IDOS Energy, Coherence, and Supporting Dynamics — Part 32

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$\|\nabla g-g\nabla\log\sqrt\rho\|^2$ on $H^1$. It is nonnegative. The weighted constant is its unique zero state, with $\int\rho\,dx=n_{\rm peak}\pi^5L^{10}/2520$. Escaping Weyl packets show $\sigma(A_\rho)=\sigma_{\rm ess}(A_\rho)=[0,\infty)$. Removing the constant does not open a gap. Bounded-amplitude annular preparations have squared weighted norm of order $R^{-6}$ and gradient energy of order $R^{-8}$, hence Rayleigh quotient of order $R^{-2}$. These statements do not classify all positive eigenvalues or scattering.

For two unused axes, $H_a=H_b=0$ makes the corresponding frame component an exactly decoupled weighted wave: $\ddot f_{ab}+c_F^2A_\rho f_{ab}=0$. Spectral calculus supplies its unique global form-domain evolution and conserved energy. The constant coordinate may evolve as $a+bt$; positive energy does not bound every coordinate. There are $28,21,15,10,6,0$ such channels at depths $2,3,4,5,6,10$. Their absence at depth ten proves nothing about the other channels' gap. rational-support-spectrum/report.md and its independent 109-check review establish this scoped transfer. Physical dimension is ten throughout; the component count and weighted function space are mathematical descriptions, not additional physical positions.

Empty Carrier Labels At A Finite Occupied Branch

An empty label needs Cartesian real-pair coordinates $u_k$, since its phase is undefined at zero amplitude. Rescale all occupied components by $\sqrt{1-\sum|u_k|^2}$. For a finite ambient carrier, the full occupation potential has diagonal quadratic coefficient

$$V^{(2)}_{\rm empty}=\sum_k C_k|u_k|^2,\qquad C_k=w^2(1+t_k+\eta b_k+B r_k),\qquad B=\frac{2dN}{d+1}\binom{N+d-1}{d}.$$

Here $t_k$ counts occupied predecessors; $b_k$ counts phase edges whose unit and predecessor endpoints are occupied; $r_k$ is total label degree in inactive axes. The last term is the second-order opening of inactive mean labels in the redistribution potential. Dropping inactive means before variation would miss it. All coefficients are positive. Same-face labels in shell $N+1$ with $h$ nonzero coordinates have $C_k=w^2[1+(1+\eta)h]$; higher same-face shells have $C_k=w^2$.

Empty pairs decouple from the occupied/support/frame/flow block at quadratic order. In spatially centered real-pair coordinates their exact linear law is

$$\epsilon\ddot u_k+2J\dot u_k+ \left(\frac\epsilon4A_\rho+2\gamma C_k\right)u_k=0.$$

The uncentered spatial derivative is $\partial_i-JL_{ki}$; its orthogonal spatial rotation is a unitary conjugacy, not a new gauge symmetry. With $u_k=e^{-Jt/\epsilon}v_k$, the equation becomes a positive massive wave, with circular disturbance frequencies

$$\Omega_\pm(\alpha)= \sqrt{\alpha/4+2\gamma C_k/\epsilon+1/\epsilon^2}\ \pm\ 1/\epsilon, \qquad \alpha\in\sigma(A_\rho).$$

The zero scalar eigenstate supplies the positive lower endpoints. Each finite nonempty set of empty labels therefore has a positive disturbance gap and bounded weighted coordinate/velocity energy evolution. An empty complement of dimension zero has no gap to minimize. This does not remove the occupied or frame neutral channels or the gapless spectrum above.

At the single origin branch $N=0$, the common completion has clock $\mu_3$: unit labels have coefficient two, and higher labels coefficient one. A combined lower clock bound must include this branch rather than invent separate origin clocks. The derivation and 29,610 author assertions are preserved in empty-label-linearization; the independent empty-label-review/review.md verifies 100 exact controls. Neither the finite-carrier result nor its classical squared amplitudes supplies an infinite-carrier limit, a Born position law or particle selection.

Simultaneous Spatial And Internal-Phase Recurrence

The scalar prescribed-weight orbit comparison can also fit internal-phase recurrence by varying a second family parameter, $s=R^2/L^2$, alongside phase momentum $P$. Both remain constant on each individual trajectory. This changes the source-to-profile size ratio across the family; it is not a recurrence theorem for the earlier fixed-source family.

After the reversible radial shooting equations are solved, let $F$ be the spatial-lock residual and $H$ the internal-cycle residual. At the zero-amplitude center limit, along the center spatial-lock curve, the internal cycles are $1/B(s)$ and the joint determinant is

$$F_PH_s-F_sH_P=F_P\frac{d}{ds}\left(\frac1{B(s)}\right).$$

At an admissible center $F_P\ne0$ when $c\ne0$. For the integrated $p=6$ weight, $B=6s/(1+6s)$, so the remaining derivative is

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