IDOS Energy, Coherence, and Supporting Dynamics — Part 41

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$K=\chi\sum_i c_iH_i(e_H)$. The charge correction is exactly $\Delta$ and its energy is $Q_2\Delta^TC^{-1}\Delta/(2\epsilon Q_1^2)$. Inactive axes remain zero. For independent pair phases instead use $\Delta_k=Q/D_H-\int\rho|e_{X,k}|^2$, whose sum is zero. The core covariance is $I/D_H-\mathbf1\mathbf1^T/D_H^2$; $c_k=D_H\Delta_k/(\epsilon Q_1)$ restores every occupied pair charge, at energy $D_HQ_2|\Delta|^2/(2\epsilon Q_1^2)$. Every empty ambient pair remains zero. No singular full covariance is inverted.

Let $\eta$ denote the resulting ten-component label-charge deficit: $\eta_i=\Delta_i$ in the affine case and $\eta_i=\sum_k k_i\Delta_k$ in the pair case. Since $\partial_i e_H=H_i(e_H)/\ell$, core compensation adds momentum density $\rho\chi\eta/(Q_1\ell)$, with integrated momentum $\eta/\ell$ and exactly zero integrated angular momentum by radial symmetry. The bubble contributes $p_0=\int\rho a$ and $B_0=\int\rho x\wedge a$. Choose

$$W_t=\chi v\cdot\nabla W,\quad v=-\frac{p_0+\eta/\ell}{g_1},\qquad (E_t)(x)=\chi F,\quad F_{ab}=-\frac{(B_0)_{ab}}{\kappa_FQ_1},\quad F^T=-F.$$

The scalar recoil has zero angular density because $W$ is radial. The frame position is initially constant, so its translation momentum vanishes even though its velocity varies spatially. Its intrinsic rotation charge is $-\kappa_F\int\rho\chi F$, one contribution per independent plane. These choices set all ten translation charges and all 45 spatial rotation charges to their branch values. Lab $Z_t=0$ requires material $z_t=-\chi Fz$; retaining this connection cancels the apparent extra support momentum. The full extra energy is

$$E_{\rm comp}=E_{\rm phase} +\frac{g_2|p_0+\eta/\ell|^2}{2g_1^2} +\frac{Q_2|B_0|_{45}^2}{2\kappa_FQ_1^2}.$$

For a fixed bubble and remote radius $R$, its weight is $O(R^{-16})$. Phase deficits and momentum are of that order, while $B_0=O(R^{-15})$. Compensation is therefore $O(R^{-30})$, smaller than the negative bulk gain $O(R^{-16})$. Strict lower energy survives at all these exact charges. Normalized velocity differences vanish outside the fixed core and bubble; physical velocities include the already controlled density derivative. The bubble position change remains order one pointwise, so convergence is in the stated weighted/physical norms, not a uniform relative-amplitude norm.

The smooth sphere-log path from the branch to the bubble can be compensated by these same formulas at every path parameter. It preserves all listed charges and stays arbitrarily close in the stated norms, with arbitrarily small upper energy excursion and a strictly lower endpoint. The endpoint gain may tend to zero. This does not prove a zero barrier to a prescribed finite release, or supply a dynamical trajectory. Eligibility still requires a lower same-action branch: $N_H\ge2$ allows $N_L=1$, while using the single shared origin requires its actual ordering against $\mu_3$. No lowest-branch minimality follows.

Sources are tail-angular-momentum, tail-phase-charges and compact-charge-compensation reports and their independent reviews, respectively 280, 659 and 719 independent checks. The conclusion fixes the stated diagonal phase family, momentum and spatial rotations; it does not classify every conserved quantity or infer persistence, decay, observed spin, or a physical charge rule.

Joint Direction And Wavelength Limit

The transition between unused-direction quadratic dispersion and occupied-direction acoustic dispersion has a controlled joint limit. Fix a finite carrier and let $u$ be an unused unit direction, $a$ an occupied vector and $r=|k|$. Set $n(r)=\sqrt{1-r^2|a|^2}\,u+ra$. For bounded $a$ and small $r$,

$$\frac{\Omega_j^2}{r^4}\longrightarrow \lambda_j\!\left[M_A^{-1} \left(K_A(a)+\frac{c_F^2}{\alpha^2}I\right)\right].$$

The frame coupling has zero rows on the nonaffine phase complement, and the affine/nonaffine static cross block vanishes exactly. Their kinetic

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