IDOS Energy, Coherence, and Supporting Dynamics — Part 33

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$-1/(6s^2)\ne0$. For the section control $p=8$, let $I_k=\mathbb E[(1+st)^{-k}]$ under the gamma$(8,1)$ measure and $r=I_6/I_7$. Changing variable to $x=st$ gives $r=1+\mathbb E_s[x]$, $r'=\operatorname{Var}_s(x)/s^2>0$. Then $B=s(2r+6)/[1+s(2r+6)]$ is strictly increasing. The joint implicit-function construction follows wherever the center positivity conditions hold. It gives a sufficiently small-amplitude family, without a certified numerical existence radius.

For spatial lock two and internal-cycle ratio $7/6$, the center size is $s=1$ for $p=6$ and approximately $0.511819803005624$ for $p=8$. The latter center positivity was checked numerically, not by rigorous interval enclosure. At $\rho/R=0.1$, numerical fits give respectively $s\simeq1.09592806464$ and $0.547157641320$. Six spatial periods then contain seven internal cycles. A separate RK45 integration verified position/velocity closure below $9.2\times10^{-11}$ and sampled field/ field-time-derivative recurrence below $7.5\times10^{-11}$ in the reported normalizations. These are bounded numerical controls, not a proof that this amplitude lies inside the analytic continuation radius. For $p=6$, the physical phase-charge ratio includes the changing $L^4$ normalization: approximately $0.616498$, whereas its normalized charge ratio is $0.740450$.

The Cartesian identity $U_t=e^{J\chi}[J\dot\chi\,\phi(x-q)-\dot q\cdot\nabla\phi(x-q)]$ shows why recurrence of position, velocity, phase and phase rate returns both the field and its time derivative. Signed radial crossings introduce no extra internal phase. full-phase-orbit-closure/report.md and its independent 31-control review give the derivation and numerical pins. Neither fitted lock is dynamically selected. This remains a rigid scalar comparison, without a full-field stability or observed-electron conclusion. The flat occupied spectrum, spatially uniform participating energy and common-action inversion requirements are not established by this result or by the rational-support energy calculations above.

Global Coupled Linear Evolution And Nonlinear Tail Competitors

For the constant positive support inertias above, the occupied-prefix quadratic system now has a specified global energy realization, so its conservation argument need not assume an unconstructed solution. Normalize weighted phase, carrier, frame and flow coordinates by $h=\sqrt\rho$ and their constant kinetic factors, and support coordinates by $M_s^{1/2}$:

$$u=(h\theta/\sqrt\beta,\ h\sqrt\epsilon X,\ h\sqrt\epsilon Y, M_s^{1/2}y,\ h\sqrt{\kappa_F}f,\ h\sqrt{\kappa_U}v).$$

The finite real component count is $2D+65$. The normalized equation is $\ddot u+G\dot u+Au=0$. Here $G$ is bounded skew multiplication: its four blocks come from carrier phase rotation, phase/support coupling, carrier/flow coupling and support/flow coupling. The last coefficient is $M_s^{1/2}D_s/(h\sqrt{\kappa_U})$, bounded by the rational derivative estimate; the phase/support coefficient is $2\sqrt{n_{\rm peak}\beta/M_W}$. No spatial derivative acts in $G$.

The positive static squares define a closed form $q$. Its exact domain $\mathcal V$ consists of ordinary $L^2$ normalized coordinates with all

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