C₀ is an integration constant. Preserve divergence form and interface ψ,bψ′; require |κ′|/κ²≪1 and |b′|/(bκ)≪1, with turning-point matching.
Chapter reference: Ch. 5 (5.3.37)–(5.3.39) · ASD-BOOK-ddbade997b45

Theory · DERIVED
ASD-WEB-DYN-025
a=1m0γ[F−(v⋅F)vc2]=F∥m0γ3+F⊥m0γ\mathbf a=\frac1{m_0\gamma}\left[\mathbf F-\frac{(\mathbf v\cdot\mathbf F)\mathbf v}{c^2}\right]=\frac{\mathbf F_\parallel}{m_0\gamma^3}+\frac{\mathbf F_\perp}{m_0\gamma}

Fixed internal state, uniform background and no net radiation for the core–accompanying-field system; 0≤v<c.
Chapter reference: Final collected manuscript · Chapter 2 §III.3 (2.20) · ASD-BOOK-ddbade997b45

Theory · EFFECTIVE MODEL
ASD-WEB-GRV-047
G0=ξGc2ρ0ℓ02G_0=\xi_G\frac{c^2}{\rho_0\ell_0^2}

ρ₀ is background volume density, ℓ₀ an intrinsic length and ξ_G dimensionless; dimensional organization alone does not determine the coupling efficiency.
Chapter reference: Final collected manuscript · Chapter 2 §II.3 (2.7c) · ASD-BOOK-ddbade997b45

Theory · EFFECTIVE MODEL
ASD-WEB-GRV-048
∇⋅urot=0,∇2urot=−4πζG0c2jm,urot(x)=ζG0c2∫jm(x′)∣x−x′∣ d3x′,urotdip=ζG02c2J×rr3,ωrot=ζG0J2c2r3.\begin{aligned}\nabla\cdot\mathbf u_{\rm rot}&=0,\quad\nabla^2\mathbf u_{\rm rot}=-\frac{4\pi\zeta G_0}{c^2}\mathbf j_m,\\\mathbf u_{\rm rot}(\mathbf x)&=\frac{\zeta G_0}{c^2}\int\frac{\mathbf j_m(\mathbf x')}{|\mathbf x-\mathbf x'|}\,d^3x',\\\mathbf u_{\rm rot}^{\rm dip}&=\frac{\zeta G_0}{2c^2}\frac{\mathbf J\times\mathbf r}{r^3},\quad\omega_{\rm rot}=\frac{\zeta G_0J}{2c^2r^3}.\end{aligned}

Stationary localized source, no bulk translation, decay at infinity and the far-field dipole approximation; ζ=4 is calibrated, not independently predicted microscopically.
Chapter reference: Final collected manuscript · Chapter 2 §V.4 (2.72)—(2.77) · ASD-BOOK-ddbade997b45

Theory · EFFECTIVE MODEL
ASD-WEB-GRV-049
Krot=1−urot⋅k^/c1−urot2/c2,T[Γ]=1c∫Γdℓopt−1c2∫Γurot⋅dx,T−−T+=2c2∮Curot⋅dx=4ωrotACc2=2ζG0JACc4r3.\begin{aligned}K_{\rm rot}&=\frac{1-\mathbf u_{\rm rot}\cdot\hat{\mathbf k}/c}{\sqrt{1-u_{\rm rot}^2/c^2}},\\T[\Gamma]&=\frac1c\int_\Gamma d\ell_{\rm opt}-\frac1{c^2}\int_\Gamma\mathbf u_{\rm rot}\cdot d\mathbf x,\\T_- -T_+&=\frac2{c^2}\oint_C\mathbf u_{\rm rot}\cdot d\mathbf x=\frac{4\omega_{\rm rot}A_C}{c^2}=\frac{2\zeta G_0JA_C}{c^4r^3}.\end{aligned}

The optical path retains first order in weak-field rotation; the last two equalities require a coaxial circular loop at fixed radius and polar angle.
Chapter reference: Final collected manuscript · Chapter 2 §V.5 (2.78)—(2.82) · ASD-BOOK-ddbade997b45

Theory · EFFECTIVE MODEL
ASD-WEB-GRV-050
c2dMdt=Pin−Pout,Ah=16πG02M2c4,Nh=Aha∗,W=g∗Nh,Sbdy=kBln⁡W=kBσhAh,σh=ln⁡g∗a∗.\begin{aligned}c^2\frac{dM}{dt}&=P_{\rm in}-P_{\rm out},\qquad A_h=\frac{16\pi G_0^2M^2}{c^4},\\N_h&=\frac{A_h}{a_*},\quad W=g_*^{N_h},\\S_{\rm bdy}&=k_B\ln W=k_B\sigma_hA_h,\qquad\sigma_h=\frac{\ln g_*}{a_*}.\end{aligned}

Nonrotating uncharged quasistatic sequence at fixed G₀; counting model of approximately independent area cells and accessible states. The microscopic a_*, g_* and correlation corrections remain to be determined.
Chapter reference: Final collected manuscript · Chapter 2 §V.7 (2.84)—(2.87) · ASD-BOOK-ddbade997b45

Theory · EFFECTIVE MODEL
ASD-WEB-GRV-051
Tbdy=c632πkBσhG02M,dEbdy=Tbdy dSbdy,σh=c34G0ℏ ⟹ Tbdy=ℏc38πG0kBM,Cbdy=−Mc2Tbdy,dEbdy=Tbdy dS+ωbdy dJ+Vbdy dQ.\begin{aligned}T_{\rm bdy}&=\frac{c^6}{32\pi k_B\sigma_hG_0^2M},\qquad dE_{\rm bdy}=T_{\rm bdy}\,dS_{\rm bdy},\\\sigma_h=\frac{c^3}{4G_0\hbar}&\ \Longrightarrow\ T_{\rm bdy}=\frac{\hbar c^3}{8\pi G_0k_BM},\qquad C_{\rm bdy}=-\frac{Mc^2}{T_{\rm bdy}},\\dE_{\rm bdy}&=T_{\rm bdy}\,dS+\omega_{\rm bdy}\,dJ+\mathcal V_{\rm bdy}\,dQ.\end{aligned}

Fixed G₀, c and σ_h with E=Mc²; the temperature coefficient adopts the Bekenstein–Hawking reference normalization. The last line assumes differentiable E(S,J,Q), not an already solved rotating strong-field microstate.
Chapter reference: Final collected manuscript · Chapter 2 §V.8 (2.88)—(2.94) · ASD-BOOK-ddbade997b45