Current Geometry of the Inferred Excited Shells

The measured masses enter the constrained spectral inference; sector depths and mode indices are its inferred assignments. This calculation tests the spatial dynamics associated with those assignments. It does not treat the indices as additional independently measured inputs or fit amplitudes to recover the masses.

The responding clock action preserves the fourteen stationary count clocks and has a complete linear orbital persistence window for ground controls. The actual inferred particles are mostly excited shells. The following exact identities identify their current channels before a larger persistence calculation.

1. The Complete Selected Massive Shell

On the conditional ordinary unit sphere put

$$h_l=l+d/2,\quad c_l=\epsilon_dP_d'(h_l),\quad x_l=\sqrt{1-c_l^2},\quad a_l^2=(1+c_l)/2,\quad b_l^2=(1-c_l)/2,$$ $$\lambda_l=h_l/c_l,\qquad m_l=h_lx_l/c_l.$$

Here $l=n-1$, and $\epsilon_d$ is one sector value within the constitutive chart. The co-rotating particle operator on an affine plateau is $\mathcal A_l=D+m_l\beta-\lambda_l I$, with $\{D,\beta\}=0$. If $D E=h_l E$, then

$$Z_lE=(a_lI+b_l\beta)E,\qquad (D+m_l\beta)Z_lE=\lambda_lZ_lE. \tag{1}$$

All positive bare-shell eigenfields enter (1), including arbitrary complex column combinations and spatial polarizations. Conversely the positive massive eigenspace at $\lambda_l$ is obtained this way from the paired bare shells $\pm h_l$. This is the complete zero space of $\mathcal A_l$, not just the selected zonal background.

The map $Z_l$ preserves the integrated canonical inner product on the positive bare shell: $a_l^2+b_l^2=1$ and the matrix element of $\beta$ between two positive bare-shell fields is zero. This is a classical field normalization statement, with no probability postulate.

2. Exact Current Suppression and the Missing Gradient Vertex

Since $\{\alpha_i,\beta\}=0$,

$$(a_lI+b_l\beta)\alpha_i(a_lI+b_l\beta)=c_l\alpha_i.$$

For a background $\Phi_l=Z_lE_l$ and a zero-shell variation $\delta\Phi=Z_l\delta E$, the complete linear vector current obeys

$$\boxed{\delta j_i=c_l\,2\operatorname{Re} \operatorname{tr}E_l^\dagger\alpha_i\delta E.} \tag{2}$$

The factor is a consequence of the selected spatial mixing, rather than a new depth-only coupling coefficient. Its physical interpretation requires the complete response and normalization; (2) alone is not an observed electric charge or interaction rate.

Let $P_Z$ be the orthogonal projector onto this selected massive shell. For any smooth real scalar $f$,

$$[D+m_l\beta,f]=-i\alpha(\nabla f), \qquad \boxed{P_Z\alpha(\nabla f)P_Z=0.} \tag{3}$$

Both sides of the commutator act at the same eigenvalue $\lambda_l$, so their compressed difference vanishes. Equivalently, integrating the linear current against every gradient gives zero. Direct differentiation also gives

$$\boxed{\nabla\cdot\delta j=0.} \tag{4}$$

The selected-shell vector response therefore occupies coexact spatial one-forms on the sphere. It cannot couple to a gradient current within that degenerate shell. Transitions out of the shell are outside (3), and must not be assigned the same zero vertex.

3. Actual Spatial Kernels at All Fourteen Assignments

An explicit positive shell centered at a unit ambient direction $p$ uses $t=X\cdot p$, $v=\nabla t$, and

$$A_l(t)=\frac{l+d-1}{d-1}C_l^{(d-1)/2}(t),\qquad B_l(t)=C_{l-1}^{(d+1)/2}(t),$$ $$E_l(X;p)=\frac{[A_l(t)I-iB_l(t)\alpha(v)]K(X)} {\sqrt{\binom{l+d-1}{d-1}}}. \tag{5}$$

For $l=0$, the polynomial coefficient $B_l$ is zero. The Killing frame has $KK^\dagger=I/r$ and $\nabla_iK=(i/2)\alpha_iK$. Ordinary sphere identities give $\nabla_i\nabla_jt=-t\delta_{ij}$, $A_l'=(l+d-1)B_l$, and the Gegenbauer ODE. Substitution into the full covariant Dirac derivative proves $D E_l=h_lE_l$.

