IDOS Energy, Coherence, and Supporting Dynamics — Part 37

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with all fixed-order derivatives bounded independently of its translated center. The exact energy and norm estimates imply, for one constant $c_*>0$ independent of $X$,

$$\|v_X\|_X\longrightarrow0,\qquad \frac{F(e(v_X))-F(e_H)}{\|v_X\|_X^2}\le-c_*.$$

Any bounded second-order expansion agreeing with the fixed-direction variations would instead have a nonnegative quadratic term and a remainder $o(\|v\|_X^2)$, a contradiction. The argument also excludes an agreeing $C^2$ extension in each fixed finite integer derivative norm with the same $\rho$ weight. It does not assert that an entire weighted Sobolev ball is an injective pointwise sphere chart, or address every possible nonlinear phase-space chart. Stronger norms imposing uniform relative smallness are outside the argument. No dynamical instability or failure of electron persistence follows. The principal weighted-energy-taylor.md and its independent 64-check review supply the exact formulation.

Low Frequencies In The Complete Coupled System

The constant-inertia coupled system has disturbance frequencies arbitrarily close to zero at every maintained depth, including ten. This strengthens the unused-frame comparison: it retains occupation, phase, support, frame and flow. It does not classify the entire spectrum or prove nonlinear stability.

For the normalized common-phase coordinate $\tau$, an escaping packet $\tau_R=R^{-5}\chi((x-R^2e_1)/R)$ with fixed unit-norm smooth compact $\chi$ has $q[\tau_R]=O(R^{-2})$ and $\|A\tau_R\|=O(R^{-1})$. The other coordinates initially vanish. Static zero states have occupation and flow zero and $D\tau=0$, where $D=\nabla-\nabla\log\sqrt\rho$; their phase part is therefore proportional to $\sqrt\rho$. Subtracting that exact phase zero state makes the packets orthogonal to the entire kernel without changing their limiting residual. Thus the static operator has zero in its essential spectrum even after its kernel is removed.

That fact alone would not prove positive disturbance frequencies tending to zero. For example, $A(k)=\operatorname{diag}(ak^2,0)$ with a nonzero two-coordinate skew gyro coupling has an exact zero branch and a positive branch bounded away from zero. The actual full pencil must be tested.

At spatial infinity the normalized coefficients have constant limits. The support/flow derivative coupling and logarithmic density gradient vanish, while the phase/support gyro coefficient remains $c=2\sqrt{n_{\rm peak}\beta/M_W}>0$. The support square retains

$$\gamma_A\left|ik\zeta_W/\sqrt{M_W} +8\zeta_Z/(L\sqrt{M_Z})\right|^2.$$

The other terms retain the carrier/frame derivative square, positive occupation Hessian, phase-edge Hessian, frame/flow gradients and the positive phase-flow completion. In the full Hermitian pencil $A_\infty(k)+i\Omega G_\infty-\Omega^2I$, at every nonzero $k$ the static kernel is the one support direction $\zeta_Z=-iL\sqrt{M_Z}k\zeta_W/(8\sqrt{M_W})$. The gyro compression on this kernel is zero; eliminating its positive complement leaves a strictly negative coefficient of $\Omega^2$. Hence the pencil has one negative eigenvalue for sufficiently small positive $\Omega$ at fixed nonzero $k$.

At $k=0$, write $B_o$ for the phase Hessian and $m_0=\dim\ker B_o+\binom{10-d}{2}$. The phase/support gyro pair has one positive and one negative eigenvalue near zero frequency. The remaining $m_0$ soft coordinates add negative directions. For positive phase edges, $\dim\ker B_o=d$, giving $m_0=30,24,19,15,12,10$ across depths $2,3,4,5,6,10$. For zero phase edges replace $d$ by $D-1$. Continuity at any sufficiently small fixed positive frequency, followed by sufficiently small nonzero $k$, therefore forces at least $m_0$ positive-frequency crossings counted with multiplicity. Their frequencies can approach zero. This argument retains both edge cases and does not infer dynamics from the static Rayleigh quotient.

For each fixed root, multiply its constant-limit plane wave by a widening cutoff translated farther into the tail. Actual coefficients and the necessary derivatives converge on its support, giving a weakly vanishing approximate eigenvector $(u,i\Omega u)$ of the closed energy-space generator. A closed-range estimate with finite-dimensional kernel would contradict these packets. Thus these imaginary frequencies belong to its Fredholm essential spectrum. They are not claimed to be localized eigenmodes. The finite-carrier proof is in coupled-zero-threshold/report.md and the independent 43-check coupled-threshold-review/report.md.

Flat-Core Approximate Standing Disturbances

For each fixed integer $k\ge4$, the changed flat-core density $\rho_k=n_{\rm peak}/(1+t^k)^2$, $t=r^2/L^2$, gives the exact scalar

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