IDOS Energy, Coherence, and Supporting Dynamics — Part 35

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$C[(1+w/a)^{10}-1]<\Delta/4$, then take $|X|$ large enough that $\rho_+/\rho_-<2$. The energy difference is strictly negative and $Q$ unchanged. Weighted carrier $H^1$ and velocity $L^2$ differences are bounded by a constant times $\rho_+(a+w)^{10}$ in squared norm, so tend to zero. The same holds in physical-field $H^1$ and velocity $L^2$, with density velocity reconstructed by continuity.

Consequently these higher prefixes are not local fixed-$Q$ energy minima in that specified weighted topology. The construction is not uniformly small in normalized carrier amplitude: it changes the pattern by order one in a weak tail patch. It therefore does not contradict the positive directional Hessian. It imposes no additional momentum or individual population constraint and proves no trajectory toward the competitor, instability rate, nucleation process or observed decay. At every maintained depth $N_H\ge2$ permits $N_L=1$; the single shared origin clock $\mu_3$ can be used instead only when its energy ordering actually gives a gain. The lowest branch is not proved minimal. tail-band-energy/report.md and the independent 255-check tail-energy-review/report.md supply the complete initial-data construction and bounds.

Flatter Cores With Full Coupled Linear Control

The earlier flat-core density can retain the complete occupied/support/ frame/flow energy estimate if the common support kinetic and gradient coefficients change together. Keep the geometric auxiliary profiles $W=(1+t)^{-4}$ and $Z_i=(x_i/L)(1+t)^{-5}$, $t=|x|^2/L^2$, and set

$$\rho=n_{\rm peak}F_k(W)=\frac{n_{\rm peak}}{(1+t^k)^2},\quad F_k(W)=[1+(W^{-1/4}-1)^k]^{-2},\quad \mu(W)=C\frac{n_{\rm peak}F_k(W)}{W^2},\quad C=\frac{L^2\omega_{\min}}{212}.$$

For each fixed integer $k\ge4$, use $M_W=M_Z=\mu(W)$ and $\gamma_A=\gamma_B=\mu(W)/4$ in SF-DW, retaining its vector residual factor $(1+W^2)^{-1}$ and its other derivative weights. This is a changed common constitutive action; $k$ is shared across depths and rungs. All changed terms are residual squares that vanish on the exact centered branch, so their full first variations and stress contributions vanish there. The current, stress and complete boosted energy retain $j=\rho U$, $T^{\mu\nu}=\rho\omega U^\mu U^\nu$ and $E=\Gamma S\omega$, with

$$S=\frac{n_{\rm peak}\pi^5L^{10}}{24k}B(5/k,2-5/k).$$

Varying $k$ at fixed $S,L$ therefore requires one common peak normalization, not a separate amplitude fit for each particle.

The coupled drift ratio becomes $\mu D_s^TD_s/\rho=C D_s^TD_s/W^2$. The existing rational-profile bound restores the same positive $223\omega/424$ Schur margin at every finite $k$. Constant nonzero support inertia would instead give a radial tail ratio of order $r^{4k-18}$, which diverges for $k\ge5$. This diagnoses failure of the old sufficient energy bound, not by itself a growing mode.

Normalize support perturbations by $\zeta=\sqrt\mu\,y$ and the other coordinates by their $\sqrt\rho$ kinetic weights. The actual entropy variation is $\alpha=(F_k'/F_k)s$. The phase/support gyro coefficient becomes

$$c_k(x)=\sqrt{\beta/C}\,W F_k'/F_k,\qquad W F_k'/F_k=\frac{k}{2}\frac{(1+t)t^{k-1}}{1+t^k}, \qquad 0\le c_k\le k\sqrt{\beta/C}.$$

The support/flow coefficient is $\sqrt{C/\kappa_U}D_s/W$, uniformly bounded in space.

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