Exact Stationary Shape Freedom at Fixed Clock and Matrix Charge

The inverse responding-clock action supports more than the original zonal matrix profile. The following family changes its actual spatial density while keeping its inferred count clock and conserved matrix integral fixed. This is a statement about exact stationary solutions, not a linear stability or particle-identification result.

1. The Same-Shell Shape Generator

Use the definitions of the horizontal orientation derivative: $E_l=MK/\sqrt N$, $M=A I-iB\alpha(v)$, $t=X\cdot p$, $w=X\cdot a$, $v=\nabla t$, and $z=\nabla w$. Let $k=l/d$,

$$D_a=A'wI-iB'w\alpha(v)-iB\alpha(z),\qquad Q_a=\alpha(tz-wv)-\frac i2[\alpha(v),\alpha(z)].$$

The constant right Clifford generator satisfies $Q_aK=K\alpha_a^{\rm right}$, $Q_a^2=I$. Define

$$\boxed{F_a=iW_a\alpha_a^{\rm right} =\frac{[iD_aQ_a-kM]K}{\sqrt N}.} \tag{1}$$

Center differentiation and constant right multiplication preserve the positive Dirac shell. Hence $DE_l=h_lE_l$ and $DF_a=h_lF_a$, where $h_l=l+d/2$ at unit radius. Moreover,

$$\int E_l^\dagger F_a=0,\qquad \int\operatorname{tr}F_a^\dagger F_a=G_l =\frac{l(d-1)(l+d)}{d^2}. \tag{2}$$

The zero in (2) is a full matrix, not merely a real traced overlap.

2. Pointwise Balance Survives Finite Shape Changes

Write $V=\alpha(v)$, $Z=\alpha(z)$, $C=[V,Z]/2$, $s_t^2=1-t^2$, and put $T=A'+tB'+B$, $L=t(A'+B)-s_t^2B'$. Direct multiplication gives

$$iD_aQ_a-kM=(Bt-B'w^2-kA)I +i[(1+k)B-Tw^2]V+iwLZ+wTC. \tag{3}$$

Every coefficient is real. The non-scalar part is anti-Hermitian, and its square is scalar: $V^2,Z^2,\{V,Z\}$ are scalar, $\{V,C\}=\{Z,C\}=0$, and $C^2=((v\cdot z)^2-|v|^2|z|^2)I$. These are ordinary Clifford identities valid at every occupied depth. Thus any real combination of $M$ and the matrix in (3) has scalar left and right Gram matrices.

For $l\ge1$, define

$$\boxed{E_\theta=\cos\theta\,E_l +\frac{\sin\theta}{\sqrt{G_l}}F_a.} \tag{4}$$

Pointwise balance and (2) imply

$$E_\theta E_\theta^\dagger=\frac{\rho_\theta(X)}r I, \qquad \int E_\theta^\dagger E_\theta=I/r. \tag{5}$$

In deriving the second identity, pointwise scalar Gram matrices turn the norm in (2) into $\int F_a^\dagger F_a=G_l I/r$. The cross matrix vanishes by exact horizontality.

3. Exact Solutions of Both Nonlinear Field Equations

Apply the same massive-shell map $Z_l=a_l I+b_l\beta$, with $a_l^2+b_l^2=1$, $a_l^2-b_l^2=c_l$, $2a_lb_l=x_l$. Put

$$\Phi_\theta=\sqrt{\mathcal A}\,Z_lE_\theta e^{-i\omega_l t}, \qquad B_0=\sqrt{\rho_B}K e^{-id t/2}. \tag{6}$$

At radius one the unchanged plateau coefficients are $m_l=h_lx_l/c_l$, $\eta_l=\omega_l-h_l/c_l$. Equation (5) gives $j_\Phi=(\mathcal A\rho_\theta,0)$ and $s_\Phi=x_l\mathcal A\rho_\theta$. Therefore the background source $C_\mu=\eta_l(j_\mu-q u_\mu)/W_B$ vanishes pointwise. The retained background remains an actual solution, and

$$(D+m_l\beta+\eta_l)\Phi_\theta=\omega_l\Phi_\theta. \tag{7}$$

No source coefficient or physical amplitude is adjusted along (4). Since $\{D,\beta\}=0$, the integrated matrix $\int E_\theta^\dagger\beta E_\theta$ vanishes: its two sides belong to opposite nonzero Dirac eigenvalues. Consequently the complete massive matrix charge is $\mathcal A I/r$ throughout the family. The complete stationary energy remains

$$\boxed{E=\omega_l\mathcal A+\frac d2 Q_B.} \tag{8}$$

Restoring radius changes the background term to $dQ_B/(2R)$. The arbitrary physical amplitude is retained explicitly.

4. A Nontrivial Density Change and an Open Constitutive Chart

For the unnormalized family $E_l+\tau F_a$, the actual density tangent is

$$\boxed{\delta\rho=\frac{2U(t)}N \left[w^2-\frac{1-t^2}{d}\right],\qquad U=BA'-AB'+B^2=(l+d)B^2-AB'.} \tag{9}$$

The shell identities $A'=(l+d-1)B$ and $dtB-(1-t^2)B'=lA$ prove (9). Its transverse average is zero. It is a quadrupolar change rather than an orientation tangent, whose zonal density derivative is proportional to $w$; a right-unitary tangent has zero density derivative altogether.

For $l\ge1$, $U$ is nonzero. If the leading coefficient of $B$ is $b_*$, its degree $2l-2$ coefficient is $(2l+d-1)b_*^2/l>0$. Thus (4) changes density at every excited assignment. At $l=0$, $F_a=0$, so this family is absent.

The base density $(A^2+B^2(1-t^2))/N$ is strictly positive on the entire sphere. Interior common zeros of $A,B$ would also give $A=A'=0$, contradicting uniqueness for the regular Gegenbauer ODE; the endpoint values of $A$ are nonzero. Compactness gives a positive minimum. Small enough $|\theta|$ therefore preserves nonzero density and the same open constitutive chart everywhere. Equation (7) then holds as a global nonlinear solution, not just a formal tangent. No claim that all finite angles remain nodeless is needed or made.

Consequence and Reproducibility

The inverse law realizes a common count clock while permitting genuine shape freedom at fixed charge and energy. This does not yet select a unique particle shape, prove persistence modulo that freedom, or establish localized support and translational inertia in unbounded space. Its symplectic conjugate has the strictly negative leading response derived in the shape-current calculation; that sign alone cannot distinguish drift along stationary shapes from an actual growing mode.

The independent checker verifies full covariant spatial jets, Clifford balance, (9), both nonlinear equations and the local complete energy at three spatial points and three angles for every assignment. Independent moment quadratures check the global norm. The report passes 1,079 evaluations; spatial eigenfield discrepancies are below $5.0\times10^{-15}$, and norm comparisons below $1.1\times10^{-12}$. The analytic arguments above supply the global identities beyond those finite point checks.

The conditional ordinary sphere, equivariant same-field block split and inverse constitutive law remain premises. Generic all-depth shape freedom does not independently favor the inferred sector assignments. These internal field shapes do not replace literal deterministic electron motion.