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spatial phase (k−qH)⋅x/ℓ(k-q_H)\cdot x/\ell, with qH,i=NH/(d+1)q_{H,i}=N_H/(d+1) on active axes. Define the smooth step S(s)=e−1/s/[e−1/s+e−1/(1−s)]S(s)=e^{-1/s}/[e^{-1/s}+e^{-1/(1-s)}] inside 0<s<10<s<1, extended by zero and one. Then

e=cos⁡θ uL+sin⁡θ uE,θ=θH[1−S(R−2)+S(R−6)],θH=arccos⁡Lc/D. e=\cos\theta\,u_L+\sin\theta\,u_E,\qquad \theta=\theta_H[1-S(R-2)+S(R-6)],\qquad \theta_H=\arccos\sqrt{L_c/D}.

The high carrier occupies R≤2R\le2 and R≥7R\ge7, the low carrier 3≤R≤63\le R\le6, and two smooth width-one layers connect them. This is a field preparation, not a stationary solution or an external physical wall. Writing t=cos⁡2θ−Lc/Dt=\cos^2\theta-L_c/D, the exact pair weights and descriptors are

u=1D+tLc,v=1D−tEc,qi=qH,i−ct,ai=−ct/ℓ,adescriptor=d,N=NH−(d+1)ct. u=\frac1D+\frac{t}{L_c},\quad v=\frac1D-\frac{t}{E_c},\quad q_i=q_{H,i}-ct,\quad a_i=-ct/\ell, \quad a_{\rm descriptor}=d,\quad N=N_H-(d+1)ct.

The full horizontal spatial residual has ∑i∣Ri∣2=θ′2\sum_i|R_i|^2=\theta'^2. With X=d(NL+d−1d−1)X=d\binom{N_L+d-1}{d-1} crossing predecessor edges, the actual potential is

V=(Ec+X)v(u−v)2+ηXvu(v−u)2. V=(E_c+X)v(u-v)^2+\eta Xvu(\sqrt v-\sqrt u)^2.

Affine phases do not remove the amplitude mismatch in the second term. The within-shell redistribution terms vanish on this preparation. At NL=0N_L=0 the separate formula is

V=(Ec+d)v(u−v)2+ηd[(NH+d−1d)−1]v2(v−u)2. V=(E_c+d)v(u-v)^2+ \eta d\left[\binom{N_H+d-1}{d}-1\right] v^2(\sqrt v-\sqrt u)^2.

The canonical smooth origin interpolation is used there, preserving its single origin clock μ3\mu_3. It cannot be replaced by a depth-specific origin value when unequal sector scales are restored.

Choose χ=e−1/(1−R2)\chi=e^{-1/(1-R^2)} for R<1R<1, zero otherwise, and retain its separate moments Q1=∫ρχQ_1=\int\rho\chi and Q2=∫ρχ2Q_2=\int\rho\chi^2. Let A=∫ntA=\int nt. Low pair deficits are −A/Lc-A/L_c and extra deficits are A/EcA/E_c; all affine label deficits are cAcA. Exact high-core horizontal velocities restore these charges. Their energy is

Ephase=Q2A22ϵQ12F,Fpair=D2LcEc,Faffine=dc2AH,AH=NH(NH+d+1)(d+1)2(d+2). E_{\rm phase}=\frac{Q_2A^2}{2\epsilon Q_1^2}F,\qquad F_{\rm pair}=\frac{D^2}{L_cE_c},\qquad F_{\rm affine}=\frac{dc^2}{A_H},\quad A_H=\frac{N_H(N_H+d+1)}{(d+1)^2(d+2)}.

The pair branch applies at η=0\eta=0 and the affine branch at positive η\eta. For NL≥1N_L\ge1 their ratio is d(d+2)(NH−NL)2Lc/[EcNH(NH+d+1)]<1d(d+2)(N_H-N_L)^2L_c/[E_cN_H(N_H+d+1)]<1. This orders the compensation coefficients, not total energies between actions whose occupation potentials also differ.

The density momentum covector is ∫nai=−cA/ℓ\int na_i=-cA/\ell; restoring the phase charges adds exactly +cA/ℓ+cA/\ell. Radial symmetry makes all 45 ordinary angular charges vanish. Thus scalar recoil and frame rotation are exactly unnecessary in this family. The full initial energy reduces to

EQ=Ewall+∫[Hr+b(rlog⁡r−r+1)] dμ+C2y2,dμ=ρ dxQ,r=nρ, \frac EQ=E_{\rm wall}+\int[Hr+b(r\log r-r+1)]\,d\mu+\frac C2y^2, \quad d\mu=\frac{\rho\,dx}{Q},\quad r=\frac n\rho, ∫r dμ=1,y=∫rt dμ,H=h+γV,C=j2Fϵj12,ji=Qi/Q,Ewall=ϵ8∫θ′2dμ. \int r\,d\mu=1,\quad y=\int rt\,d\mu,\quad H=h+\gamma V,\quad C=\frac{j_2F}{\epsilon j_1^2},\quad j_i=Q_i/Q,\quad E_{\rm wall}=\frac\epsilon8\int\theta'^2d\mu.

The flattened count clock is evaluated throughout each layer. The wall and compensation use ρ\rho, not nn. The common phase velocity is

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