IDOS Energy, Coherence, and Supporting Dynamics — Part 74

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phase densities are exactly $j_{l_*}=0$, $j_k=w_0$, and the auxiliary density is exactly $w_0$. The common spatial phase covector is nonzero and its cost $\mathcal D$ must not be discarded.

Let $\chi_R$ be a nonnegative smooth correction cutoff strictly inside the transition annulus where $f_R$ is bounded away from zero. Define

$$\mathcal C=\int\frac{w_0\varepsilon_h(P)a^2|\nabla f_R|^2} {8(1-P)}\,dx,\qquad \mathcal K=\int\frac{w_0\varepsilon_h(P)g(P)a^2f_R^2\chi_R} {2(1-P)}\,dx,$$ $$\delta_R=\mathcal C/\mathcal K,\qquad \dot A=\nu A\sqrt{g(P)}\sqrt{1-\delta_R\chi_R}, \qquad\nu\in\{+1,-1\}.$$

The velocity reduction removes exactly $\delta_R\mathcal K=\mathcal C$ of kinetic energy, cancelling the cutoff gradient cost. After cancelling $a^2$ and rescaling the annulus, $\mathcal C/a^2=O(R^{-8})$ and $\mathcal K/a^2\asymp R^{-6}$. Thus $\delta_R=O(R^{-2})$ with every required fixed amplitude derivative, and the curve extends smoothly to $a=0$. No division by a vanishing outer cutoff is used.

Radial parity makes all ten total momenta zero; the radial momentum density has zero angular moment, and the full initial support, frame and flow time momenta vanish. The complete energy and all 45 angular momenta match the branch. At $\eta=0$ every independent pair-phase charge matches. At $\eta>0$ only the actual affine phase combinations are asserted to be conserved; no broken pair symmetry is promoted to a conservation law.

The inherited full relative group bound remains $C_s(1+|t|)^{14}e^{\Gamma_*|t|}$. The compact first coefficient has an $O(R^{-2})$ velocity/forcing error, giving

$$\|Y_{1,R}(t)\|_s\le C_s e^{\Gamma_*t} [1+R^{-2}(1+t)^{15}].$$

Choose $T_a=\Gamma_*^{-1}\log(\theta/|a|)$ and $R(a)=R_*(1+T_a)^8$. A finite Taylor hierarchy with orders $r_j=s+N-j+2$ retains the extra derivative needed by the quasilinear comparison. First fix the exact and approximate chart balls and their comparison constant $K_s$; then choose $(N+1)\Gamma_*>K_s$; only then decrease $\theta$. The full-space error is $O((|a|e^{\Gamma_*t})^{N+1})$ through $T_a$. This controls both annuli before finite propagation is used to identify the fixed core.

Consequently, for a fixed core $B_b$ and $\mathcal A_b=\int_{B_b}w_0>0$,

$$\|Y_a(0)\|_{s,\rm ordinary} \le C|a|[1+\Gamma_*^{-1}\log(\theta/|a|)]^{40}\longrightarrow0,$$ $$\|\Psi_{l_*}(T_a)\|_{L^2(B_b)} \ge\frac\theta2\sqrt{\mathcal A_b}>0\quad(\nu=+1). \tag{S1}$$

The branch family used for this comparison retains the empty label; internal label permutations are not included. The negative radial sign gives the corresponding statement at $-T_a$, not a claim of forward departure along its contracting tangent. The compact support radius grows as $a\to0$; a common fixed radius is not established.

A Common Perturbation-Budget Comparison

The two theorems can be compared without identifying their distinct time constants with measured lifetimes. Fix one admissible action, one prefix, one non-origin singleton in its growing alternative, a common finite carrier and $s\ge10$. Fix their chart equivalence constants. Define a permitted initial budget

$$\epsilon(a)=C_0|a|(1+T_a)^{40},$$

with $C_0$ large enough to bound the singleton's ordinary initial norm. An ordinary Sobolev bound controls the corresponding uniformly local norm; fixed changes of chart affect only constants. Take arbitrary compatible prefix data whose relative norm is at most this same budget.

For sufficiently small $|a|$, $\epsilon(a)(1+T_a)^{12}=O(|a|(1+T_a)^{52})\to0$, so (P1) holds through the singleton departure time and gives

$$\sup_{|t|\le T_a}B_s^{\rm prefix}(t) \le C|a|(1+T_a)^{49}\longrightarrow0. \tag{C1}$$

At the same positive time, the specially prepared singleton has the fixed physical lower bound (S1). Furthermore its logarithmic time is smaller than the prefix's guaranteed sufficient horizon in this budget:

$$\frac{T_a}{T_{\epsilon(a)}} =O\left(|a|^{1/12}[\log(1/|a|)]^{13/3}\right) \longrightarrow0. \tag{C2}$$

This is a comparison of a universal small-data persistence bound near a fixed prefix with existence of departing data near a fixed singleton.

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