IDOS Energy, Coherence, and Supporting Dynamics — Part 29

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are disturbance frequencies around the supplied particle clock. Curvature occurs already to first order: $F^{(1)}=d(C^TX)$. Explicit first-prefix transverse standing waves realize ranks $2,2,4,4,6,10$ across depths $2,3,4,5,6,10$; this is generic even rank, not a special depth-five effect. Their plane strengths are equal only at chosen points. Constant-support plane waves are not localized rational- support eigenmodes, center-position curvature or deterministic electron orbits.

The complete derivation and explicit waves are in the 15 October spatial-carrier-linearization/report.md (315 author checks). audit-team-20261015/spatial-carrier-review/review.md independently expands the actual sparse polynomials and verifies 791 exact assertions. The localized reference still has energy density $n_0(x)\omega$; the distinct uniform participating-energy inversion requirement remains unresolved.

Modulated Rational-Weight Orbits And Their Energy Domain

The prescribed-weight scalar product-vortex comparison now has a local finite-radius continuation. This remains a different action from SF-DW. Its source and rigid internal profile are stipulated, and its center is a literal position in ordinary six-dimensional space. With the exact weighted norm $N(q)$, connection $a(q)$, derivative Gram matrix $G(q)$, and $V=\Lambda\operatorname{tr}G/2$, the fixed-phase-momentum action is

$$R_P=\frac\Lambda2\dot q^T M(q)\dot q- P\frac{a(q)}{N(q)}\cdot\dot q-U(q),\quad M=G-\frac{aa^T}{N},\quad U=V+\frac{P^2}{2\Lambda N}.$$

The positive rational source gives exact smooth Gaussian-moment integrals for all these coefficients. Eliminating both physical plane angular momenta at zero leaves a regular two-signed-radius mechanical system; apparent angular-coordinate singularities cancel. At each parameter point with $\delta^2>0$, the specified equal-amplitude quadrature two-plane petal continues locally. In scaled radii, shoot from $(1,0)$ with velocity $(0,w)$ to $r_1=r_2$, $\dot r_1=-\dot r_2$. The harmonic-limit root is $w=\delta$, $T=\pi/(4\delta)$ and its two-variable shooting Jacobian has determinant $2\delta\ne0$. Reflection and interchange symmetries extend the segment to a reduced period $8T$.

Both plane angles acquire the same advance per reduced period. At nonzero $P$ and nonzero gradient-curvature coefficient $c$, a second implicit-function argument adjusts the conserved charge with family amplitude to retain a specified rational angular advance. Each charge remains constant during its own trajectory. This establishes locally closed spatial curves; it does not select a rational lock dynamically or establish internal-phase recurrence, orbital stability or full-field profile persistence.

The separate numerical script solves four amplitudes $\rho/R=0.05,0.1,0.2,0.3$ for lock two at $R/L=1$. It retains both the $p=6$ weight obtained by integrating uniform unused directions in ordinary ten-dimensional space and the distinct $p=8$ section control. At $\rho/R=0.3$, sampled radius modulation is respectively 2.3783% and 5.2247%. The selected charges are 0.900118 and 0.908678 times their zero-amplitude limits. The internal phase advances by 1.27931055560 and 1.23296141858 cycles over the closed spatial period, so it does not recur there. Twenty-four quadrature/tolerance controls and two Cartesian integrations support the reported curves. A separate reviewer changed the integrator and maximum step and recovered closure/agreement at about $10^{-11}$ in normalized coordinates. These are numerical controls, not interval certification or proof that the local existence neighborhood extends to every tested amplitude.

The exact conserved energy also supplies a global statement within this rigid comparison:

$$E=\frac\Lambda2\dot q^TM\dot q+V+\frac{P^2}{2\Lambda N}.$$

For $P\ne0$, $N(q)\to0$ at spatial infinity and $N(q(t))\ge P^2/(2\Lambda E)>0$. This confines $q$ to a compact set; strict positivity of $M$ there bounds velocity and yields global smooth ODE continuation. It is confinement, not stability of neighboring orbits.

For $P=0$, the metric remains $M$, not $G$. On an invariant outward radial ray starting at $r_0>0$, parity gives $a_r=0$ and thus $M_{rr}=G_{rr}$.

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