A Responding Same-Field Clock and Its Complete Ground Current Channels

Measured masses are the inputs to the constrained spectral inference; depths and mode indices are its inferred assignments under shared count and scale rules. Those assignments enter the subsequent dynamical test here. The constructed action realizes the supplied spectrum rather than independently deriving it. The additional question is whether its spatial structure persists when the field defining its local time direction also responds.

1. A Covariant Continuation with Both Equations Retained

Let $\Phi$ and $B$ be two separately retained matrix blocks of $\Psi_\infty$, each with the canonical commuting Dirac kinetic action. Their Lorentz-equivariant decomposition and independent phase symmetries are additional premises. They are field components, with no independent mediator particle. Use signature $(+,-,\ldots,-)$, and define

$$s=\operatorname{tr}\bar\Phi\Phi,\quad j^\mu=\operatorname{tr}\bar\Phi\gamma^\mu\Phi,\quad J_B^\mu=\operatorname{tr}\bar B\gamma^\mu B,$$ $$W_B=\sqrt{J_B^\mu J_{B,\mu}},\qquad u^\mu=J_B^\mu/W_B,\qquad q=u_\mu j^\mu.$$

Work on the future timelike background domain $J_B^0>0,W_B>0$. The proposed potential is

$$V_d=\frac qR f_d^{\rm flat}(s/q). \tag{1}$$

The same shell-plateau prescription as in the persistence construction defines $f_d^{\rm flat}$. At a selected shell, its affine neighborhood reduces (1) to

$$V_d=m_l s+b_l q,\qquad m_l=\frac{h_lx_l}{Rc_l},\quad b_l=\frac{\omega_lR-h_l/c_l}{R},\quad h_l=l+d/2. \tag{2}$$

Here $b_l<0$ for the maintained inverse count branch. The letters $m_l,b_l$ denote source coefficients; $m_l$ is not a newly inferred observed particle mass. The physical mass-frequency target remains $\omega_l=\mu_d\binom{l+d}{d}$.

Vary both blocks. On the plateau, the complete covariant equations are

$$i\not\nabla\Phi=(m_lI+b_l\not u)\Phi, \qquad i\not\nabla B=\not C\,B,$$ $$\boxed{C_\mu=\frac{b_l}{W_B}(j_\mu-q u_\mu).} \tag{3}$$

In a time slicing the sources are $G_\Phi=[m_l\beta+b_l(u^0I-u^i\alpha_i)]\Phi$ and $G_B=(C_0I+C_i\alpha_i)B$, with the displayed lower-index convention. The sign of every spatial contraction follows from the stated metric. The new density is Lorentz scalar when both fields transform; a particular nonzero solution still selects a physical flow frame. It retains the background's equation and kinetic energy.

2. Exact Stationary Spatial Clocks and Complete Energy

On the conditional ordinary round sphere, let $K$ be a positive Killing matrix frame, normalized pointwise by $KK^\dagger=I/r$, with $D_RK=dK/(2R)$. Choose

$$B_0=\sqrt{\rho_B}\,K e^{-i\Omega t},\quad \Omega=\frac d{2R},\qquad \rho_B>0.$$

It has $J_B=(\rho_B,0)$. Each of the fourteen maintained matrix shell solutions has $j=(\rho_\Phi(q),0)$. Consequently $q=\rho_\Phi$ and $C=0$ pointwise, even when the particle density varies in space. The background is therefore an actual free stationary solution of (3), and the original $\Phi_l$, its clock and its complete local source are unchanged. This statement uses the same inverse prescription at every occupied depth; it does not force one particle to be another's restriction.

The full canonical energy ledger at this stationary pair is

$$\boxed{E=\omega_l Q_\Phi+\Omega Q_B.} \tag{4}$$

The interaction has degree two in $\Phi$ and degree zero in $B$, so its Euler contractions are $\Phi^\dagger G_\Phi=V_d$ and $B^\dagger G_B=C\cdot J_B=0$. The latter does not set the free background kinetic energy to zero. Relative to this same unchanged stationary background the excess is $\omega_l Q_\Phi$. The sphere has finite volume; an infinite-space background subtraction is a different boundary problem.

