IDOS Energy, Coherence, and Supporting Dynamics — Part 28

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when its component of $a$ is constant almost everywhere in the weighted measure. Horizontal carrier and supporting-field inertia are separate. For constant $b$, this reduces to $(Q/b)\operatorname{Cov}(a)$ under the normalized density $n/Q$; no single partition-function factorization is generally available when $b$ varies.

If $n_0,b,H,a$ translate jointly on the whole spatial domain, a change of integration variables makes the charge equation and collective one-form $C_i(X)=-\int n_Qa_i$ independent of $X$. Hence $d_XC=0$, even when the local physical-space curvature $da$ is nonzero. Relative source dependence, boundary terms or deformation must be computed to obtain a center force.

The same restricted Legendre transform yields

$$E_{\rm clock}=uQ-v\cdot\int na\,dx-\int b(n-n_0)\,dx =\int\left[nH+b\left(n\log\frac n{n_0}-n+n_0\right)\right]dx.$$

If $H\ge\mu_{\min}>0$, pointwise relative-entropy positivity gives $E_{\rm clock}\ge Q\mu_{\min}$. This is a restricted energy contribution, not full-field coercivity or a separate conservation law when other dynamic sectors exchange energy with it. Time-dependent prescribed sources also require their own dynamics for a conserved total energy.

The complete parameter-derivative formula, including derivatives of $b$ and the implicit normalization, is in principal/variable-b-collective-charge.md under the 15 October clock-action run. Its checker passes 27 exact controls. The separate audit-team-20261015/variable-b-review supplies 38 independent derivative and normalization checks and a separately pinned energy addendum with 10 independent checks. No physical charge assignment, full particle inertia, precession lock or special depth-five effect follows from this reduction.

Spatial Carrier Disturbances And Curvature Dynamics

The centered equal-prefix SF-DW branch has a positive quadratic carrier/phase/density energy when its support, frame and rest flow are prescribed. This restriction is generally not invariant under the full equations. Let $D=\binom{N+d}{d}$, $w=D^{-1}$, $L_{ki}=(k_i-N/(d+1))/\ell$, $C=2\sqrt w L$ and

$$p_k=w+2\sqrt w X_k,\quad \phi_k=\omega t+L_k\cdot x+\theta+Y_k/\sqrt w,\quad n=n_0(1+r),\quad \sum_kX_k=\sum_kY_k=0.$$

At a branch with vanishing first and second mass jets, the quadratic action divided by $n_0$ is

$$r\dot\theta-\frac b2r^2+2X\cdot\dot Y +\frac\epsilon2(|\dot X|^2+|\dot Y|^2) -\frac\kappa2|\nabla\theta+C^TX|^2 -\frac\epsilon8(|\nabla X|^2+|\nabla Y|^2) -\gamma(X^TAX+Y^TBY).$$

Here $A,B$ are quadratic-form matrices, so their force Hessians are $2\gamma A,2\gamma B$. The actual sparse occupation and phase gates give

$$A=4w^2(G_0+G_1)+4wq^2G_m+\eta w^2G_p, \qquad B=\eta w^2G_p,\qquad q=N/(d+1).$$

$G_0$ is the origin-anchor graph, $G_1$ the predecessor graph, $G_m$ the directed within-shell redistribution graph, and $G_p$ the sum of outer products of phase-edge vectors $h=e_0+e_k-e_{e_j}-e_{k-e_j}$. All are nonnegative forms. On the sum-zero occupation tangent, $A\ge4D^{-2}I$. The kernel of $G_p$ consists exactly of affine label functions and has dimension $d+1$. The phase-edge term contributes to both $A$ and $B$.

The mean-label variation cancels in the spatial residual, leaving $|\nabla X|^2+|\nabla Y|^2$; the homogeneous variance potential must not be imported into this spatial calculation. The conserved constrained quadratic energy is the sum of the corresponding nonnegative kinetic and potential terms. For fixed nonconstant $n_0$, spatial derivatives use $n_0^{-1}\operatorname{div}(n_0\nabla)$ and zero integrated boundary flux. This excludes finite-energy exponentially growing linear eigenmodes in the restricted block, but supplies neither a gap nor nonlinear persistence.

For constant coefficients and nonzero physical wave vector $k$, eliminating the static common-phase block leaves the occupation form

$$\frac\epsilon4|k|^2I+2\gamma A+ \kappa C\left(I-\frac{kk^T}{|k|^2}\right)C^T>0.$$

Together with the positive kinetic form and antisymmetric gyroscopic term, this gives nonzero purely oscillatory finite-carrier frequencies. These

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