Completed Coulomb petals

Physical premise (IDWT): 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. Depth counts a pattern's bound spatial involvement within that space. “Observed” and “unobserved” describe an observer's access, not different kinds of dimension. Older projective and nested-geometry calculations below retain their additional hypotheses; they do not establish a separate physical manifold or compactified direction. See spatial meaning and mathematical scope.

Summary: The quadratically completed Coulomb–angular-source Hamiltonian has exact autonomous signed-rose relative equilibria; the one-electron L=2 and L=4 petal branches have no resolved exponential mode, while every tested antipodal pair branch is unstable.

Correction to the first completion test

The preceding Coulomb-angular calculation added the quadratic term required by its uniform-connection interpretation to the initial conditions of the linear-coupling solution. Those data departed strongly from the old orbit. That test correctly rejected the old preparation, but it did not test whether the completed Hamiltonian has different petal solutions.

Solving the completed relative-equilibrium equations answers that question: it does. During exploration, the canonical momentum equation was once transcribed with the wrong column-vector sign. The production Hamiltonian was correct. Re-deriving the equation from its tested energy gradient and correcting the solver changed the stability verdict materially; only the corrected calculation is recorded below.

Completed Hamiltonian and target rotation

Use the same six-coordinate classical Hamiltonian as the previous note, including the quadratic term:

$$H=\sum_i\frac{|p_i-m\lambda Cq_i|^2}{2m} -\sum_i\frac{k}{|q_i|}+\sum_{i<j}\frac{\gamma}{|q_i-q_j|} +\frac{\langle C,C\rangle}{2I}.$$

The source remains a skew 6 by 6 coadjoint variable. Its magnitude and spectrum are conserved; this is not yet a mechanism that induces source angular momentum from zero. The test uses m=k=I=1, lambda=0.2 and, for two electrons, gamma=0.4. These are dimensionless model parameters rather than fitted atomic constants.

Let Eab denote the unit skew generator and define

$$A_L=-\beta\{E_{12}+E_{45}+L(E_{14}+E_{25})\}.$$

A relative equilibrium satisfies

$$\dot q_i=A_Lq_i,\qquad \dot p_i=A_Lp_i, \qquad\dot C=[A_L,C],$$

together with the autonomous Hamilton equations. At q1=e1, canonical momentum is fixed by p1=m(A_L+lambda C)q1. For the antipodal pair, q2=-q1 and p2=-p1. The remaining vector and commutator equations determine C and beta numerically. No target trajectory occurs in the RHS; it occurs only as the relative-equilibrium ansatz whose residual must vanish.

The computed residual is at most 1.52e-13. Independent integrations over a full clock turn agree with the exact group orbit to 8.62e-9, conserve energy to 4.52e-9, and return the full state to 8.62e-9.

Actual projected trajectories

The first electron's first two Cartesian coordinates are exactly

$$(x,y)=\cos(L\theta)(\cos\theta,\sin\theta), \qquad\theta=\beta t,$$

while its full six-dimensional radius is one. This is an actual smooth six-coordinate orbit with a signed linear projection, not a stationary orbital and not a curve assigned after integration. Even L gives the same planar image as the visualization's modulus-folded construction: L=2 has four petals and L=4 has eight. Odd L remains a different image: L=1 is an offset circle and L=3 is a three-petal rose rather than the displayed two and six petals. The visualization's height readout is not derived here.

The corrected solved branches are:

Source norm here is <C,C> in model units. These particular unconstrained solutions differ in magnitude, but that is not necessary. Constraining <C,C>=15 and resolving produces spectrally stable one-electron branches:

Their equilibrium residuals are below 3e-13 and apparent maximum real linear parts below 8e-7. Therefore different total source strength is not required for L=2,3,4. Their full conserved spectra still differ, so integer L remains encoded in prepared source data rather than quantized dynamically by this model.

Stability result

Linearize the full Hamiltonian equations in the co-rotating frame. The Jacobian contains Cartesian orbit variables and all fifteen source components. An independent central difference checks random Jacobian directions to 7.38e-10. For branches with a positive real eigenvalue, repeating the Jacobian with a five-times smaller step gives the same growth rate within 2e-6; direct perturbation integrations reproduce exp(sigma t) within 1.98e-6.

For the one-electron L=2,3,4 branches, the apparent real part shrinks approximately in proportion to the finite-difference step and reaches about 1e-7 at the finer step. There is no resolved exponential mode. This is linear spectral stability, not automatically nonlinear orbital stability: symmetry zero modes can produce drift.

As finite nonlinear probes, deterministic perturbations of norm 1e-4 were integrated for ten clock turns. L=2 and L=3 had rotating-frame deviations 0.01161, 0.01148, and 0.02396 respectively, with radius changes below 8e-5. These finite tests support bounded behavior on that interval but are not a theorem for all time or all perturbation directions.

All four antipodal two-electron branches have resolved exponential modes. The completed magnetic term therefore does not stabilize that particular classical pair configuration. This does not exclude non-antipodal correlated motion or the separately checked antisymmetric carrier model.

What is finished and what is not

The existence question for a consistent one-electron completion now has a positive answer: the quadratically completed autonomous Hamiltonian supports actual four- and eight-petal projected orbits, and those two tested branches have no resolved exponential linear instability. The earlier claim that quadratic completion simply destroys the petals was too broad; it destroys the old preparation but admits new relative equilibria.

