IDOS Energy, Coherence, and Supporting Dynamics — Part 39

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finite prefix, let $T$ be the $(D-1)$-dimensional carrier tangent space, $H_a$ the $d$ active columns of $H$, and $\delta=\omega-1/\kappa$. Write $A_o>0$ and $B_o\ge0$ for the actual occupation and phase Hessians. The following matrices retain the occupation and independent-flow response:

$$B=[2H^T/\sqrt\epsilon,\ \delta I/\sqrt{\kappa_U}],\qquad J=\begin{pmatrix}2I_T/\epsilon\\\sqrt{\epsilon/\kappa_U}H^T\end{pmatrix},$$ $$R=\frac{\delta I-\epsilon H^TH}{\kappa_U}>0,\quad M_0=\operatorname{diag}(2\gamma A_o/\epsilon,R)+\kappa B^TB, \quad M_\eta=I_T+J^TM_0^{-1}J, \quad M_A=H_a^TM_\eta H_a>0.$$

In particular the effective phase inertia is not the identity. Its sign follows from a nonzero-frequency reduction of the actual support gyro pair. Put $a_s=\gamma_A/M_W$, $m_s=64\gamma_A/(L^2M_Z)$ and $c=2\sqrt{n_{\rm peak}\beta/M_W}$. Eliminating vector support and then scalar support adds a positive term to the common-phase entry:

$$D_s=1+\frac{a_s|k|^2}{m_s-\Omega^2},\qquad P_{\tau\tau}^{\rm reduced} =\kappa\beta|k|^2-\Omega^2+\frac{c^2}{D_s}.$$

It tends to $c^2$, not zero. The reduction is for $\Omega\ne0$; the exact static support branch remains at zero frequency.

For positive phase edges, $\ker B_o=\operatorname{range}H_a$. Along a wholly unused physical direction, mixed frame rotations cancel the leading affine-phase gradient. Their remaining spatial stiffness is fourth order in wave number. With $\alpha^2=\epsilon/\kappa_F$, the $d$ positive affine-phase branches satisfy

$$\Omega_j(k)^2=|k|^4 b_j+o(|k|^4),\qquad \{b_j\}=\sigma\!\left(M_A^{-1}\frac{c_F^2}{\alpha^2}I_d\right)>0.$$

For an occupied unit direction $n$, let $S=H_a^TH_a$ have positive eigenvalues $\lambda_i$. Express $n_i$ and $v_i$ in an orthonormal eigenbasis of $S$. The relaxed affine stiffness is

$$4v^TK_A(n)v=\sum_i\lambda_i n_i^2v_i^2+ \sum_{i<j}\frac{\lambda_i\lambda_j}{\lambda_i+\lambda_j} (n_iv_j+n_jv_i)^2.$$

This follows by minimizing $\operatorname{tr}[(nv^T-F)S(nv^T-F)^T]$ over skew $F$. A nonzero rank-one matrix cannot be skew, so $K_A(n)>0$. The $d$ branches are acoustic: $\Omega_j^2=|k|^2a_j+o(|k|^2)$, with $\{a_j\}=\sigma(M_A^{-1}K_A(n))>0$. For a mixed direction use its occupied projection in this stiffness. Every fixed nonzero occupied projection is acoustic; the limit is not uniform as that projection tends to zero.

At zero phase-edge coefficient, all $D-1$ phase directions are soft. Wholly unused propagation still has the $d$ quadratic branches, together

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