IDOS Energy, Coherence, and Supporting Dynamics — Part 54

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with explicit label enumeration and a different core partition confirms all 72. These certify feasible relaxed comparisons, not the optimum. Smooth recovery proves negativity for sufficiently small positive $\tau$, with a common sufficiently small value for this finite set, but this argument alone supplies no numerical threshold. The original-action $\tau=1$ result remains 48 of 72 numerical comparisons.

principal/density-convex-envelope.md, its two executed checkers and the independent 665-check review preserve the theorem and distinct numerical and rigorous evidence. No trajectory, decay, persistence, observed mass or special depth-five mechanism is established by this variational limit.

A Smooth Uniform Core With Its Complete Positive Exterior

The strict ordinary-volume clock and the conserved-charge clock are distinct premises. On the retained coherent branch, the actual frequency is $m>0$, $h=\rho m$ and $E=mQ$. Pointwise $m=h/\sigma(\rho)$ requires $\sigma(\rho)=\rho$ throughout the positive density range. A continuous, nonzero, integrable energy density cannot equal one positive constant on its entire positive set in connected $\mathbb R^{10}$: that set would be both open and closed. A finite participating region remains possible, but excluding a positive exterior requires an additional dynamical gate.

A changed common constitutive law instead retains independent auxiliary $q>0$, uses $q_0(W)$ for the reference charge, and replaces the nonclock SF-DW derivative weights by $q_0$. Define

$$\psi(s)=\begin{cases}0&s\le0,\\s e^{-1/s}&s>0,\end{cases} \quad f(s)=e^{-\psi(s)},\quad s=\frac{W^{-1/4}-2}{\varepsilon_p}, \quad q_0=\sigma f(s),\quad \mu=\frac{C_s q_0}{W^2},\quad C_s=\frac{L^2\omega_{\min}}{212}.$$

Here $\sigma,\varepsilon_p,L>0$ are common constants. On the unchanged rational geometry $W=(1+t)^{-4}$, $Z=(x/L)(1+t)^{-5}$, $t=|x|^2/L^2$, this gives $s=(t-1)/\varepsilon_p$. The charge is exactly $\sigma$ for $|x|\le L$, strictly positive outside, and asymptotically $\sigma e^{1+1/\varepsilon_p}e^{-|x|^2/(\varepsilon_p L^2)}$. Both $q_0$ and $\mu$ extend smoothly and flatly through $W=0$; the join at $W=1/16$ is $C^\infty$, rather than the earlier plateau's $C^7$.

Use $a_\alpha=\langle Je,\partial_\alpha e\rangle$, $P=g+UU$, $A=Pa$, $v_*=1/2$, and

$$h_c=m+\frac{v_*^2}{\kappa}\left(\sqrt{1+\kappa^2 A^2/v_*^2}-1\right), \quad F=U\cdot a-h_c-\gamma V,\quad \mathcal L=qF-b[q\log(q/q_0)-q+q_0]+\mathcal L_{\rm res}+\mathcal L_{\rm con}.$$

The retained support residuals use temporal weight $\mu$ and spatial weights $\mu/4$, including the vector factor $(1+W^2)^{-1}$; their algebraic square retains its original positive coefficient. Carrier, frame and

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