IDOS Energy, Coherence, and Supporting Dynamics — Part 30

Contents · Previous · Next

If $E>\sup_{r\ge r_0}V(r)$, the center escapes. Since the Gaussian convolution gives $G_{rr}\sim C r^{-2p}$ and $V\to0$, its remaining travel time is asymptotic to a constant times $\int^\infty r^{-p}dr$, finite for both $p=6,8$. This demonstrates incompleteness of that zero-charge rigid model, not a physical superluminal trajectory or a full-field singularity. $P$ is internal phase momentum, not established electric charge.

Exact sources are the 15 October rational-orbit-modulation, modulated-orbit-computation, and principal/rigid-orbit-global-energy reports and scripts, with separate audit-team-20261015 proof, numerical and energy reviews. The proof's opening scope was narrowed by review; the energy report's full zero-charge metric was corrected and rechecked.

Large Occupied Counts And Slow Carrier Disturbances

For the frozen-support spatial carrier above, large occupied count produces actual slow positive disturbance frequencies, not merely a small stiffness lower bound. This is a constant-coefficient Fourier result, separate from the localized coupled spectrum. At zero physical wave number, positive fixed $\epsilon,\gamma,\eta,\ell,\kappa$ and $N\ge5$, at least $N-4$ positive frequencies obey

$$0<\Omega\le\frac{\gamma(2+8d+4d\eta)}{D^2}, \qquad D=\binom{N+d}{d}.$$

To see why, use shell-constant label vectors $f_s$, with shell multiplicity $m_s=\binom{s+d-1}{d-1}$, satisfying $f_0=f_1=0$, $\sum m_sf_s=\sum sm_sf_s=0$. This space has dimension $N-3$, obeys $C^Tf=G_mf=0$, and has $f^TG_0f=\|f\|^2$, $f^TG_pf=f^TG_1f\le4d\|f\|^2$. It supplies a trial space, not an invariant shell truncation. The full permutation-fixed sector is invariant and has only one phase-kernel direction. Its Hermitian gyro pencil has one negative eigenvalue near $\Omega=0$, but at least $N-3$ at the displayed upper bound, using trial vectors $(X,Y)=(if,f)$. Continuity therefore forces the stated number of positive frequency roots.

The condition $\eta>0$ matters: when phase edges are absent, the nonzero zero-wave-number frequencies instead satisfy $\Omega^2=h/\epsilon+4/\epsilon^2$ and stay above $2/\epsilon$. Small occupation stiffness alone would have missed this distinction.

At fixed nonzero physical wave number along the active all-ones direction, common-phase/density screening gives an $O(1/N)$ upper bound on a positive frequency for constant $b>0$. For $b=\mu_dD/128$, fixed positive sector units and other fixed coefficients, the same fixed-wave-number system instead admits a positive bound uniform in $N$ at each $d\ge2$. Neither is a gap uniform as wave number tends to zero. Count asymptotics compare increasing finite carriers, not infinitely many labels in a fixed finite cutoff. No special depth-five effect, particle lifetime or new particle mass-frequency follows. The actual-count report has 215 checks; its independent review proves the inertia argument and supplies 61 checks.

Coupled Support Energy And Spatial Spectral Limits

The support, physical frame and rest-flow variables can now be retained in the occupied-prefix quadratic calculation. All spatial derivatives act in

Contents · Previous · Next