IDOS Energy, Coherence, and Supporting Dynamics — Part 80

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| $N\ge1$, $\eta=0$ | $55+D$ | $54+D$ | $109+2D$ | | $N\ge1$, $\eta>0$ | $56+d$ | $55+d$ | $111+2d$ | | Common origin $N=0$ | $56$ | $55$ | $111$ |

Here a length-two chain is the pair consisting of a static state and its time-linear generalized state. The common phase supplies the one unextended static state. Nonnegative conserved quadratic energy excludes polynomial solutions of degree two or higher, because a nonzero highest velocity coefficient would force that energy to grow. This argument does not exclude nonpolynomial continuum behavior.

For independent eligible directions $k_i$, their complete inertia is

$$(M_{\rm eff})_{ij}=\langle k_i,M_{\rm kin}k_j\rangle+ \mathfrak a((X_i,\chi_i),(X_j,\chi_j)). \tag{N13}$$

It is finite, symmetric and strictly positive: the kinetic Gram is positive on independent directions and the static form is nonnegative. For boosts it reduces exactly to $Q\Omega I$, and for unused-frame rotations to $\kappa_EQI$. At $\eta=0$, a constant phase $\gamma$ with $\mathbb E\gamma=0$, $B\gamma=0$ has explicit corrector $u=2\mathsf H^{-1}\gamma$, $\chi=-2(Bu)\cdot x$, $v=0$, giving

$$M_{\rm phase}(\gamma,\delta)=Q\left[ \varepsilon\mathbb E(\gamma\delta)+ 4\mathbb E(\gamma\mathsf H^{-1}\delta)\right].$$

The newly completed rotation and registration entries remain the exact response integrals (N13); they have not been assigned numerical values or set to zero by symmetry without proof. The ellipsoidal multiplier retains the actual susceptibility,

$$K=(\kappa^{-1}I+4B\mathsf H^{-1}B^*+\nu^2R_\infty^{-1})^{-1}.$$

Permutation covariance yields two active eigenvalues of $B\mathsf H^{-1}B^*$, on the all-ones axis and its orthogonal complement. They are distinct from the covariance eigenvalues $A_d/\ell^2$ and $(d+1)A_d/\ell^2$, where $A_d=N(N+d+1)/[(d+1)^2(d+2)]$. The unused-space eigenvalue of $K$ is $\Omega^{-1}$. No continuous rotation symmetry of the label tensors or radial-polynomial solution of (N8) is assumed.

In canonical variables $p=M_{\rm kin}\dot{\mathbf q}+\mathcal B\mathbf q$, the conserved linear Noether functionals are

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