IDOS Energy, Coherence, and Supporting Dynamics — Part 42

Contents · Previous · Next

mixing remains: eliminating the complement contributes $O(r^8)$ for fixed positive phase edges and $O(r^6)$ for zero edges, both smaller than the displayed term. Uniformity holds on bounded $a$ sets at fixed carrier and coefficients, not through a vanishing positive edge gap or an infinite carrier limit. At $N=1$, a tilt along the parallel covariance axis gives separate coefficients $B_j+|a|^2 A_j$, with $B_\parallel/A_\parallel=16(d+1)^2$ and $B_\perp/A_\perp=16(d+1)(d+2)$. This is a wavelength-dependent disturbance crossover, not a particle encounter acceptance angle. There is no unused direction at depth ten. The principal directional-crossover.md and independent 112-check review establish this limit.

Finite Replacement Fractions And Charge Costs

The small-tail comparison does not decide the cost of replacing a finite fraction of a pattern. A two-region algebraic control separates that question from the smooth-wall problem. Give the high and lower uniform prefixes background weights $1-t,t$, keep $n=\rho$, and initially omit their joining layer. Let $D_H,D_L$ be their counts, $\Delta=m_H-m_L>0$, and

$$A_N=\frac{N(N+d+1)}{(d+1)^2(d+2)},\qquad B=\frac{d(N_H-N_L)^2}{(d+1)^2}.$$

At fixed occupations, the phase-compensation Gram matrix is the integral of the local covariance. It is not the covariance of the spatially averaged occupations. The exact minimum horizontal kinetic energies per $Q$ are

$$\mathcal C_{\rm pair}(t)= \frac{t^2(D_H/D_L-1)}{2\epsilon(1-t)},\qquad \mathcal C_{\rm affine}(t)= \frac{t^2B}{2\epsilon[(1-t)A_H+tA_L]}.$$

For independent pair phases, complete proper-prefix replacement cannot preserve the deleted pair charges: their canonical pairing vanishes when the pair amplitude is exactly zero. The displayed cost diverges as $t\to1$. For affine charges, the endpoint cost is finite if $N_L>0$; the origin $N_L=0$ has a singular covariance and cannot restore the nonzero deficit. These statements distinguish the two phase-edge choices.

Put $s=2\epsilon\Delta$ and $R=D_H/D_L-1$. The bulk energy $-\Delta t+\mathcal C(t)$ is negative precisely for positive $t$ below

$$t_{\rm pair}=\frac{s}{R+s},\qquad t_{\rm affine}=\frac{sA_H}{B+s(A_H-A_L)}.$$

The affine threshold can exceed one. On the flattened positive-prefix law, $x_d=2\epsilon\mu_d$ gives $t_{\rm pair}=x_dD_L/(1+x_dD_L)$, independent of $N_H$. For a zero prefix use the actual shared origin clock $\mu_3$; the same-depth simplification applies only when its clock really equals that sector unit. These sharp-region formulas are not admissible finite-gradient spatial preparations or universal bounds on smooth walls. A wall changes both the deficit and the covariance.

There is a separate smooth full-space endpoint for positive phase edges and $N_L>0$. Use the centered lower texture everywhere, retain the rational support, and add $K=H_L\operatorname{Cov}_L^{-1}(q_H-q_L)/\epsilon$. Then $e_t=m_LJe_L+K$ preserves $Q$ and the original affine charges. It carries momentum $p=Q(q_H-q_L)/\ell$, which the scalar support velocity $W_t=-(p/g)\cdot\nabla W$ cancels, with $g=Q\omega_{\min}/159$. Radial symmetry makes the integrated carrier angular moment zero, and scalar recoil has zero angular density. The complete endpoint energy is

$$\frac{E_{\rm end}}Q=m_L+\frac{B}{2\epsilon A_L} +\frac{159B}{2\omega_{\min}\ell^2}.$$

It is lower than the high branch exactly when the two positive costs sum to less than $\Delta$. Equality and the reversed inequality are retained; the unresolved physical $\ell$ has not been fitted to force success. With $\epsilon=\omega_{\min}\ell^2/(4M^2)$ the condition becomes

$$x_d>\frac{B}{D_H-D_L} \left(\frac1{A_L}+\frac{159}{4M^2}\right).$$

This finite-energy endpoint is not close to the high branch pointwise at infinity, is not a stationary compensated particle, and supplies no connecting trajectory. Both entropy prescriptions give the same initial energy because $n=\rho$; their evolution need not agree. The source finite-fraction-phase-cost/report.md has 4220 author checks and an independent 483-check review, including eleven maintained same-depth pairs and thirteen positive-prefix controls. No second tau, integer bottom assignment or photon mass branch is introduced.

Density Redistribution At Fixed Total Charge

Contents · Previous · Next