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with D−1−dD-1-d acoustic branches. If Q⊥Q_\perp is an orthonormal basis of the complement of range⁡Ha\operatorname{range}H_a in TT, their squared speeds are the eigenvalues of (1/4)MB−1(1/4)M_B^{-1}, where

MB=Q⊥TMηQ⊥−Q⊥TMηHaMA−1HaTMηQ⊥>0. M_B=Q_\perp^TM_\eta Q_\perp- Q_\perp^TM_\eta H_aM_A^{-1}H_a^TM_\eta Q_\perp>0.

The mass Schur complement cannot generally be replaced by its first term. For a fixed nonzero occupied projection, all D−1D-1 phase branches are acoustic. Both edge cases also retain the (10−d2)\binom{10-d}{2} independent unused-frame waves Ω=cF∣k∣\Omega=c_F|k|. At depth ten there are neither unused propagation directions nor unused frame planes; its affine phase branches remain acoustic. None of these counts or coefficients singles out depth five.

The coefficients describe the constant limit at infinity and its established approximate-spectrum transfer, not localized particle eigenfrequencies or observed interaction strengths. soft-directional-dispersion/report.md and its independent review give the full Schur sequence. The accepted corrected version has 101 author checks; review includes 90 independent checks and 76 correction checks. The numerical N=1N=1 control requires the occupation/flow cross entry 28λ28\sqrt\lambda, not 14λ14\sqrt\lambda. For example its common-unit depth-two longitudinal acoustic speed squared is 407/5904407/5904, and its parallel quadratic coefficient is 407/41407/41. The correction changes example coefficients, not the directional powers.

Tail Comparisons At Exact Phase, Momentum And Rotation Charges

The lower-prefix comparison can preserve more than common QQ and total PP. For positive phase-edge coefficient, the exact diagonal pair rotations of the full finite carrier obey θk=α+∑i=110βiki\theta_k=\alpha+\sum_{i=1}^{10}\beta_i k_i. Every edge enforces θ0+θk=θei+θk−ei\theta_0+\theta_k=\theta_{e_i}+\theta_{k-e_i} modulo 2π2\pi; induction proves necessity and sufficiency. For ambient cutoff M≥1M\ge1 this is eleven independent continuous phase generators. At zero phase edges every real pair rotates independently. These mathematical Noether charges are not identified with electric charge or spatial angular momentum.

On the independent-flow rest slice, set et=(hclock+γV)Je+Ke_t=(h_{\rm clock}+\gamma V)Je+K, with K⊥e,JeK\perp e,Je. SF-DW still gives n=ρn=\rho and πe=ρJe+ϵρK\pi_e=\rho Je+\epsilon\rho K. For diagonal TT, write qT=⟨e,Te⟩q_T=\langle e,Te\rangle and HT=J(T−qT)eH_T=J(T-q_T)e. Its exact current is

jT0=ρqT+ϵρ⟨K,HT⟩,jTi=−ρqT(hclock)ai−ϵρ4⟨Ri,HT⟩. j_T^0=\rho q_T+\epsilon\rho\langle K,H_T\rangle,\qquad j_T^i=-\rho q_T(h_{\rm clock})_{a_i} -\frac{\epsilon\rho}{4}\langle R_i,H_T\rangle.

For T=IT=I, HT=0H_T=0 and the common current is unchanged. Continuity still determines nt=div⁡(ρha)n_t=\operatorname{div}(\rho h_a); it does not impose nt=ρtn_t=\rho_t on this initial slice.

All compensating velocities can be confined to one fixed unchanged core, disjoint from the remote lower-prefix bubble. Choose a nonnegative smooth radial compact cutoff χ\chi and define

Q1=∫ρχ,Q2=∫ρχ2,gj=MW10∫χj∣∇W∣2(j=1,2). Q_1=\int\rho\chi,\quad Q_2=\int\rho\chi^2,\qquad g_j=\frac{M_W}{10}\int\chi^j|\nabla W|^2\quad(j=1,2).

These positive constants remain fixed as the bubble moves outward. Charges involve χ\chi; energies involve χ2\chi^2. On the high core, the active label covariance and its inverse are

C=A[(d+1)I−11T],A=NH(NH+d+1)(d+1)2(d+2),C−1=I+11TA(d+1). C=A[(d+1)I-\mathbf1\mathbf1^T],\qquad A=\frac{N_H(N_H+d+1)}{(d+1)^2(d+2)},\qquad C^{-1}=\frac{I+\mathbf1\mathbf1^T}{A(d+1)}.

Let Δi=QqH,i−∫ρqi(eX)\Delta_i=Qq_{H,i}-\int\rho q_i(e_X) be the active phase-charge deficit. Set c=C−1Δ/(ϵQ1)c=C^{-1}\Delta/(\epsilon Q_1) and

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