Covariant Motion of Bounded Flat-Space Count Patterns

The common flat-space count action admits a rest-hyperplane realization with contracted stationary profiles, the same proper-time count clocks, complete positive energy and bounded localization. This is a changed physical realization of that action. It includes an explicit center and a rest profile on ordinary ten-dimensional spatial hyperplanes. It is nonlocal on those hyperplanes; no unrestricted local laboratory scalar-field action or interacting completion is asserted.

Internal Energy and Its Spatial Equation

Let $\phi(\xi)$ be a nonzero classical Schwartz field on $\mathbb R^{10}$, with zero centroid and mean gradient momentum. Keep $N=\int|\phi|^2$ arbitrary and varied. Define

$$M_{ab}=N^{-1}\int\xi_a\xi_b|\phi|^2,\qquad T_{ab}=N^{-1}\operatorname{Re}\int(\partial_a\phi)^*\partial_b\phi, \qquad RMR=T,\quad R>0.$$

Use the same width functions $g,V$, constants and matrix functional calculus as the count action:

$$t(r)=\frac{r-b_0}{b_1-b_0},\quad g(r)=\frac{t(r)^2}{t(r)^2+[t(r)-1]^2},\quad V(r)=\lambda t(r)^2[t(r)-1]^2,$$ $$D=\operatorname{tr}g(R),\quad K_a=\operatorname{tr}[Rg(R)M]-D/2,\quad K_n=\operatorname{tr}(RM)-5-K_a,$$ $$W=\mu\frac{\Gamma(K_a+D+1)}{\Gamma(K_a+1)\Gamma(D+1)} +\Lambda K_n+\operatorname{tr}V(R),\qquad U=NW>0.$$

Write $dW=\operatorname{tr}(C\,dM+E\,dT)$, including the full variation of $R$ through $RMR=T$. The explicit Sylvester formulas are given in the linked action. With $s=\operatorname{tr}(ET+CM)$,

$$G_U=\frac{\delta U}{\delta\phi^*} =\left[W-s-\partial_a E_{ab}\partial_b+\xi_a C_{ab}\xi_b\right]\phi. \tag{1}$$

The coefficients are constant across a given rest slice but depend on its complete field. Normalization has not been frozen: $\int\phi^*G_U=U$. The profile is an internal classical field, not a position probability or a replacement for a deterministic electron trajectory.

One Complete Action, Including the Center Shift

Separate the canonical field as $\psi(x)=e^{i(P/N)\cdot(x-X)}\phi(x-X)$. Its exact one-form is

$$\Theta_\psi=P\cdot dX+\Theta_\phi,\qquad \Theta_\phi=\frac i2\int(\phi^*d\phi-d\phi^*\phi).$$

Terms containing $d(P/N)$, including $dN$, vanish by the two centering conditions. The internal field has the inherited reduced symplectic structure. Define its classical angular momentum and energy by

$$S_{ab}=\operatorname{Im}\int\phi^* (\xi_a\partial_b-\xi_b\partial_a)\phi,\qquad H=\sqrt{U^2+c^2|P|^2}.$$

Rotational covariance of the full matrix law gives $[M,C]+[T,E]=0$, hence $S$ is conserved by (1). Translations, uniform phase gradients and common phase leave the centered energy invariant, so $\{U,X\}=\{U,P\}=\{U,S\}=\{U,N\}=0$.

Put

$$a=\frac{c^2SP}{U(H+U)},\qquad Y=X-a.$$

Since $P\cdot a=0$, the complete changed-variable action is

$$\boxed{I=\int dt\,[P\cdot\dot Y+\Theta_\phi(\dot\phi) -a\cdot\dot P-H].} \tag{2}$$

All $U,S,P,N$ dependence of $a$ must be varied. In particular, $Y$ and the field are not independent canonical factors. For isolated motion the resulting equations give

$$\dot P=0,\quad \dot Y=c^2P/H,\quad i\partial_\tau\phi=G_U,\qquad \tau=t/\gamma,\quad\gamma=H/U. \tag{3}$$

The term $-a\cdot\dot P$ cannot be discarded before variation, even though it vanishes on these isolated trajectories. A future interacting law must retain its effects when momentum changes.

The boost generators

$$B_i=HX_i/c^2+(SP)_i/(H+U)=HY_i/c^2+(SP)_i/U$$

obey $\{B_i,H\}=P_i$, $\{B_i,P_j\}=H\delta_{ij}/c^2$ and $\{B_i,B_j\}=-J_{ij}/c^2$, with $J_{ij}=X_iP_j-X_jP_i+S_{ij}$. The shifted center satisfies $\{Y_j,B_i\}=Y_iP_j/H$, the required inertial worldline law.

