The Charge-Fixed Dynamical Background Current

The assigned-shell calculation leaves the first coexact background harmonic dynamical because its static field operator is singular. Its current response can nevertheless be calculated. The distinction follows from the equations and conserved matrix charges; it does not require removing the field mode or prescribing a response. Here “charge” denotes a Noether matrix integral, not electric charge.

1. The Conserved Matrix Integrals

On an affine plateau the complete retained equations have Hermitian left-acting Hamiltonian sources. Consequently each matrix block obeys a matrix-valued continuity equation, before taking a trace:

$$\partial_t(B^\dagger B)+\nabla_i(B^\dagger\alpha_iB)=0, \qquad \partial_t(\Phi^\dagger\Phi)+\nabla_i(\Phi^\dagger\alpha_i\Phi)=0. \tag{1}$$

The cancellation uses the same source and its adjoint in the two terms; spatial variation of the source does not introduce a derivative here. On the closed conditional sphere,

$$\mathsf Q_B=\int B^\dagger B,\qquad \mathsf Q_\Phi=\int\Phi^\dagger\Phi \tag{2}$$

are separately conserved Hermitian matrices. They are the integrals associated with the separate constant right-unitary symmetries of the stated two-block action. The block split remains an additional premise.

For the positive Killing background $B_0=\sqrt{\rho_B}K$, a free zero-shell variation is $\delta B=K C$ with constant matrix $C$. Its linear charge variation is proportional to $C+C^\dagger$, since $K^\dagger K=I/r$. Fixing the full matrix (2) to first order removes its Hermitian component. The remaining anti-Hermitian component is an actual tangent $\delta B=B_0T$ to the right-unitary orbit. Variations orthogonal to the background Dirac shell contribute zero to this linear integrated charge. Fixing only the scalar trace would be insufficient.

2. Current Response with the Zero Field Mode Retained

Set $R=1$; all disturbance frequencies below are in inverse-radius units, distinct from the selected particle oscillation frequency. For a coexact harmonic $L\ge1$, define

$$A_B=\begin{pmatrix}1-d&\sqrt{(L+1)(L+d-2)}\\ \sqrt{(L+1)(L+d-2)}&-2\end{pmatrix},\qquad e_1=(1,0)^T.$$

In the actual anti-Hermitian Clifford-form basis, write the background disturbance as $u+i v$. With an imposed particle current $j_L(t)$, the leading free-background equations and measured current are

$$\dot u=A_Bv-\eta_l e_1j_L(t),\qquad \dot v=-A_Bu,\qquad \delta J_{B,L}=2e_1^Tv. \tag{3}$$

Here the source normalization follows from the canonical background kinetics; the factors of $\rho_B$ cancel between its source and current. The reciprocal particle equation remains required for complete dynamics. For squared disturbance frequency $z=\nu^2$, the susceptibility in $\delta J_{B,L}=2\eta_l F_B(L,z)j_L$ is

$$\boxed{F_B(L,z)=e_1^TA_B(A_B^2-zI)^{-1}e_1 =\frac{2(L-1)(L+d)+(d-1)z} {((L-1)^2-z)((L+d)^2-z)}.} \tag{4}$$

The equivalent retarded initial-value response is

$$\delta J_{B,L}(t)=2\eta_l\int_0^t e_1^T\sin[A_B(t-s)]e_1\,j_L(s)\,ds. \tag{5}$$

At $L=1$, $A_B$ has eigenvalues $0,-(d+1)$. The sine in (5) annihilates its zero mode. The current susceptibility therefore has a removable zero-frequency factor:

$$\boxed{F_B(1,z)=\frac{d-1}{z-(d+1)^2},\qquad F_B(1,0)=-\frac{d-1}{(d+1)^2}.} \tag{6}$$

This does not eliminate the field zero mode. Its $v$ coordinate is conserved and corresponds to a matrix-charge variation. At fixed (2) it vanishes initially and remains zero. Its $u$ coordinate is driven by the integral of the source and is the retained right-unitary tangent. A nonzero initial conjugate charge gives an additional persistent current preparation; (6) is the source-induced response on the fixed-charge level.

