IDOS Energy, Coherence, and Supporting Dynamics — Part 63

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count term is $3\mu D$, so flatness is an occupied-spectrum statement. Equal action $n/D$ and equal branch-energy allocation $n\mu$ follow from this interacting Hamiltonian. Rescaling $n$ changes energy without changing the frequency. A common integrated action $\mathcal A$, including a proposed calibration $\mathcal A=\hbar$, remains an additional normalization premise.

The count law has a nonzero and initially unfavorable radial second jet. For a normalized sphere tangent with occupied radial part $x$, occupied phase part $y$ and empty part $h_E$, let $Q$ denote the coefficient of the squared path parameter, half its second derivative. Direct expansion gives

$$Q_m=\mu D(-4\|x\|^2+2\|h_E\|^2),\qquad Q_V\ge D^{-2}(4\|x\|^2+\|h_E\|^2),$$ $$Q_{m+\Xi_q}\ge\mu D\left[ 4(\lambda-1)\|x\|^2+(\lambda+2)\|h_E\|^2\right] +\lambda\mu D^3\eta Q_\phi.$$

Thus $\lambda=2$ works at all finite counts. It retains neutral allowed phases, translations and frame directions. This inequality does not provide a uniform spatial gap or an infinite-carrier limit. The supplied mean-label unit law has vanishing first and second jets at the six maintained depth knots $d=2,3,4,5,6,10$ and the common origin. The unchanged-curvature claim for that variable unit applies there. Other faces require the actual unit derivatives; the constant-unit theorem covers every finite prefix face.

In the complete support, density, frame and flow action retain $F=U\cdot a-h-\Xi_q$, the saturated $h$, all constraints and all support residuals. Use the smooth plateau reference $\rho_0(W)$ above. On the centered texture

$$c_k=D^{-1/2}\exp\!\left[J\left(\Omega t+ (k-\bar k)\cdot x/\ell+\phi_k\right)\right],\quad k\in A, \qquad\bar k_i=\frac N{d+1},$$

all support and horizontal residuals vanish. The auxiliary density is $\rho=\rho_0$, $j=\rho_0U$, $T=\rho_0\Omega\,UU$, and $E_{\rm rest}=M_{\rm boost}=\mathcal A\Omega$ in characteristic-speed-one units, with $\mathcal A=\int\rho_0\,d^{10}x$. The positive exterior remains part of the complete energy. The ordinary-volume clock $\mathcal H/\sigma$ agrees with $\Omega$ only in the flat core; this construction does not supply an exterior participation gate.

A Field-Dependent Horizontal Coefficient

Define the label covariance and its spatially scaled trace by

$$C(p)=\sum_kp_k(k-\bar k)(k-\bar k)^T,\quad \mathcal V=\operatorname{tr}C/\ell^2,\quad \epsilon(p)=\frac{\omega_*}{4[\omega_*\mu(p)+\mathcal V]}.$$

Replace only the horizontal term by $\mathcal L_{\rm hor}=\rho_0\epsilon(p)(|K|^2-|\mathscr R|^2/4)/2$. The two entropy alternatives remain $(b,\kappa)=(\omega_*/128,8/\omega_*)$ and $(m/128,8/m)$. Require $0<\omega_*\le\min_d\mu_d$ for the supplied unit law. This coefficient contains no particle index, carrier cutoff or special depth-five parameter. Its covariance is passive: a relabeling must also transform the unit's origin and face data and the occupation tensors. It is not an additional active symmetry of those fixed tensors.

The exact centered-gradient Gram is $H^TH=C/\ell^2$, hence

$$\epsilon\|H\|_{\rm op}^2\le \frac{\omega_*}{4}\frac{\mathcal V}{\omega_*\mu+\mathcal V} <\frac{\omega_*}{4}.$$

The coefficient is smooth and positive for a fixed finite carrier; its lower bound need not survive increasing the carrier. On a uniform prefix,

$$C=A[(d+1)I-\mathbf1\mathbf1^T],\qquad A=\frac{N(N+d+1)}{(d+1)^2(d+2)},\qquad \operatorname{tr}C=d^2A.$$

This follows from weak-composition counting, including the slack coordinate. The active eigenvalues are $A$ once and $(d+1)A$ with multiplicity $d-1$; inactive axes have zero eigenvalues. Since $K=\mathscr R=0$ on the branch, coefficient derivatives first affect the cubic variation. The branch clock, stress and positive quadratic form therefore survive, with horizontal coefficient $\epsilon$ and full flow margin at least $(223/424)\rho_0\Omega I$. Off-branch response and disturbance frequencies are changed, and every derivative of $\epsilon(p)$ remains in the equations.

All Non-Origin Singletons Have A Growing Spatial Ground Block

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