3. Responding Current Hessian

At a rest-current background put $g=\rho_\Phi/\rho_B$. This is a physical density ratio, currently uncalibrated; it is not a mass-fitting parameter. For arbitrary complex matrix disturbances,

$$\delta J_{B,i}=2\operatorname{Re}\langle\alpha_i B_0,\delta B\rangle, \quad \delta j_i=2\operatorname{Re}\langle\alpha_i\Phi_0,\delta\Phi\rangle.$$ $$\delta G_\Phi=(m_l\beta+b_lI)\delta\Phi -\frac{b_l}{\rho_B}\sum_i\alpha_i\Phi_0\,\delta J_{B,i},$$ $$\boxed{\delta G_B=\frac{b_l}{\rho_B}\sum_i\alpha_iB_0 \left(-\delta j_i+g\,\delta J_{B,i}\right).} \tag{5}$$

The current-dependent second-order potential is

$$-\frac{b_l}{\rho_B}\delta J_B\cdot\delta j +\frac{b_l\rho_\Phi}{2\rho_B^2}|\delta J_B|^2. \tag{6}$$

Independent joint action variations, with actual full-rank Clifford matrices, verify (3), (5) and (6) across depths two, three, four, five, six and ten. They also verify simultaneous boosts of both fields. Holding $u$ fixed would remove precisely these terms and would describe a different persistence calculation.

4. Spatial Geometry Exposes the Missing Current Channel

The following spectral comparison uses ground-shell controls, not the assigned excited electron, muon, tau or neutrino states. Set $R=1$, $h=d/2$, $c=3/5$, $x=4/5$, $a=2/\sqrt5$, $b=1/\sqrt5$, $\lambda=h/c$, $m=hx/c$, and $b_0=c/H_d-\lambda$. Here $H_d=\sum_{j=1}^d1/j$, and $b_0$ is the coefficient in (2), distinct from the mixing coefficient $b$. This ground control uses $\epsilon_d=c/H_d$. It is a separate preparation from the smaller sector values used to keep all fourteen maintained excited assignments inside their open constitutive charts.

Write a matrix field as $M K$. In an orthonormal tangent frame,

$$D(MK)=\left[-i\sum_i\alpha_i\nabla_iM +\frac12\sum_i\alpha_i M\alpha_i\right]K. \tag{7}$$

On an exterior Clifford grade $p$, the second term is $(-1)^p(d-2p)M/2$. Full even-depth Clifford forms span the matrix carrier; odd depths require their additional $\beta$-multiplied copy. Their trace characters are checked at all retained ranks.

Physical vector-current sources occupy the vector grade and its $\beta$ partner. A vector harmonic can be a gradient or be divergence free. The latter, coexact family is absent from a zonal radial vector ansatz. The two ordinary Hodge eigenvalues are

$$\Lambda_S=L(L+d-1),\qquad \Lambda_T=(L+1)(L+d-2),\quad L\ge1.$$

The coexact formula is the $p=1$ specialization of Dowker, equation (1), after shifting his nonnegative label to $L\ge1$. This standard spectrum does not supply the coupled dynamics derived here.

For a normalized scalar harmonic, its gradient and scalar span

$$C_S=\begin{pmatrix}h&\sqrt{\Lambda_S}\\ \sqrt{\Lambda_S}&1-h\end{pmatrix}.$$

For a normalized coexact one-form $v$, use $M_1=-i\alpha(v)$ and $M_2=-\sum_{ij}\alpha_i\alpha_j\nabla_i v_j/\sqrt{\Lambda_T}$. Equation (7) gives

$$C_T=\begin{pmatrix}1-h&\sqrt{\Lambda_T}\\ \sqrt{\Lambda_T}&h-2\end{pmatrix}. \tag{8}$$

An independent spatial check constructs the actual tangent vectors $v(X)=AX$ and $v(X)=X_2 AX$, with $A$ rotating ambient axes zero and one. These are the first two coexact harmonics on the ordinary unit sphere. Tangent first and second covariant jets verify their divergence, Hodge eigenvalues and both full matrix Dirac equations in (8). In particular, the second-harmonic instability below uses a verified spatial field, rather than just a guessed eigenvalue.

Append the $\beta$ partners, giving $D_F=\operatorname{diag}(C_F,-C_F)$ and $\beta_F=\left(\begin{smallmatrix}0&I_2\\I_2&0\end{smallmatrix}\right)$. The uncoupled co-rotating matrices are

$$A_\Phi=D_F+m\beta_F-\lambda I_4,\qquad A_B=D_F-h I_4.$$

Let $r_B=e_2$ in the scalar family, and $r_B=e_1$ in the coexact family; put $r_\Phi=a r_B-b\beta_F r_B$. The real and imaginary quadrature equations are $\dot p=Yq,\ \dot q=-Xp$, with

$$X=\operatorname{diag}(A_\Phi,A_B), \quad Y=X+\begin{pmatrix} 0&-2b_0\sqrt g\,r_\Phi r_B^T\\ -2b_0\sqrt g\,r_Br_\Phi^T&2b_0g\,r_Br_B^T \end{pmatrix}. \tag{9}$$

At depth two, the Clifford dual vector is another physical vector current. It adds the corresponding term to $X$, with $s_B=\beta_F r_B$ and $s_\Phi=a s_B-b\beta_Fs_B$. It already belongs to the scalar four-component family; it must not be counted again as an independent coexact block. The scalar characteristic polynomial from (7)–(9) agrees exactly with the independently derived earlier radial basis at every occupied depth.

