IDOS Energy, Coherence, and Supporting Dynamics — Part 16

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$$\langle p_M,p_N\rangle=R_{00}^{|N-M|},$$

not the count-ratio rule. Coefficients change between cutoffs, and truncating the higher polynomial generally destroys pointwise uniformity. Hence this does not contradict the earlier inherited-prefix rank bound. Both spatial phase current and normalized Gram tensor generally vary. Conjugate duplication cancels the current but leaves Gram variation, at a cost of $4(d+1)$ real components, up to 44 at depth ten. Consequently the constant-descriptor count-action proof does not transfer to this family. Its periodic cell is an averaging device, not compactified physical space. Full band occupation, equal modal energy, local complete-energy uniformity, selected clock frequency and reciprocal mixing remain distinct tests. The theorem and 466 exact checks are preserved in small-carrier-full-band.

A Retained Carrier With Exact Uniform Simplex Occupation

A common finite carrier and algebraic occupation potential close a gap left by the low-moment constructions. This is an inverse-designed candidate with explicit component cost, not a derivation of the fundamental field. Fix once the ambient ten-axis labels $K_M=\{k\ge0:|k|\le M\}$. In real paired components, let $J$ rotate each pair and define Newton companion matrices

$$(T_jW)_\alpha=\alpha_jW_\alpha+W_{\alpha+e_j}$$

below the terminal degree, omitting the second term at $|\alpha|=M$. The triangular interpolation matrix $\mathsf V_{\alpha k}=\prod_j(k_j)_{\alpha_j}$ is invertible and obeys $T_j\mathsf V=\mathsf V\operatorname{diag}(k_j)$. Thus the matrices commute and are self-adjoint for the positive metric $H=\mathsf V^{-T}\mathsf V^{-1}$. The complete first-order residual system has spatial principal symbol $|\xi|^2H$ and exactly the finite joint labels. It uses $2\binom{M+10}{10}$ real physical components, not merely the number occupied by one lower-depth preparation. No large matrix is needed to prove these identities.

Use the physical multiplet $\Psi=\sqrt{2n}e$, $\|e\|_H=1$, and the same timelike phase current as in the covariant support construction. Put $\mu_j=\langle e,T_je\rangle_H$. With a supplied orthonormal spatial material frame $E_j$ orthogonal to $u$, define

$$\mathcal R_\mu=P_\mu{}^\nu(\partial_\nu e-a_\nu Je) -J\sum_jE_{j\mu}(T_j-\mu_j)e, \qquad \mathcal K=u\cdot\partial e-sJe.$$

The added action is $\epsilon n\|\mathcal K\|_H^2/2- \epsilon n\|\mathcal R\|_H^2/8$. Both residuals vanish on the centered rest preparation

$$e(t,x)=e^{J\omega t}\mathsf V \{e^{J(k-\bar k)\cdot x}c_k\},\qquad \sum_k\|c_k\|^2=1,\quad \bar k=\sum_k k\|c_k\|^2.$$

Spatial units here set the label wave-number scale to one. Rest current requires this centering. The spatial residuals imply constant $\mu_j$; the proper-time residual also fixes their time dependence. Spatial residuals alone allow a common local phase, which is removed by imposing the actual rest current. The full action has no arbitrary local-phase symmetry.

Physical localization is not lost: $\int\|\Psi\|_H^2=2S$ for the finite support profile. In physical coordinates the horizontal derivative is $\partial_\mu\Psi-a_\mu J\Psi- \tfrac12(\partial_\mu\log n)\Psi$. Keeping this density term gives exactly the same full first variation and principal cone at $n>0$. The normalized carrier has a finite centered label spectrum; the localized physical field has an envelope-broadened ordinary Fourier spectrum. These are different statements. The normalized jet relations must not be asserted for bare derivatives of the localized physical field.

A Common Occupation Potential

Let $\Pi_k$ be the fixed joint spectral projectors, and set $p_k=\|\Pi_ke\|_H^2$, $q_i=\sum_k k_ip_k$. These are classical component norms, not position probabilities. Define

$$V_{\rm down}=\sum_kp_k\sum_{\alpha\le k}(p_\alpha-p_k)^2,$$ $$V_{\rm mix}=\sum_{i\ne j}q_iq_j \sum_{k_i\ge1}(p_k-p_{k-e_i+e_j})^2, \qquad V=V_{\rm down}+V_{\rm mix}.$$

All predecessor inequalities are componentwise. The zero set is exactly uniform occupation of a complete cumulative simplex $K_N$ in an active coordinate face, together with the constant-label state $N=0$. Indeed an occupied label forces all its predecessors to have the same positive weight as label zero. Positive $q_iq_j$ then forces equality throughout every connected composition shell in the active coordinates. The highest occupied shell and all its predecessors form the full prefix. Conversely every such uniform prefix makes both sums zero. This proves both directions without a separate coefficient for each depth or cutoff. The law allows other depths as unidentified branches; it does not select the observed assignments.

At a uniform branch with $D=\binom{N+d}{d}$, the constrained amplitude expansion $V(e+\eta h)=\eta^2Q(h)+O(\eta^3)$, with normalization understood, obeys

$$Q(h)\ge\frac4{D^2}\sum_{k\ { m occupied}}|h_{k,\rm radial}|^2 +\frac1{D^2}\sum_{k\ { m unoccupied}}\|h_k\|^2.$$

All predecessors are essential: the label-zero term provides stiffness even for a newly excited label far outside the occupied band. The exact kernel consists of the occupied pair-phase directions. The potential has zero first variation in physical amplitudes even though its unrestricted gradient with respect to boundary weights $p_k$ need not vanish. Adding $-\gamma nV$ to the action adds positive quadratic restoring energy and no principal derivatives. It supplies neither dissipation nor attraction, and does not lock the remaining relative phases.

On positive-cutoff zeros, $q_i=N/(d+1)$ on the active face. The algebraic descriptors $z_1=\sum_iq_i$, $z_2=\sum_iq_i^2$ recover

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