The checker uses two independently chosen centers and arbitrary complex right-column variations, retaining $\nabla K$ in both fields. It tests (1), (2) and (4) for W, Z, Higgs, down, strange, up, charm, top, the three maintained neutrino comparisons, electron, muon and tau. Each depth uses $\epsilon_d=0.3/\max P_d'(h_l)$ over its maintained assignments. This is a declared common sector chart control; it does not derive a physical radius or independently calibrate a field amplitude.

4. A Positive Static Background Response and Its Resonant Exception

For the retained positive Killing background, its coexact harmonic $L\ge1$ has the Dirac matrix

$$C_T=\begin{pmatrix}1-d/2&\sqrt{\Lambda_T}\\ \sqrt{\Lambda_T}&d/2-2\end{pmatrix},\qquad \Lambda_T=(L+1)(L+d-2).$$

The co-rotating background matrix is $A_B=C_T-dI/2$. For the unit vector-current source $e_1$,

$$\boxed{F_B(L)=e_1^TA_B^{-1}e_1 =\frac2{(L-1)(L+d)}>0,\qquad L\ge2.} \tag{6}$$

This identity holds at every occupied depth, including two. At $L=1$ the determinant is zero. That harmonic contains an actual free-background zero mode and must remain in the coupled dynamics. Replacing its pole by zero or deleting that channel would alter the action's response.

To state the leading static consequence, assume a source compatible with this first-harmonic solvability condition, and expand at small particle to background density ratio. Let $\eta_l=\omega_l-\lambda_l$ denote the affine density coefficient, distinct from the mixing coefficient in (1). The background quadratic energy to leading order is

$$\langle\delta B,A_B\delta B\rangle -\frac{\eta_l}{\rho_B}\int\delta J_B\cdot\delta j.$$

Eliminating its nonzero-frequency static response gives the conditional quadratic form on the particle zero shell

$$\boxed{\mathcal Q_{\rm eff} =-\frac{\eta_l^2}{\rho_B}\sum_{L\ge2} \frac{2\|\delta j_L\|_2^2}{(L-1)(L+d)}\le0.} \tag{7}$$

The exact background self-current term contributes at the next order in this small-load expansion. With a variable excited-state density it mixes angular harmonics and must be retained beyond leading order.

Equation (7) is a signed response result. It is not a persistence theorem: the resonant $L=1$ sector requires the dynamical charge-fixed treatment linked below, a statically compatible subspace has not been proved dynamically invariant, and nonzero-frequency resonances with the surrounding spectrum are not covered. The induced quadratic form is nonpositive wherever this static elimination is valid; its kernel and its complete Hamiltonian dynamics need further analysis.

5. Locality of the Retained Dynamics

The inverse operator in (6) is a static response of retained modes of the same field. It is not an additional mediator or a physical assertion of instantaneous propagation. The full two-block action contains no derivative interaction. On its smooth timelike and constitutive domain the principal matrices are the canonical $\alpha_i$ for each block, with $\|\alpha(n)\|=1$ for every unit tangent direction.

For two smooth solutions whose values stay in a bounded subset separated from the domain boundaries, their nonlinear sources have a finite local Lipschitz constant $L_N$. The difference $\Delta\Psi$, including both retained blocks, satisfies

$$\partial_t|\Delta\Psi|^2+\nabla\cdot j_\Delta \le2L_N|\Delta\Psi|^2,\qquad |j_\Delta\cdot n|\le|\Delta\Psi|^2.$$

Integrate over a region whose boundary retreats inward at unit speed. The boundary term is nonpositive, and Gronwall's inequality preserves zero difference if the solutions initially agreed there. Thus changes have finite geodesic propagation speed at most one, while the stated smooth solutions and domain separation persist. A static reduction must not replace this complete time evolution when interpreting timing. This conditional finite-propagation statement does not establish global regularity, preserved timelikeness or a vacuum extension.

Reproducibility and Scope

The single checker passes 210 spatial evaluations and exact all-depth background resolvent identities. The maximum positive bare-shell residual is $9.3\times10^{-15}$; the massive-shell, current scaling and divergence errors are below $4.9\times10^{-16}$. It retains the actual source hash and the dated result.

The commutator and response identities apply to the inferred excited assignments without a particle-by-particle adjustment. They provide a smaller, geometric persistence problem. They do not yet establish complete excited-state persistence, localization, charge, observed spin, decay channels or independent preference for the sector assignments.

The charge-fixed dynamical current calculation now resolves the first-harmonic background current susceptibility. Its zero field mode remains retained; fixing the conserved full matrix charge removes its conjugate current preparation. The resulting first-harmonic term has the opposite sign to (7). The accessible selected-shell currents and their reciprocal dynamics still require calculation.