The physical theory is not finished in three respects. First, C=0 remains C=0, so the model does not induce the required source magnitude. Second, the integer frequency ratio is encoded in conserved initial source data. Third, the tested two-electron pair is unstable. A source with canonical amplitude and momentum can in principle be driven from zero, but its action, spectrum, and stable relative equilibria must be derived and tested rather than appended to force the desired answer.

Cylindrical and Breathing Views of the Same Trajectory

An ordinary orthogonal change of observed axes connects this trajectory's petals to cylindrical and annular-breathing projections. Its two circular eigenplanes have coordinates $u=e^{i(1-L)\theta}/\sqrt2$ and $v=e^{i(1+L)\theta}/\sqrt2$. For the observed plane $X+iY=\cos\chi\,u+\sin\chi\,v$,

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

At $\chi=0$, the three-dimensional projected curve lies on a cylinder. For $0<\chi<\pi/4$, its end-on radius breathes between two nonzero turning radii. At $\chi=\pi/4$, it recovers the signed petals. The actual full radius stays one throughout. This changes the view, not the Hamiltonian or its stability. A radial dwell-time enhancement at the turning radii does not by itself establish complete concentric circles in the spatial trace.

The full projection derivation and reproducible checks verify the same branch at depths four, five, six and ten, alongside lower-depth controls. This particular solution spans four spatial directions; projected breathing alone does not select depth six. Noninteger ratio examples also satisfy the autonomous equations. A separately constructed six-direction breathing family and its physical detector response remain to be identified.

A Canonical Response Can Turn On From Zero

A separate deterministic extension replaces the coadjoint source with skew canonical coefficients $B,\Pi$. They are proposed collective modes of the same field, not mediator particles; their physical reduction remains open. On a flat ordinary spatial patch, take

$$H=\frac{|p-m\lambda Bq|^2}{2m} +\frac{\langle\Pi,\Pi\rangle}{2M} +\frac K2\langle B,B\rangle-\frac{k}{|q|}, \qquad m,M,K>0.$$

With $v=(p-m\lambda Bq)/m$, the response obeys $\dot B=\Pi/M$, $\dot\Pi=-KB-m\lambda(q\wedge v)$. Thus $B=\Pi=0$ can be driven by a nonradial actual orbit. Energy and the complete rotational charge $q\wedge p-[B,\Pi]$ are conserved. This changes the source phase space: $B$ itself is not the old source angular momentum.

The isotropic central-force preparation has an exact limitation. If the response starts at zero, all motion stays in the plane spanned by the initial $q,p$, at every occupied depth. Induction alone therefore does not generate the higher-dimensional precession required by the petal picture. The chosen $-k/r$ remains a mechanical control rather than a derived cross-depth nuclear kernel.

Fast response gives

$$\mathcal L_{\rm eff}=\frac m2|\dot q|^2 +\frac{m^2\lambda^2}{2K}|q\wedge\dot q|^2+\frac{k}{r}.$$

The extra term supplies angular inertia. A circle's small radial frequency has $\kappa_r^2/\Omega^2=(1+3x+6x^2)/(1+x)$, with $x=m\lambda^2r^2/K$. The ratio is continuous; its four-to-one value at $x=3$ is not dynamical integer selection or a mass-frequency prediction.

The full finite-response system is more restrictive. In units $m=K=r=1$, write $x=\lambda^2$ and $\eta=M\Omega^2$. Each of the $d-2$ transverse blocks has a growing mode precisely for

$$\frac{1+x}{1+2x}<\eta<1.$$

The complete radial/source block has positive squared frequencies. Thus checking radial stability alone misses the transverse instability. Depth two has no such transverse block; depths three, four, five, six and ten share its same instability interval. The endpoints have extra zero modes, and spectral stability outside the interval is not nonlinear stability.

The standalone calculation and recorded results verify 18 exact conditions, 72 complete Jacobians, conservation and rotation covariance at all six occupied depths, six zero-source integrations, and five nonlinear growth tests. The latter agree with the predicted growth to relative error below $2.1\times10^{-8}$. Zero-source runs retain their initial plane and conserve energy to $1.9\times10^{-10}$ and rotational charge to $4.2\times10^{-10}$ over the tested interval. This supplies a conditional induction mechanism and its limits, not evidence favoring the assigned sector depths or a completed electron--nucleus interaction.

Reproduction and related notes

The final normal and optimized runs completed 5,377 evaluations across 27 conditions. Maximum equilibrium residual was 1.52e-13, full-turn trajectory error 8.62e-9, and perturbed stable-branch energy drift 3.97e-9. Three deliberately incorrect variants—wrong canonical force sign, omission of the quadratic force, and reversed source commutator—were rejected by independent equilibrium checks. The complete regression passed all 51 diagnostic scripts.

• completed_coulomb_petals_checks.py — solves, integrates, and linearizes the completed relative equilibria.
• Coulomb angular response — baseline Hamiltonian and the old-preparation completion test corrected here.
• Two-electron shared response — antisymmetric shared-source carrier result, distinct from the unstable Cartesian pair.
• Magnetic bivector master bridge — candidate field-theoretic origin and its remaining identification problem.
• Research status — current project boundary.
• Multicentre orbit axes — extension: connects actual-orbit directions to trigonal and tetrahedral nuclear geometry and separates visible coincidence from full-depth collision.