Cylindrical, Breathing and Petal Views of One Actual Orbit

The Physical Question

A deterministic electron trajectory can appear cylindrical from one view and show radial expansion and contraction from its end. This observation does not require an electron probability cloud. The distinction to test is between a changing full spatial radius and a changing projected radius.

The existing completed Coulomb-petal model supplies an explicit example. This calculation changes the ordinary spatial observation axes, not its Hamiltonian or physical trajectory. It connects several appearances within that model; it does not identify every cylindrical six-dimensional orbit with this example.

The Existing Autonomous Trajectory

The one-electron Hamiltonian, in the maintained dimensionless units, is

$$H=\frac12|p-\lambda Cq|^2-\frac1{|q|} +\frac12\langle C,C\rangle, \qquad \lambda=\frac15, \qquad \langle U,V\rangle=-\frac12\operatorname{tr}(UV).$$

Here q and p are actual ordinary spatial position and canonical momentum. C is the existing skew coadjoint source, proposed as a same-field collective description. It is not a mediator particle. Its physical origin and prepared spectrum remain assumptions of the model.

The completed relative equilibrium has, in its four active coordinates,

$$q(\theta)= \bigl(\cos L\theta\cos\theta, \cos L\theta\sin\theta, \sin L\theta\cos\theta, \sin L\theta\sin\theta\bigr), \qquad \theta=\beta t.$$

The original six-coordinate labels for these entries are 1, 2, 4 and 5. Their squared norm is identically one.

Two Physical Rotation Planes

Define ordinary orthogonal coordinates

$$u_1=\frac{q_1+q_4}{\sqrt2},\qquad u_2=\frac{q_2-q_3}{\sqrt2},\qquad v_1=\frac{q_1-q_4}{\sqrt2},\qquad v_2=\frac{q_2+q_3}{\sqrt2}.$$

The subscripts in this display refer to the four active entries above. Writing a pair of real coordinates as one complex coordinate is only notation for an ordinary spatial plane. Trigonometric addition gives

$$u_1+i u_2=\frac1{\sqrt2}e^{i(1-L)\theta},\qquad v_1+i v_2=\frac1{\sqrt2}e^{i(1+L)\theta}.$$

The source-supported orbit is therefore the sum of two equal-radius circular motions in mutually orthogonal physical planes. The completed force law, rather than a later drawing rule, supplies the checked trajectory.

An Ordinary Three-Axis Projection

Choose the orthonormal observed axes

$$X=\cos\chi\,u_1+\sin\chi\,v_1,\qquad Y=\cos\chi\,u_2+\sin\chi\,v_2,\qquad Z=-\sin\chi\,u_1+\cos\chi\,v_1.$$

The unobserved fourth axis is W=−sin(χ)u₂+cos(χ)v₂. These are four mutually orthogonal axes; X, Y and Z constitute a conventional rank-three spatial projection. No different kind of dimension or inherent electron 3+3 split is introduced. This selects a view in a flat comparison; it does not derive an apparatus or nuclear frame.

The end-on coordinate and its squared radius are

$$X+iY=\frac1{\sqrt2} \left[\cos\chi\,e^{i(1-L)\theta} +\sin\chi\,e^{i(1+L)\theta}\right],$$

$$\boxed{r_{\rm end}^2=X^2+Y^2 =\frac12\bigl[1+\sin(2\chi)\cos(2L\theta)\bigr].}$$

This projected breathing leaves |q|²=1 exactly. Over a complete breathing cycle, for 0≤χ≤π/4,

$$r_{\min}^2=\frac{1-\sin(2\chi)}2,\qquad r_{\max}^2=\frac{1+\sin(2\chi)}2.$$

Three views make the distinction concrete:

The first three-dimensional curve lies on a cylinder and oscillates along its axis. Intermediate views have nonzero inner and outer end-on turning radii. At the aligned view the inner turning radius reaches zero, producing the signed-petal projection. The observer formula is different from the modulus-folded projective display for odd L, as in the maintained orbit note.

