Centroid-relative self-binding

Scope and reason for this addition

Binding a particle's structure about its own centre differs from trapping its centre at an external origin. This distinction fits the V2 spatial premise: an electron has six ordinary macroscopic spatial dimensions, treated alike by its self-binding. Its shared directions with a lower-depth observer acquire no preferred status.

The existing self-consistent functional explicitly leaves a correctly varied centroid prescription open. Here is that prescription for its flat scalar model.

Define the centre inside the functional

Let $x\in\mathbb R^d$ be ordinary spatial coordinates and assume a decaying field with finite kinetic energy and second moment. Write

$$\rho=|\psi|^2,\qquad Q=\int\rho\,dx>0,\qquad R=\frac{1}{Q}\int x\rho(x)\,dx,\qquad z=x-R,$$

$$M=\int zz^{\mathsf T}\rho(x)\,dx.$$

$M$ is the unnormalized central second moment. For $g>0$, define

$$E[\psi]=\frac{\hbar^2}{2m}\int|\nabla\psi|^2\,dx+U[\rho], \qquad U=\frac g2\operatorname{tr}(M^2).$$

Equivalently,

$$U=\frac g2\iint \big[(x-R)\cdot(y-R)\big]^2\rho(x)\rho(y)\,dx\,dy.$$

The same density-dependent $R$ is used throughout. Translation of the whole field translates $R$ and leaves $M$ and $E$ unchanged. Under a spatial rotation $O$, $M$ becomes $OMO^{\mathsf T}$ and $\operatorname{tr}(M^2)$ is unchanged. No subset of the $d$ directions is selected.

Variation and self-force

One must vary $R$ as well as the explicit density factors. Its variation cancels because $\int z\rho\,dx=0$:

$$\delta M =\int zz^{\mathsf T}\delta\rho\,dx -\delta R\left(\int z\rho\,dx\right)^{\mathsf T} -\left(\int z\rho\,dx\right)\delta R^{\mathsf T} =\int zz^{\mathsf T}\delta\rho\,dx.$$

This holds even for variations that change $Q$, since $R$ is defined using the current $Q$. Therefore

$$\frac{\delta U}{\delta\rho(x)}=g z^{\mathsf T}Mz, \qquad i\hbar\partial_t\psi =\left[-\frac{\hbar^2}{2m}\Delta+g z^{\mathsf T}Mz\right]\psi$$

for the usual scalar Schrödinger variational dynamics. At each fixed time, $R$ and $M$ are spatially constant coefficients in this potential. Its total self-force is

$$-\int\rho\,\nabla_x\!\left(g z^{\mathsf T}Mz\right)\,dx =-2gM\int z\rho\,dx=0.$$

With the stated decay and regularity assumptions, the continuity equation and momentum balance give

$$P=\int\psi^*(-i\hbar\nabla)\psi\,dx,\qquad \dot P=0,\qquad \dot R=\frac{P}{mQ}.$$

Thus the centre moves freely while the self-potential acts on the shape. In the rest frame $R=0$, choosing $\hbar^2/(2m)=1$ recovers the existing isotropic stationary calculation. Translating that solution gives a family of equally energetic resting centres. The time-dependent equation also admits the usual Galilean boost

$$\psi_v(x,t)= \exp\!\left[\frac{im}{\hbar} \left(v\cdot x-\frac{|v|^2t}{2}\right)\right]\psi(x-vt,t).$$

This establishes the moving-centre claim for this specified nonrelativistic model. It does not derive relativistic dispersion, stability of every excited profile, particle mass counts, or cross-sector forces. Separately centering each species in a coupled model would require a fresh variation of the complete interaction.

Verification and related work

The spatial structure checks compare the analytic density derivative with finite differences of the full recentered pair energy, including norm-changing perturbations. They also check translations, rotations, zero net force, and the boost operator identity.

• Self-consistent functional — supplies the original scalar closure and its normalization assumptions.
• Nested spatial tools — companion tools for spatial evolution and normalization.