IDOS Energy, Coherence, and Supporting Dynamics — Part 17

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$d=z_1^2/z_2$ and $N=(d+1)z_1/d$. A single positive smooth function $m(z)$ can interpolate the supplied target count clocks with zero first and second jets, without using derivative descriptors. For example, for distinct finite target points $z_A$ and frequencies $\omega_A>0$, put

$$L_A(z)=\prod_{B\ne A}\frac{\|z-z_B\|^6}{\|z_A-z_B\|^6}, \qquad m(z)=\frac{\sum_A\omega_AL_A(z)}{\sum_AL_A(z)}.$$

The denominator is positive everywhere and $m(z)=\omega_A+O(\|z-z_A\|^6)$ near each target. All $N=0$ descriptors coincide and cannot receive distinct frequencies. Unlisted branches need not obey the desired count formula. The supplied scales and selected target values remain inputs; the occupation theorem does not derive them.

Complete Branch And Disturbance Scope

All new terms and their full first variations vanish on these coherent uniform branches, preserving $E_{\rm rest}=S\omega$, $Q=S$ and the full Lorentz energy-momentum relation. The retained carrier now has positive second-order horizontal kinetics. Including it in the coupled phase/support cone adds $\epsilon nI$ to its kinetic block, $\epsilon nI/4$ to its spatial block and $\nabla e/\omega$ to its drift matrix. Since $\|\nabla e\|_H^2\le M^2$, choosing $\epsilon=\omega_{\min}/(4M^2)$ together with the earlier rational-support coefficients gives $\mathsf H\ge nI/(4\omega)$ and satisfies the complete causal Schur bound. The occupation potential and algebraic mass descriptor add no principal derivative terms. This is a local linear result on compact positive-density regions; vacuum and generic nonlinear evolution remain open.

Equal branch norms give action $S/D$ and the corresponding branch energy allocation $S\omega/D=S\mu_d$ per occupied label. This does not turn the nonlinear field into independent free oscillators. In particular the local total energy remains $n(x)\omega$, so the separate constant-density local clock requirement is not solved by uniform component occupation. Independent physical orientations and reciprocal encounters still require a same-field realization of the supplied material frame. No exceptional depth-five response follows. The full proofs and 732 and 2,056 exact checks are preserved in causal-band-constraints and uniform-simplex-occupation under the current research run.

Rest Centering Changes A Spatially Integrated Overlap

The occupied-band result does not automatically supply the inverted mixing rule as a physical response. For aligned nested faces, common support and matching shared-label phases, coefficient overlap is $C_{ab}=D_{\cap}/\sqrt{D_aD_b}$. But each isolated rest preparation requires its own centering vector $q_{a,j}=N_a/(d_a+1)$ on its active directions. Their pointwise inner product therefore carries the relative phase $(\omega_b-\omega_a)t-(q_b-q_a)\cdot x/\ell$, where $\ell$ sets the label wave-number unit. The two particle clocks are not synchronized.

For coincident rational profiles in ordinary $\mathbb R^{10}$, integrating the complete physical inner product and dividing by $2S$ gives

$$C_{ab}F(z)\cos[(\omega_b-\omega_a)t+\phi],\qquad z=\frac L\ell|q_b-q_a|,\qquad F(z)=\frac{z^3K_3(z)}8.$$

The Gaussian-mixture representation $F(z)=\Gamma(3)^{-1}\int_0^\infty t^2e^{-t-z^2/(4t)}dt$ derives this factor from the full density, including its tail. $F(0)=1$, $F'(z)=-z^3K_2(z)/8<0$, and $F(z)=1-z^2/8+O(z^4)$ near zero. For the maintained electron/muon pair, $C_{e\mu}^2=17/3515$ and $|q_\mu-q_e|^2=2904/49$. At $L/\ell=0.1$ the factor is about $0.930666$; at $L/\ell=1$ it is $0.0198625$. Neither length ratio is an empirical assignment.

Thus small and broad support give different overlap diagnostics under the same centered construction. General shared-label phases can suppress even the coefficient overlap; equal occupation does not select them. Separated centers and independent frames require further calculations. This is not a force, mixing angle or transfer rate: the actual response kernel must come from the common action. The exact identities and 575 checks over all 91 maintained target pairs are in principal/centered-band-overlap.md and its executable companion.

Sparse Occupation Interactions And A Dynamic Physical Frame

The occupation potential admits a genuine simplification. Replace its all-predecessor sum by

$$V_{\rm anchor}+V_{\rm edge} =\sum_kp_k(p_0-p_k)^2+ \sum_k\sum_{j:k_j>0}p_k(p_{k-e_j}-p_k)^2.$$

Keep the previous shell redistribution term. Nearest predecessors give downward closure by induction; the anchor retains quadratic stiffness

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