The result demonstrates continuity between cylindrical, breathing and petal appearances of this one full trajectory. It does not prove that a separately constructed six-direction breathing trajectory is merely another view.

What Concentric Rings Would Establish

For uniform θ and sampling over full breathing cycles, let s=r_end². Changing variables from cos(2Lθ) gives the deterministic dwell-time density

$$f_s(s)=\frac1{\pi\sqrt{(s-s_{\min})(s_{\max}-s)}}, \qquad s_{\min}<s<s_{\max}.$$

Consequently

$$f_r(r)=\frac{2r} {\pi\sqrt{(r^2-r_{\min}^2)(r_{\max}^2-r^2)}}.$$

The enhancement near either turning radius is the slower radial motion there. It is a time histogram of a definite position, not a probability cloud or a Born law. Its identification with a detector count rate requires a separate interaction and sampling model.

A radial histogram does not determine the angular distribution. A single commensurate orbit reaches its turning radii at finitely many angles per period; the exact trace need not contain complete concentric circles. Coloring by Z can reveal the depth of each traced point, but it does not add positions to that trace. Distinguish ring-like turning enhancements, complete circles, and an annulus filled by long-time phase sampling.

For irrational L, the two plane clocks are incommensurate and their phase pair has dense closure on a two-torus. Its end-on closure is the annulus between the stated radii. This is a mathematical infinite-time statement; a finite trace remains a curve. Rational L instead gives a closed trajectory. The model has no rule preferring an integer L.

Occupied-Depth Comparison

The same active four-coordinate solution and source embed without changing the equations into depths 4, 5, 6 and 10. Depths 2 and 3 cannot contain its two independent physical rotation planes. This limited requirement does not exclude other breathing trajectories at those depths.

Indeed, a tilted ordinary circular orbit in three dimensions,

$$q(\theta)=(\cos\theta,\cos\iota\sin\theta,\sin\iota\sin\theta),$$

has full radius one while its end-on radius obeys

$$r_{\rm end}^2=\cos^2\theta+\cos^2\iota\sin^2\theta.$$

It breathes in projection already at depth three. The same control embeds in every larger occupied depth. An ordinary full-plane projection of a depth-two circle has constant radius. Neither comparison is a theorem about general nonlinear dynamics at depth two.

Breathing or a cylindrical appearance alone therefore supplies no special depth-six selection and no new support for the assigned sector structure.

Reproducible Checks and Limits

breathing-orbit-checks.py uses the exact database versions of the existing completed-petal solver and its Hamiltonian helper. It verifies the coordinate identities, orthonormality, full radius, end-on law, energy, total angular charge and common-rotation covariance. Six occupied depths are included. The available autonomous cases use L=√2, 3/2, 2 and 4 at depths 4, 5, 6 and 10. All sixteen equilibrium residuals are below 7.6×10⁻¹⁴. A midpoint time histogram independently checks the stated radial dwell-time cumulative law.

The recorded run passes 1,601 evaluations across nineteen conditions. This unit does not repeat a stability campaign. Merely changing an observer map cannot alter the stability of an unchanged full-state solution; earlier stability results retain their original tested branches and dimensional scope. The existence of the noninteger cases also confirms that source preparation does not dynamically select an integer lock.

The recent canonical cylindrical-response model is a different Hamiltonian and a different relative-equilibrium ansatz. Its tested instabilities do not reject these existing coadjoint-source projections, nor do they reject all breathing cylindrical trajectories. Conversely, a useful projected shape does not supply the missing same-field source origin, physical detector law, six-direction dynamics, or simplex mass-frequency mechanism.

Related

• Completed Coulomb petals supplies the autonomous trajectory and its scoped earlier stability results.
• Actual orbit research correction distinguishes deterministic position, field coefficients and accumulated registration.
• Cylindrical petal continuation changes the potential rather than merely the view; it is a separate dynamical problem.
• Recorded projection checks retains the executed source hashes and complete comparison.