Every physical current is covered by the gradient/coexact decomposition. The orthogonal complement of these closed source families has only the complex-linear self-adjoint free-plus-mass generators. Constant scalar harmonics give their separate two-component free blocks. Thus the current-active ground calculation is broader than a radial test. The reported finite probes do not by themselves certify all harmonic indices or bounded evolution of its zero modes.

5. An Exact Instability Threshold

For $d\ge3,L\ge2$ in the coexact family, define $\Delta=(L-1)(L+d)$ and $k=-2b_0>0$. Two resolvent identities give

$$r_B^TA_B^{-1}r_B=\frac2\Delta, \qquad r_\Phi^TA_\Phi^{-1}r_\Phi =c\,r_B^TA_B^{-1}r_B.$$

The second follows from $\lambda^2-m^2=h^2$, $a^2-b^2=c$, $2ab=x$, and $\lambda-xm=hc$. A block determinant then gives the exact identity

$$\boxed{\frac{\det Y}{\det X}=1-\frac g{g_*(d,L)},\qquad g_*(d,L)=\frac{\Delta^2}{2k\Delta+4k^2c}.} \tag{10}$$

When $g>g_*$, $\det(YX)<0$. Its real characteristic polynomial has at least one negative real squared-frequency root, hence an exponentially growing disturbance. A positive determinant alone does not prove stability. Since $g_*$ increases with positive $\Delta$, the second harmonic has the smallest such threshold among $L\ge2$.

At the common control $g=1/5$:

Direct exact Sturm counts at $L=1,2,3,5,10,20$ retain nonnegative squared frequencies in every sampled scalar family. In the coexact family the second harmonic at five, six and ten has growth approximately $0.11615/R,0.21177/R,0.32575/R$, respectively. The other reported coexact samples have nonnegative squared frequencies. These statements concern the stated density ratio and ground controls.

At the first scalar harmonic, the generator's zero-power nullities are $(4,4,4)$ above depth two, while depth two has $(2,4,4)$. The first coexact harmonic has $(2,4,4)$ at every tested higher depth. Those latter values exhibit secular zero-frequency Jordan blocks. Real squared frequencies therefore do not suffice for bounded linear norm persistence.

6. A Common Small-Load Spectral Window and the Zero Directions

At $g=1/100$, with the same $c=3/5$ ground preparation at every occupied depth, an exact all-harmonic certificate succeeds. Factor out the two independent background factors above depth two; the remaining coupled polynomial has degree six. Depth two has a degree-eight coupled polynomial. For its monic polynomial $p(z)=z^n+a_1z^{n-1}+\cdots+a_n$, use Newton identities to obtain the power sums $s_k$, and form

$$H_{ij}=s_{i+j},\quad 0\le i,j<n.$$

Every leading minor of $H$ is strictly positive for $L\ge2$. The certificate either has nonnegative numerator and denominator coefficients after $L=t+2$, or uses an exact zero Sturm-root count on $t\ge0$ and positive values at zero. The companion matrix satisfies $HC=C^TH$, so its roots are real. All elementary symmetric coefficients have strictly positive signs. Consequently the roots are positive. The first harmonic is checked separately by exact Sturm counts after stripping its zero multiplicity. The free background factors and uncoupled spatial complement are real by their self-adjoint generators.

This proves absence of exponentially growing ground harmonic eigenmodes under the stated action and common load. It is not a nonlinear persistence theorem or an identified electron result. A uniform bound on the complete spatial evolution additionally requires control of the diagonalizing maps across the infinite harmonic family; the following argument supplies that bound after the secular symmetry directions are removed.

The first coexact zero directions have a specific symmetry meaning. For $L=1$, put

$$P=M_1+\sqrt{(d-1)/2}\,M_2.$$

Direct spatial derivatives give $P^\dagger=-P$ and $\nabla_iP=(i/2)[\alpha_i,P]$. Since $\nabla_iK=(i/2)\alpha_iK$, this means $P K=K T$ for a constant anti-Hermitian right-column generator $T$. The corresponding particle and background zero vectors are independently $\delta\Phi=\Phi_0T$ and $\delta B=B_0T$. They are tangents to the exact separate right-unitary symmetries. At the tested loads these two directions exhaust each secular block's kernel. Depth two contains the same directions in its scalar/dual-vector block, with the quadrature exchanged.