Actual Spacetime Histories

Use signature $(+,-,\ldots,-)$, $x^0=ct$, $p^\mu=(H/c,P)$ and $n^\mu=cp^\mu/U$. Choose

$$e_a^0=cP_a/U,\qquad e_a^i=\delta_{ia} +\frac{c^2P_iP_a}{U(H+U)}.$$

Then $n\cdot n=1$, $n\cdot e_a=0$ and $e_a\cdot e_b=-\delta_{ab}$. The complete physical history is

$$x^\mu=Y_0^\mu+cn^\mu\tau+e_a^\mu\xi_a,\qquad \Psi(x)=\phi(\tau,\xi). \tag{4}$$

The moments in (1) are evaluated on the actual rest hyperplanes of (4). With $D_a=e_a^\mu\partial_\mu$, its spacetime equation is

$$icn^\mu\partial_\mu\Psi =[W-s-E_{ab}D_aD_b+C_{ab}\xi_a\xi_b]\Psi.$$

A proper orthochronous Lorentz transformation changes the chosen rest frame by an ordinary spatial rotation. The demonstrated rotational covariance of $R,C,E$ therefore gives $\Psi'(Lx)=\Psi(x)$. The frame connection also reproduces the action's essential term: $\tfrac12 S_{ab}e_a\cdot de_b=-a\cdot dP$.

For a stationary count profile $\phi=\sqrt N u(\xi)e^{-i\omega\tau}$, the physical moving field is

$$\Psi(x,t)=\sqrt N\, u(x_\perp,\gamma(x_\parallel-vt)) e^{-i\omega\gamma(t-v\cdot x/c^2)}. \tag{5}$$

Its longitudinal variance contracts by $1/\gamma^2$, its center phase clock is $\omega/\gamma$, and its complete action energy is $H=N\gamma\omega$, with $H^2-c^2|P|^2=N^2\omega^2$. The orthogonal shell counts, frequencies and arbitrary common norm normalization are unchanged. Gaussian profiles have finite energy and decaying, noncompact spatial support in all ten directions.

Nonstationary Profiles and Persistence

The phase action $N$ is defined on a rest slice. A laboratory slice samples different rest times:

$$\int|\Psi(x,t)|^2dx=\frac1\gamma\int |\phi(t/\gamma-v\cdot\xi/c^2,\xi)|^2d\xi. \tag{6}$$

This equals $N/\gamma$ for stationary intensity, but generally varies for a nonstationary field. The laboratory intensity centroid also need not be $Y$; it is not the definition of the covariant center in (2).

An exact example uses $b_1=1,b_0=4,c=1,\mu=1/7,d=2$. Let $h_j$ be width-one Hermite functions on the first active axis, and $G$ the ground product in the other nine coordinates. Then

$$\phi=\sqrt{N/2}\,e^{i\tau/7} [h_0+e^{-2i\tau}h_4]G$$

solves (1): its constant moments give $K_a=2,W=6/7,F_a=1/2$. At $v=3/5$, $\gamma=5/4$, equation (6) becomes

$$\frac{4N}{5}[1+I_0\cos(8t/5)],\qquad I_0=\frac{162e^{-9/25}}{625\sqrt6}\simeq0.07382668.$$

This is positive-energy nonstationary dynamics of the stated action, not a prescribed breathing profile. It leaves the invariant phase action $N$ intact.

The bounded-localization theorem transfers because (3) has exactly the same rest equation as the canonical action. On its trapped components, every rest Schwartz seminorm is uniformly bounded for all $\tau$. Substituting the tilted time in (6) preserves uniform spatial decay. The compact smooth moment flow and (1) also bound time derivatives, so the fixed-boost chain rule transfers all weighted spatial derivative bounds. Constants depend on the finite boost; no uniform limit as $|v|\to c$ is claimed.

Verification and Scope

The covariance checker tests 72 conditions, including noncommuting width feedback, torque cancellation, complex circulating profiles and contracted variances. The independent review checker tests 41 symbolic conditions, including nonzero-angular boost signs and the exact nonstationary example. These accompany the derivations; they do not establish empirical particle identification.

Rest-hyperplane and center constructions have established mathematical precedents in Lusanna and Materassi and Alba, Lusanna and Pauri. The equations above apply that structure to the specified classical count-pattern energy without importing their underlying particle theories.

The achieved result is an isolated covariant realization with exact clocks and bounded spatial patterns. Physical scale selection, local propagation, separated-particle interactions, observed spin and charge, and a complete electron-orbit derivation remain separate questions.