IDOS Energy, Coherence, and Supporting Dynamics — Part 70

Contents · Previous · Next

$e^{\Gamma t}e^{-J(t/\epsilon_0+x_1/\ell)}$ and zero companion components attains the physical rate. No isolated spectral gap is used.

There is an exact initial curve on the original energy and every phase, translation and spatial-rotation charge level. Put $P_a=a^2$ and initially take $c_k=\sqrt{1-P_a}$, $c_0=a e^{-Jx_1/\ell}$, all other pairs zero, with the exact rest support, frame and flow. Define

$$r_a=(1-P_a)^2+P_a^2,\qquad V_a=2(1-P_a)(1-2P_a)^2,\qquad H_a=r_a^{-1}+2(V_a+V_a^2)r_a^{-3},$$ $$\epsilon_a=\frac1{4[1+P_a(1-P_a)/\ell^2]} \quad\hbox{or}\quad\frac14,\qquad \kappa_a=8\quad\hbox{or}\quad8r_a,$$ $$\mathcal D_a= \frac{\sqrt{1+4\kappa_a^2P_a^2/\ell^2}-1}{4\kappa_a}, \quad \bar\omega=H_a+\mathcal D_a,\quad \omega_{\rm rel}=-\frac1{\epsilon_a(1-P_a)}.$$

The displayed mean and relative phase rates set $F=0$, hence $\rho=w_0$, and give $j_0^0=0$, $j_k^0=w_0$ pointwise. Only the empty-origin phase has a spatial gradient, so its zero canonical charge gives zero momentum; the full spatial angular momentum also vanishes. The common phase gradient still contributes $\mathcal D_a$ to energy and is not discarded. For amplitude rate $\dot a=\gamma a$, exact energy conservation initially requires

$$\gamma^2(a)= -\frac{2(1-P_a)(H_a+\mathcal D_a-13)}{\epsilon_a P_a} -\epsilon_a^{-2}.$$

This is smooth at zero, with limit $192$. Both signs $\gamma_\pm=\pm\sqrt{\gamma^2}$ exist for sufficiently small signed $a$. These relative disturbances are noncompact, but their physical fields and complete energy changes are finite because $w_0$ is integrable.

The nonlinear comparison keeps the quasilinear derivative loss. First fix the small chart and error/approximation balls, hence their difference-estimate constant $K_s$. Choose finite order $N$ with $(N+1)\Gamma>K_s$ and construct the Taylor approximation $\sum_{j=1}^N a^jY_j(t)$ using the actual equations and exact initial curve. The regularity ladder $r_j=s+N+1-j$ leaves one extra derivative for each recursive source and for the final approximation. The integral

$$\int_0^\infty(1+t)^{14}e^{-(j-1)\Gamma t}\,dt<\infty \qquad(j\ge2)$$

gives $\|Y_j(t)\|_{r_j}\le C_j e^{j\Gamma t}$. The symmetrized difference estimate then yields

$$\|Y-Y_{\rm app}\|_s \le C_N(|a|e^{\Gamma t})^{N+1}, \qquad 0\le t\le T_a=\Gamma^{-1}\log(\theta/|a|),$$

after choosing a fixed sufficiently small $\theta>0$ last. This order of choices avoids circular dependence of $N$ on the approximation. The full solution continues through $T_a$, even as $T_a\to\infty$ when $a\to0$. At that time the origin component has size at least $\theta/2$, and $\rho\ge w_0/2$. For the classical field $\Psi=\sqrt{2\rho}\,e$,

$$\|\Psi_0(T_a)\|_{L^2(dx)} \ge\frac{\theta}{2}\sqrt{\mathcal A}, \qquad \|\Psi_0(0)\|_{L^2(dx)} =\sqrt{2\mathcal A}\,|a|\longrightarrow0.$$

Every translated, rephased or physically frame-rotated member of that singleton family has zero origin component. This is fixed physical departure from the unwanted branch. The positive velocity sign gives forward departure; the negative sign gives the corresponding backward-time statement, with no forward departure inferred for its contracting tangent. The full Noether levels remain conserved: their cutoff flux errors vanish with the integrable tail, including the one additional spatial factor in the angular current.

What The Construction Establishes

The common tail and the uniform vector square preserve exact prefix frequencies, equal participating allocation, complete stress and common action normalization across the maintained depths. The uniform positive chart includes the complete support, density, frame, flow and finite carrier. The fixed-departure theorem above has the narrower constant-unit $k=e_1$, square-selector and free-phase scope. It does not automatically extend to other singletons, unequal units, $q=4$, occupied multimode branches, the original vector square, vacuum or infinite carrier rank. An instability's eventual channel, products and lifetime remain separate questions; the prototype is not identified with an observed particle.

The strict ordinary-volume clock remains uniform only in the flat core: $\mathcal H/\sigma=\Omega$ there and $\mathcal H/\sigma=(w_0/\sigma)\Omega$ in the positive exterior. The exterior is still part of the object. These support improvements do not supply a participation gate, physical finite-component realization, selected indices or a derivation of the unequal sector units. They make the mathematical compatibility and its evolution domain more concrete, without adding empirical evidence or selecting depth five. Real field components remain distinct from literal electron position in six ordinary spatial dimensions.

The uniform-vector independent checker has 1,325 assertions, including 655 literal empty-label cases, one exact matrix and four rigorous real enclosures. Finite checks support the displayed proofs; they do not measure physical evidential strength.

Occupied-Prefix Persistence And Compact Singleton Departure Under One Law

The preceding polynomial-density and uniform-vector completion now admits two complementary nonlinear statements. Every fixed occupied simplex prefix has a quantitative small-disturbance persistence interval. Every non-origin singleton under the actual covariance-dependent horizontal law has some compact relative disturbances that preserve its complete conserved levels and cause a fixed physical departure. These statements concern different branches of the same supplied action. They distinguish a class of patterns dynamically without selecting the observed occupied indices.

One Fixed Action And Its Two Branch Families

Fix the whole finite carrier $\mathcal K_M=\{k\in\mathbb N^{10}:|k|\le M\}$, all action parameters and

Contents · Previous · Next