For example, with $\eta_l=1$, zero initial disturbance and constant unit source, the full equations give

$$\delta J_{B,1}(t)=-\frac{2(d-1)}{(d+1)^2} [1-\cos((d+1)t)]. \tag{7}$$

They simultaneously retain the linearly drifting zero field coordinate. For cosine forcing at frequency $\nu$, (7) becomes $2F_B(1,\nu^2)[\cos(\nu t)-\cos((d+1)t)]$. The finite static coefficient is a zero-frequency response or a symmetry-reduced stationary response. An undamped initial-value solution need not converge to a static limit.

3. Exact Background Self-Current Feedback in Ground Controls

For a spatially constant particle density ratio $g=\rho_\Phi/\rho_B$, the retained background self-current term changes (3) to

$$\dot u=A_Bv-\eta_l e_1j_L+2\eta_l g e_1e_1^Tv, \qquad \dot v=-A_Bu.$$

Thus its current response with the particle current supplied is

$$F_g(L,z)=\frac{F_B(L,z)}{1+2\eta_l g F_B(L,z)}. \tag{8}$$

At the first harmonic this is exactly

$$\boxed{F_g(1,z)=\frac{d-1}{z-\omega_g^2},\qquad \omega_g^2=(d+1)^2-2\eta_l g(d-1).} \tag{9}$$

For the maintained ground controls $\eta_l<0$, this response pole moves upward and stays real. The zero field direction remains present. This is a background susceptibility with its self-feedback included, not the spectrum of the fully reciprocal particle-background system. An excited zonal particle has spatially varying density, which mixes harmonics in this feedback term; (8) cannot be applied to it independently in each harmonic as an exact finite-load result.

4. The Leading Excited-Shell Form Has Two Signs Available

For $L\ge2$, (4) at zero frequency is the previously derived positive $2/[(L-1)(L+d)]$. On the fixed background matrix-charge level, eliminating the nonzero-frequency background response to leading small load gives

$$\boxed{\mathcal Q_{\rm eff}=\frac{\eta_l^2}{\rho_B} \left[\frac{d-1}{(d+1)^2}\|\delta j_1\|_2^2 -\sum_{L\ge2}\frac{2\|\delta j_L\|_2^2}{(L-1)(L+d)}\right].} \tag{10}$$

The first term is positive and the higher terms are negative. Equation (10) extends the earlier higher-current-only compatible static calculation. It does not establish that the form on the actual particle zero shell is indefinite: the accessible currents, their kernel and their symplectic structure must be determined. Even a definite reduced energy does not settle resonances with the omitted nonzero-frequency particle modes. The complete equations retain both blocks and their conserved charges.

5. A Geometric Ordering to Test Further

The magnitude of the first-harmonic free static susceptibility is $(d-1)/(d+1)^2$. Its occupied-depth values are

The exact identity

$$\frac18-\frac{d-1}{(d+1)^2} =\frac{(d-3)^2}{8(d+1)^2}\ge0$$

places the maximum at depth three, with four next among the maintained occupied depths. Every other depth has a nonzero response. Radii, $\eta_l$, source normalization and the actual particle current maps are absent from this ordering, so it is a response lead rather than a derived color rule, electric charge, force hierarchy or independent support for the assigned sectors.

Reproducibility and Physical Scope

The single checker verifies (4), (6), (8) and (9) symbolically at every occupied depth and independently integrates the complete four-dimensional real background ODE for constant and cosine sources. It also tests self-feedback for the common ground preparations $c=3/5$ at $g=1/100$ and $1/5$. Its final report passes 84 evaluations; the maximum current discrepancy is $3.9\times10^{-11}$, and the conserved charge-coordinate discrepancy is below $7.9\times10^{-15}$. Both successful source versions and dated reports are preserved with exact hashes.

These are currents of retained components of the single field. There is no extra mediator or particle-exchange interpretation. The full local Dirac system supplies the conditional finite-propagation result already proved in the assigned-shell note; a per-harmonic static reduction does not replace it. Excited-shell persistence, nonlinear domain preservation, unbounded-space localization and the origin of the effective action remain unresolved.

The exact inferred-shell orientation currents now supply an actual charge-horizontal family for (10). Pure orientation tangents carry zero current; conjugate disturbances have an exact finite coexact expansion and a rational signed response at every assignment. The electron and muon responses in this family are negative, which alone establishes neither instability nor decay. The full accessible current map and coupled Hamiltonian evolution remain open.