Literal Orbit Precession in Flat Spatial Coordinates

The premise is that the six-dimensional electron itself follows a literal deterministic trajectory around the nucleus in the surrounding higher-dimensional physical space. The position q(t) below is relative to the nuclear center, taken as the origin; this external motion is distinct from the electron's internal classical field oscillations. A useful corrected result identifies the equation obeyed by the maintained orbit curve in flat coordinates, while keeping it distinct from the older curved-space action that generated the same curve.

Let $J^2=-I$, $J^T=-J$ on four active real coordinates, and let $e_1,e_3$ span an orthogonal real plane with the corresponding $Je_1,Je_3$ directions. Put

$$q(t)=\rho e^{\beta Jt} (\cos\delta t\,e_1+\sin\delta t\,e_3).$$

Direct differentiation gives

$$\boxed{q''=2\beta Jq'-(\delta^2-\beta^2)q.}$$

Indeed, if $q=e^{\beta Jt}u$, then $q''-2\beta Jq'=e^{\beta Jt}(u''+\beta^2u)$, and $u''=-\delta^2u$. This proof uses ordinary flat coordinates and no position probability. Extending $J$ as three equal orthogonal rotation blocks is compatible with the six-direction picture, but this curve alone tests only two blocks.

A local action and its frequencies

With positive inertia $m$, a flat collective action is

$$L=\frac m2|\dot q|^2 +m\beta\,q^TJ\dot q -\frac m2(\delta^2-\beta^2)|q|^2.$$

The sign follows explicitly from $J^T=-J$. With $k=\delta^2-\beta^2$, the canonical momentum and position derivative are

$$p=\frac{\partial L}{\partial\dot q}=m\dot q-m\beta Jq, \qquad \frac{\partial L}{\partial q}=m\beta J\dot q-mkq.$$

Consequently

$$\frac{dp}{dt}-\frac{\partial L}{\partial q} =m\ddot q-2m\beta J\dot q+mkq=0,$$

which gives the boxed trajectory equation. A negative sign on the velocity term would instead reverse the $2\beta J\dot q$ force. The antisymmetric velocity term supplies no instantaneous work. It is a Lorentz-type mathematical force from a uniform isoclinic two-form, not a mediator particle. In the rotating frame at rate $\beta$, the orbit is the fixed circle $u(t)$. The two signed normal frequencies are $\beta+\delta$ and $\beta-\delta$. This is the classical Larmor/Fock–Darwin interpretation of the flat surrogate.

A commensurability $\delta=L\beta$ requires

$$\omega_0^2=\delta^2-\beta^2=(L^2-1)\beta^2.$$

Thus $L=1$ has no harmonic restoring term, $L=2$ requires $3\beta^2$, and $L=3$ requires $8\beta^2$. The action admits continuously variable ratios. It does not choose an integer lock. The signed projection $\rho\cos(\delta t)e^{i\beta t}$ differs from the folded readout $\rho|\cos(\delta t)|e^{i\beta t}$; for an odd integer ratio their petal counts differ. Neither readout is an atomic position probability density. The interactive trajectory distinguishes the preparations, and the actual-orbit correction states their physical interpretation.

Why the earlier curved equation cannot be substituted

$$-\frac{2(z^\dagger\dot z)\dot z}{1+|z|^2}.$$

For the same curve, its connection coefficient is

$$b=\frac{2\beta}{1+\rho^2},$$

rather than $2\beta$. Dropping the denominator changes the equation except in the small-radius limit. The exact flat identity is therefore a separate realization of the same coordinate curve, not a proof that the old curved action was already flat. The current physical hypothesis is ordinary flat containing space; older projective spectra and orbit dynamics remain conditional mathematics under their stated metric.

A corrected density-gradient route

A conserved internal phase can produce a translation connection after Routh reduction only if the profile's current depends on its center. A rigid pattern with constant current yields a total derivative and no center force. Identifying a spinor's internal complex structure with a spatial $J$ is another additional assumption.

For the tested rigid product-vortex profile in $D$ real dimensions and an exactly quadratic kinetic density $w(x)=w_0+w_2|x|^2/2$, the full weighted norm contains

$$C(q)=w_0+\frac D2w_2R^2+\frac{w_2}2|q|^2, \qquad C_0=w_0+\frac D2w_2R^2.$$

Eliminating the phase at fixed charge $P$ also adds $P^2/(2\Lambda N(q))$ to the amended potential. The inertia is the weighted gradient Gram tensor. Keeping all three effects gives the exact finite-radius conditions

$$\beta=\frac{w_2\Omega R^2}{2C_\rho},\qquad \delta^2=\frac{\beta^2C_0+(w_2/2)(D-\Omega^2R^2)} {C_0+w_2\rho^2},$$ $$\frac{P}{\Lambda N_0}=C_\rho\Omega-\frac{w_2}2\beta\rho^2.$$

Here $R$ is the rigid profile width, $\Omega$ its phase rate, $N_0$ the unweighted profile norm, and $C_\rho=C(q)$ at $|q|=\rho$. As one exact algebraic example, $D=6$, $w_0=w_2=R=\rho=1$, $\Omega^2=486/113$ give $\beta^2=6/113$, $\delta^2=24/113$, hence lock two. This is an existence example under prescribed coefficients, not a prediction of the lock.

The linear restoring sign depends on $w_2(D-\Omega^2R^2)$. Negative curvature can therefore be restoring at sufficiently large phase rate. However, a globally quadratic weight with $w_2<0$ becomes negative in the far tail. Positivity over a few profile widths does not repair an infinite Gaussian tail. Such negative-curvature examples require a globally positive density with its complete integrals before they are physical support models. The loaded carrier supplies a particular anisotropic first-mode density response; it does not fix the sign for every nuclear source.

The same flat oscillator mechanism works in four coordinates of the external trajectory and embeds in higher-dimensional particle structures. It establishes useful higher-dimensional capability, not a unique preference for six. Deriving the position-dependent current, global support, nuclear recoil, integer lock and detector response under one same-field law remains the substantive orbit problem. The internal count field and its spin-density planes remain distinct from this literal position trajectory.