A Charge-Horizontal Shape Current at Every Excited Assignment

The orientation calculation constructs the normalized positive shell $E_l=M_lK/\sqrt N$, its horizontal center derivative $W_a$, and its exact canonical norm $G_l=l(d-1)(l+d)/d^2$. Here $l=n-1$, where the index and depth are inferred from measured masses under the shared spectral rules. Right multiplication by the constant Hermitian Clifford matrix $\alpha_a^{\rm right}$ preserves the shell and horizontality. Consider

$$Y_a=W_a\alpha_a^{\rm right}. \tag{1}$$

The full matrix integral $\int E_l^\dagger Y_a$ vanishes, and $\|Y_a\|^2=G_l$. This is a conserved-matrix-charge constraint, not an electric-charge assignment. The conjugate $iY_a$ has the same norm and canonical symplectic pairing $2G_l$ with $Y_a$.

1. An Explicit Current and Its Harmonics

Use $t=X\cdot p$, $w=X\cdot a$, $v=\nabla t$, and $z=\nabla w$, with $p,a$ orthonormal. The real Gegenbauer polynomials and normalization are

$$A=\frac{l+d-1}{d-1}C_l^{(d-1)/2}(t),\quad B=C_{l-1}^{(d+1)/2}(t),\quad N=\binom{l+d-1}{d-1}.$$

Set $B=0$ at $l=0$. Direct full Clifford traces give the bare linear current of (1):

$$\boxed{J_{\rm sh}=[\phi-(t\phi'+d\phi)w^2]v +[-(1-t^2)\phi'+dt\phi]wz,\qquad \phi=\frac{2B(A+tB)}N.} \tag{2}$$

It is the codifferential of the closed two-form $\phi w\,dt\wedge dw$. In particular its divergence vanishes. For the massive selected shell, multiply (2) by $c_l$, and by $\mathcal A$ if the base field and disturbance both carry physical amplitude $\sqrt{\mathcal A}$. That amplitude remains uncalibrated.

The polynomial $\phi$ is odd with degree at most $2l-1$. The coexact basis in this angular family uses $\phi_L=C_{L-2}^{(d+3)/2}$, $L\ge2$, in (2). Its exact Hodge eigenvalue and normalized squared current norm are

$$\Lambda_L=(L+1)(L+d-2),\qquad \boxed{\|V_L^{\rm sh}\|^2= \frac{\Lambda_L(d-1)}{(d+1)(2L+d-1)}\binom{L+d}{d+2}.} \tag{3}$$

The Gegenbauer ODE is $(1-t^2)\phi_L''-(d+4)t\phi_L' +(L-2)(L+d+1)\phi_L=0$. Substituting it into the ordinary sphere Hodge operator verifies (3)'s eigenvalue in general dimension. For the norm, the two-form has squared pointwise norm $\phi_L^2 w^2(1-t^2-w^2)$. Conditional sphere moments give its transverse average $(d-1)(1-t^2)^2/[d(d+2)]$; integration by parts and the Gegenbauer norm give (3).

Thus the complete expansion is

$$J_{\rm sh}=\sum_{L=3,5,\ldots,2l+1}q_L V_L^{\rm sh}. \tag{4}$$

There is exactly no first coexact harmonic. Exact rational polynomial elimination computes every $q_L$, independently of numerical spatial quadrature. Since $B(A+tB)$ is a nonzero polynomial for $l\ge1$, at least one current power is strictly positive.

2. A Strictly Negative Leading Response

The charge-fixed background response therefore gives

$$\boxed{S_{\rm sh}=-\sum_{L=3,5,\ldots,2l+1} \frac{2q_L^2\|V_L^{\rm sh}\|^2}{(L-1)(L+d)}<0, \qquad l\ge1.} \tag{5}$$

At $l=0$ the direction itself vanishes. The physical reduced form is $\eta_l^2\mathcal A^2 c_l^2 S_{\rm sh}/\rho_B$. Its normalization cannot be omitted in a physical rate comparison.

Representative exact values are $S_{\rm strange}=-233/500$, $S_{\rm up}=-2440/7007$, and $S_e=-354830314897901983/14445741281688310$. The report preserves all exact fractions and harmonic powers.

These signs apply throughout the occupied excited sectors; they are not a depth-three/four selector or independent support for their assignments. A negative constrained energy direction does not by itself establish exponential growth, observed decay or negative complete particle inertia. In fact the conjugate $iY_a$ belongs to an exact stationary shape family. The full symplectic dynamics, reciprocal frequency response and off-shell modes remain necessary for a persistence conclusion.

3. A Broader Bounded Polarization Check

The same script retains two transverse center directions $a,b$, right matrices $I,\alpha_a,\alpha_b,i\alpha_a\alpha_b$, and both real phases. It computes their full real Gram matrix, canonical symplectic form, current Gram matrix and first-harmonic moments. Two independent cubatures agree. After removing null field combinations, the real ranks are eight at depth two and sixteen at the other excited depths; the symplectic map squares to $-4I$ in the physical metric. These finite ranks are numerical, whereas (2)–(5) are exact polynomial results.

The numerical span also contains higher-current witnesses orthogonal to both orientation/conjugate pairs, with zero first-harmonic current. This is broader than a two-direction calculation, but it is still not the entire degenerate shell or an invariant dynamical truncation.

Reproducibility

The single checker combines actual Clifford contractions, full finite-span symplectic geometry, symbolic Hodge identities, rational expansions and two independent spatial cubatures. Its final report passes 300 evaluations across fourteen assignments. The exact profile agrees with actual matrices within $5.1\times10^{-13}$ normalized error. Successful versions and corrected pilot errors retain their exact source bytes. These are internal classical field deformations, distinct from the electron's literal deterministic position trajectory.