Projected Electron Orbits in the 6D Sector

The attractive interaction between an electron and a nucleus may constrain the external orbit to four spatial directions. The nucleus’s depth-three and depth-four quark patterns motivate this hypothesis, but their depth numbers do not establish a common alignment or the direction of the force. The displayed law explicitly assumes four-direction restoring and gyroscopic terms. Its exact trajectory turns at one rate while passing between two orthogonal planes at a second rate. The electron’s distance from the nucleus stays constant. A three-coordinate projection has an apparent radius that contracts and expands as the angular position advances, giving a ring or petals.

The electron itself remains a six-dimensional pattern. Its other two position coordinates are zero in this preparation; that does not show that those directions are dynamically inert or unreachable by the nuclear interaction. A projected force alone would leave free motion in its complement unless initial data or another mechanism confines it. Both nuclear and electron–electron responses need the coupled field dynamics. The relation between this orbit and three-dimensional bulk matter is examined in a separate note with scoped energy and orientation checks.

The Flat Action law supplies the baseline trajectory and clock rates. The default display applies a separate three-plane construction to those rates, producing the nubbed readout described below. That readout is not itself an established solution of the flat action. The external trajectory remains distinct from internal field oscillations; the coefficients and integer frequency ratios are supplied inputs.

The electron follows literal deterministic motion in six ordinary spatial dimensions. Physical spacetime has infinitely many ordinary spatial dimensions plus time. Matter occupies up to ten spatial dimensions; all remaining spatial directions are unbound for every particle. The working matter-domain geometry is flat. Particle depth describes dimensional involvement of the same field.

The Spatial (Signed) setting shows three real coordinates of the baseline external position around the nucleus, with an adjustable display-height scale. The default Folded setting instead shows a normalized three-plane readout. The chosen trajectory uses four directions and has two remaining position coordinates set to zero. It embeds in six or more containing dimensions without selecting the electron’s physical depth. Here $L$ labels the chosen frequency ratio between the orbit and its precession. Their relation to observed atomic states, energies and transition strengths requires the physical interaction and observer response. Squared coordinates and ray overlaps do not assign a probability cloud to the electron.

Let $q\in\mathbb R^4$, $J=\operatorname{diag}(J_2,J_2)$, with $J_2(x,y)=(-y,x)$, so $J^T=-J$. A real classical collective action is

The plus sign of the gyroscopic term matters: its Euler–Lagrange equation is

Writing pairs of real coordinates as complex shorthand, its exact trajectory is

In the frame rotating at $\beta$, this is circular oscillator motion at $\delta$; the two signed normal frequencies are $\beta+\delta$ and $\beta-\delta$. The default petal examples set $m=\rho=\beta=1$, $\delta=L$, and $k=L^2-1$. Thus changing L changes the restoring coefficient; this is a family of prescribed candidate laws, not integer selection by one fixed action. L1 has k = 0. The Ring is a separate harmonic-circle preparation with $\beta=0,k=1$, parametrized by $q=(\cos t,\sin t,0,0)$.

The signed plane image is $\cos(L\theta)e^{i\theta}$, with $\theta=\beta t$: it has L distinct petals for odd integer L and 2L for even L. Its radial maxima over a full clock turn are not always distinct petals. Folded and Smooth use nonnegative radius and have 2L petals. Their illustrative height is a separate readout, not the signed physical coordinate. These one-plane petal counts do not specify the additional extrema of the three-plane readout. None of these readouts is a probability cloud or a calibrated detector response.

The four components of $q$ are a premise of this candidate law. Their proposed connection to the nucleus’s quark depths needs a derivation of the attractive interaction and its directional constraint. Along the stated trajectory $|q|=\rho$ is constant: the electron keeps a fixed distance from the nucleus while its orbit precesses. The flat force coefficient is $2\beta$. The exact curve establishes compatibility with the prescribed law; its rate, restoring coefficient, frequency lock, position-dependent current, complete same-field reduction and reciprocal nuclear response remain to be derived. Nor does the flat action derive a physical isoclinic field from the electron’s internal phase. Corrected density-weighted reductions have position-dependent norm, inertia and amended charge potential; an unweighted uniform-field formula cannot substitute for them.

The default Three Phased Planes setting starts with the normalized real vector $v=(1,10^{-3},10^{-3},10^{-3})/\sqrt{1+3\times10^{-6}}$. It applies rotations in the coordinate planes $(0,1)$, then $(0,2)$, then $(0,3)$, using angles $\alpha_j=L\theta+js$, where $s$ is the Plane Phase Spread slider and $\theta=\beta t$. Each rotation replaces $(v_0,v_j)$ by $(v_0\cos\alpha_j-v_j\sin\alpha_j,\ v_0\sin\alpha_j+v_j\cos\alpha_j)$. These are real rotations in four coordinates; the displayed radius is a readout of their result.

The code takes $a_0=|v_0|$ and $a_1=|v_1|$, scales each by its own sampled maximum over an exchange cycle, and calls the resulting values $\widehat a_0,\widehat a_1$. Folded uses $R=\widehat a_0$; Smooth uses $R=\widehat a_0^2$. The displayed coordinates are

where $p=1/2$ for Folded and $p=1$ for Smooth. The height $\eta$ and the factor $\cos(3\theta)$ are illustrative display choices, not coordinates derived from the flat-action solution. The small seeded components and phase spread alter peak heights and can introduce secondary maxima—the nubs. Their number and relative size depend on the preparation and readout, so the L buttons label the exchange-rate ratio rather than a guaranteed total lobe count. The default phase spread of 0.90 radians was chosen to resemble the relative sizes of the smaller lobes in textbook orbital diagrams. This is a visual size comparison; it does not establish agreement of the complete angular shapes or identify these readouts with textbook orbitals.

One Plane restores the simpler Folded or Smooth readout described above. Spatial (Signed) bypasses the three-plane readout and uses the baseline position coordinates; its phase-spread control therefore has no effect. The Ring also bypasses the three-plane construction. State-distance information refers to the baseline trajectory, even while the canvas shows the nubbed readout. One clock turn takes 8.5 seconds of animation time; physical clock units remain uncalibrated.