IDOS Energy, Coherence, and Supporting Dynamics — Part 64

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At a single occupied nonzero label $k$, set $j=\sum_i k_i$, $s=\#\{i:k_i>0\}$, $\delta=\mathbf1_{j=1}$, $\Sigma=2(j^2-|k|^2)$ and $v=1+s+\Sigma$. Mixing the origin with intensity $u$ gives the exact polynomials

$$V(u)=(1-u)(1-2u)^2+s(1-u)^3 +\delta(1-u)(3u^2-2u)+\Sigma(1-u)^4, \quad V_\phi(u)=(s-\delta)u(1-u)^2.$$

The constant unit has zero derivative along this path. So does the supplied mean-label unit: its depth descriptor stays fixed and its cutoff step is already one near $u=0$, whether or not that descriptor is a depth knot. The actual origin-direction energy coefficient is

$$\frac{Q_0}{\mu_k}=2+\lambda\left[ 6F_q(v)+F_q'(v)(-5-3s-2\delta-4\Sigma)+\eta(s-\delta)\right].$$

Direct inequalities, not a fitted finite scan, yield $Q_0/\mu_k\le2-4\lambda$ for $q=2$ and $Q_0/\mu_k\le2-\lambda(156+159\eta)$ for $q=4$. The latter is sharp at a nonunit pure-axis label. At a singleton $C=0$ and $\epsilon_k=1/(4\mu_k)$. The complete empty-pair equation is

$$\epsilon_k h_{tt}+2Jh_t+ (\epsilon_k A_0/4+2Q_0)h=0.$$

Removing its fixed spatial phase conjugates $A_0$ to $A_\rho=-\rho_0^{-1}\operatorname{div}(\rho_0\nabla)$ in $L^2(\rho_0dx)$. The closed weighted gradient form contains the constant in its operator domain with eigenvalue zero: smooth cutoffs approach it in the form norm because $\rho_0$ is integrable. No artificial boundary is introduced at the plateau join. Removing the gyro rotation with $h=e^{-Jt/\epsilon_k}y$ gives

$$y_{tt}+[A_0/4+\epsilon_k^{-2}+2Q_0/\epsilon_k]y=0, \quad\Delta=1+2\epsilon_kQ_0.$$

Consequently $\Delta\le-2(\lambda-1)<0$ for $q=2$ and $\Delta\le2-78\lambda-(159/2)\lambda\eta<0$ for $q=4$. Every non-origin singleton has an actual growing weighted spatial ground component. The companion phase, support, flow and frame block is retained, with its positive flow margin $(329/424)\rho_0\Omega I$. The origin has positive empty coefficients and neutral companion symmetry directions; this is not a nonlinear stability theorem for an electron.

The old constant-$\epsilon$ action has a distinct alternative: $\Delta>0$ gives oscillation, $\Delta=0$ the Jordan threshold and $\Delta<0$ exponential growth. A negative energy coefficient alone does not exclude gyro stabilization. The new coefficient is an explicit change of action that removes this stabilization for every non-origin singleton. It has not classified all unwanted multimode states or selected the observed particle indices. Its growth rates are not measured decay lifetimes.

Exact Initial Constraints And The Nonlinear Scope

For a retained diagonal phase generator $T$, write $q_T=\langle e,Te\rangle$ and $H_T=J(T-q_T)e$. The exact current is

$$j_T^\beta=\rho q_T[U^\beta-P^{\beta\gamma}h_{A,\gamma}] +\rho_0\epsilon(p)\left[U^\beta\langle K,H_T\rangle -\tfrac14P^{\alpha\beta}\langle\mathscr R_\alpha,H_T\rangle\right].$$

At $\eta=0$ all pair phases are symmetries; at positive $\eta$ use the retained affine family. The common phase has $H_I=0$. Coefficient derivatives

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