IDOS Energy, Coherence, and Supporting Dynamics — Part 38

Contents · Previous · Next

weighted operator

$$\sqrt{\rho_k}A_k\frac1{\sqrt{\rho_k}}=-\Delta+V_k,\qquad V_k=\frac{4k t^{k-1}[(k-4)t^k-(k+4)]} {L^2(1+t^k)^2}.$$

Its domain is $\{f:\sqrt{\rho_k}f\in H^2\}$, its spectrum and essential spectrum are $[0,\infty)$, and its only eigenvalue is the simple constant zero state. The positive-eigenvalue exclusion follows by spherical-harmonic reduction: the transformed radial potential is $[(\ell+4)^2-1/4]/r^2+V_k$, integrable on every exterior half-line. At positive energy, variation of constants has an invertible limiting matrix, so a nonzero solution has a nonzero oscillatory amplitude and cannot be square-integrable. No classification of singular continuous spectrum is needed for this statement.

Nevertheless there are approximate standing disturbances with controlled finite-time errors. Normalize a spherical harmonic by $\int_{S^9}|Y_\ell|^2=1$, set $a=\ell(\ell+8)$, and choose a fixed positive root of $qJ'_{\ell+4}(q)-4J_{\ell+4}(q)=0$. Define

$$g(s)=\frac{s^{-4}J_{\ell+4}(qs)}{J_{\ell+4}(q)},\quad f(x)=\begin{cases}g(r/L)Y_\ell,&r\le L,\\Y_\ell,&r>L,\end{cases} \qquad \lambda=q^2/L^2.$$

Since $g(1)=1$ and $g'(1)=0$, this continuation is $C^1$ across the matching sphere and lies in the actual operator domain. The sphere is a comparison surface, not a physical wall. Its residual $R=(A_k-\lambda)f$ is exactly

$$L^2R/Y_\ell=\begin{cases} 4k s^{2k-1}g'(s)/(1+s^{2k}),&s<1,\\ a/s^2-q^2,&s>1. \end{cases}$$

The ball norm identity is $\int_0^1s^9g^2ds=(1-a/q^2)/2>0$. Evaluating the inner and outer transition layers gives, at each fixed mode,

$$\frac{\|R\|^2}{\|f\|^2} =\frac{2Cq^2(q^2-a)}{kL^4}+o(k^{-1}),\qquad C=\frac{\pi^2}{18}+\frac{\log2}{2}-\frac7{12}.$$

For example the inner layer integral is $\int_0^\infty u^2e^{2u}(1+e^u)^{-4}du=\pi^2/36-1/6$; the outer layer contributes $(\log2-1/2)/2$ to $C$. These are fixed-mode asymptotics, not a uniform high-mode limit. Physical amplitude and $n_{\rm peak}$ cancel in the relative residual.

For the scalar wave $u_{tt}+c_F^2A_ku=0$ with data $(f,0)$, spectral calculus yields

$$\frac{\|u(t)-\cos(c_F\sqrt\lambda t)f\|}{\|f\|} \le\frac{c_F|t|\|R\|}{\sqrt\lambda\|f\|}.$$

The corresponding energy-error norm is bounded by $c_F^2|t|\|R\|$. After $m$ periods, the relative bound is $2\pi m\|R\|/(\lambda\|f\|)$, so the guaranteed number of cycles at fixed tolerance grows as $\sqrt k$. This is an approximation interval, not a calculated resonance width or actual decay time. For the first positive radial mode, computed relative residuals at $k=174,1803$ are approximately $0.0601746,0.0186006$; the ten-percent bound certifies only $0.2645,0.8556$ cycles. For the first $\ell=1$ mode at $k=1803$, it certifies about $1.945$ cycles. These are checked double-precision quadratures, not interval-certified values. A nearly flat density alone therefore does not certify long coherence.

The scalar wave transfers exactly to frame planes with two unused axes, giving $28,21,15,10,6,0$ invariant linear channels at the six depths. It does not transfer to the depth-ten coupled block, and the constant-inertia full-system threshold proof above is a separate result. No Bessel root is identified with the supplied simplex particle clock. The author flat-core-resonances/report.md and independent 210-check flat-resonance-review/report.md preserve the exact formulas, numerical controls and finite-$k$ bounds. Nonlinear vacuum regularity, uniform participating energy and physical clock selection remain unresolved.

Directional Dispersion Of The Coupled Tail Channels

The complete constant-inertia limiting pencil also determines how its low-frequency disturbances depend on propagation direction. For each fixed

Contents · Previous · Next