IDOS Energy, Coherence, and Supporting Dynamics — Part 9

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Separate phase symmetries conserve both block actions but permit spatial interaction through this mixture. For nonorthogonal patterns, all energy and canonical interference must remain. In collective coordinates, $W_{ab}=\int\langle J\partial_a\psi,\partial_b\psi\rangle$ and $W\dot q=\nabla H$. A nonzero raw position derivative can cancel in this full equation: $H=\omega S$ generates a common phase rotation even when interference makes $S$ position dependent. Six-depth initial-response controls confirm this cancellation and show no exceptional depth-five suppression. They are not full PDE encounters or physical acceleration predictions. Unlike particles keep their own isolated frequencies.

Selecting An Envelope Without Adding Rest Energy

One explicit changed action adds the positive local density square

$$K_{\log}=\frac\beta2\int\rho\, \|\operatorname{Hess}\log\rho+\alpha 1\|^2\,dx, \qquad \alpha,\beta>0.$$

Its zero set on a connected positive region is $\rho=\rho_{\rm peak}\exp[-\alpha|x-X|^2/2]$. The center and total action are free; $\alpha$ is one supplied common inverse length squared. The square and its full first variation vanish on this profile, including every translated and phase-boosted member. Thus the exact clock, complete energy and inertia survive. At fixed action and center its density-shape Hessian is $\beta\int\rho\|\operatorname{Hess}(\delta\rho/\rho)\|^2$, with Gaussian weighted gap $2\beta\alpha^2$. This does not control every polarization direction or establish nonlinear stability. The logarithm introduces a vacuum/domain problem outside the bounded $H^1$ theorem.

A polynomial alternative is

$$F_{ij}=\rho\partial_i\partial_j\rho -(\partial_i\rho)(\partial_j\rho)+\alpha\rho^2\delta_{ij},\qquad K_{\rm poly}=\frac\gamma2\int\sum_{ij}F_{ij}^2\,dx.$$

It has the same zero-energy Gaussian branch without density division, but requires a stronger function domain. Its Hessian weights shape changes by $\rho^4$, so it has no corresponding uniform tail gap; arbitrarily broad normalized Gaussians approach zero added energy. For an action-preserving dilation by $\lambda$, the two support energies are

$$K_{\log}(\lambda)=\frac{\beta A n\alpha^2}{2}(1-\lambda^{-2})^2, \qquad K_{\rm poly}(\lambda)=\frac{\gamma n\alpha^2I_4}{2} \lambda^{-3n}(1-\lambda^{-2})^2,$$

where $n=10$ for full ambient localization and $I_4=\int\rho^4$ at the selected Gaussian. Both have positive local size curvature. Reduced $n=d$ controls must not be confused with full ten-dimensional support. Gaussian tails remove the earlier exact disjoint compact-support family. Uniform local participating energy, physical occupied-band selection and the action unit are still not supplied by either support term.

The alternative of strictly positive net added support energy cannot retain unchanged $H_0$, $A$, $\omega$ and $E=A\omega$ simultaneously. A compensating additive pedestal can restore rest energy and clock but misses its share of translation inertia unless its moving contribution is derived afresh. Zero-energy squares avoid that particular obstruction; they do not complete the wider inverse problem.

The reviewed derivations and executable controls are preserved under this run's count-current-inertia, bounded-clock-variation-audit, common-gradient-selector, small-carrier-moment-control, count-current-pair-response, current-action-independent-review and envelope-stiffness-compatibility paths. The original current variation has 368 checks, the common selector 509, the few-component control 114 and the pair calculation 141. Independent off-rung variation checks agree within $1.64\times10^{-10}$; principal adds 48 conservation checks and the support calculation 64 exact checks. Their analytic scopes, rather than assertion counts, delimit the results. Bottom's double count, tau adjacency and neutrino mixing remain unresolved.

How Much Spatial-Moment Information Certifies Occupation?

For balanced standing textures, represent their normalized scalar spectral weights by

$$\mu_N=D^{-1}\delta_0+(2D)^{-1} \sum_{k\in K_N\setminus\{0\}}(\delta_k+\delta_{-k}).$$

There are $D$ linked classes $\{k,-k\}$, including zero, and $2D-1$ signed atoms. Odd total moments vanish. The full derivative Gram through spatial order $r$ determines the scalar moments through degree $2r$ in this orthogonal standing ansatz. Duplicate polarization copies at one wavevector remain invisible to these moments at every order.

If competitors are restricted to the prescribed finite $K_N$, moments through degree $2N$ determine every class weight. The polynomials $P_\alpha(q)=\prod_i\prod_{j=0}^{\alpha_i-1}(q_i^2-j^2)$, $\alpha\in K_N$, have a triangular nonzero-diagonal evaluation matrix on $K_N$. This degree is necessary: below $2N$, an axis-supported signed weight perturbation annihilates all retained even moments, and its positive endpoint removes a label. Imposed equal weights would be additional information not supplied by these moment constraints.

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