For each one, $\delta(\Phi\Phi^\dagger)=0$ and $\delta(BB^\dagger)=0$ pointwise. Thus the linear-in-time Jordan contribution changes neither density matrix nor any traced scalar or vector current to first order. It should not be described as a proved growing physical-density disturbance. The field norm still has secular linearized motion, and nonlinear persistence modulo these symmetries requires its own calculation.

There is nevertheless a complete linear orbital bound. In the physical orthonormal form basis, write the free-plus-source matrices as

$$X_L=L\Gamma+X_0+O(L^{-1}),\qquad Y_L=L\Gamma+Y_0+O(L^{-1}),\qquad \Gamma^2=I.$$

Thus $Y_LX_L=L^2I+L B_\infty+O(1)$, where $B_\infty=\Gamma X_0+Y_0\Gamma$. For each coupled factor of degree $n$, the exact polynomial

$$p_\infty(t)=\lim_{L\to\infty}L^{-n}p_L(L^2+Lt)$$

agrees with the corresponding actual physical limit matrix. It has $n$ distinct real roots in all eleven current families. The background's other two modes are separately uncoupled above depth two. Consequently the coupled eigenvector matrices tend to a nonsingular limiting eigenvector matrix and have uniformly bounded condition numbers for sufficiently large $L$. The earlier positive Hermite certificates give simple roots at every remaining $L\ge2$; the finitely many remaining integer blocks have a finite maximum condition number.

For a squared-frequency eigenvector $v$ with $\nu^2>0$, the first-order generator eigenvectors are $(v,\pm iX_Lv/\nu)$. Since $X_L/L\to\Gamma$ and $\nu/L\to1$, this conversion and its inverse are uniformly bounded at high harmonics. The complete nonzero-frequency evolution therefore has a time-independent uniform norm bound. The first-harmonic Jordan blocks become semisimple zero blocks on the quotient by their identified right-unitary tangent directions. All other low harmonics and the free complement have bounded evolution.

Let $\mathcal T$ be the finite-dimensional real tangent space of the separate constant right-unitary actions at $(\Phi_0,B_0)$. In the quotient Sobolev norm this gives, for every fixed Sobolev order $s$,

$$\boxed{\|[\delta\Phi(t),\delta B(t)]\|_{H^s/\mathcal T} \le C_{d,s}\|[\delta\Phi(0),\delta B(0)]\|_{H^s/\mathcal T}, \qquad t\in\mathbb R.} \tag{11}$$

The Hodge harmonic weights are equivalent to the ordinary Sobolev norm on this fixed compact sphere, and the Killing-frame map is smooth. Equation (11) is a conditional full linear orbital persistence result for these ground controls at the common load. It does not transfer the earlier exactly quadratic nonlinear plateau proof: the responding flow $u(B)$ remains nonlinear even on an affine particle plateau.

7. What This Adds

The constructed clock removes the frozen preferred time from the action, preserves the stationary count solutions and includes the background's complete source and energy. The spatial form calculation identifies the additional channel that a radial test misses and gives an exact condition for its growth. It does not demonstrate a stable electron, observed decay, localized support in unbounded space, complete boost inertia, a selected electric charge or the common master-field origin of the block split. Nonlinear preservation of the timelike background domain and continuation to a zero-field vacuum also remain unestablished.

The ground threshold ordering is not a color or neutrino mechanism. The physical density ratio is uncalibrated, and the actual assigned excited states have different spatial coefficients. These results constrain this inverse realization; they do not supply independent dimension-selective support for the sector evidence document.

Reproducibility

The independent full variation checker passes 264 evaluations. Its maximum joint source error is $1.8\times10^{-10}$, background Hessian error $2.3\times10^{-10}$, and second-action error $9.5\times10^{-8}$ at the recorded double-precision steps. The actual geometry checker passes 3,908 evaluations; spatial Dirac and Hodge discrepancies are below $4.3\times10^{-16}$. The angular checker retains exact polynomials, Sturm counts and Jordan nullities, and the threshold checker verifies (10) symbolically. The uniform checker retains all eleven current-family certificates at the common small load; the zero-mode checker identifies the complete secular eigendirections at both tested loads. The asymptotic checker independently verifies all omitted free factors, agreement with the actual physical high-harmonic matrix, and its simple real limiting roots. Dated reports retain exact source hashes, including the corrected serialization-only failed pilots.

The inferred excited-shell current calculation identifies the complete selected massive shell, its vanishing gradient vertex and its coexact response at all fourteen assignments. Its signed leading static form retains the resonant first background harmonic and does not transfer the ground persistence theorem to those particles.