Coulomb angular response

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: A six-coordinate Kepler model with reciprocal source angular momentum has exact signed-rose trajectories, including repelling two-electron solutions, but the pair is unstable and magnetic quadratic completion changes the motion.

What is restored, and what is a new premise

The visualization history contains Coulomb binding plus a prescribed precession construction, not a single implemented force-law integration joining them. The old c50ef57 prose used a visible 1/r potential plus a separate sector Hamiltonian; it did not derive the subsequent frequency lock. This test supplies actual Cartesian positions and canonical momenta q_i,p_i in R6, without appending another independent six-dimensional electron position.

Use the explicitly chosen Kepler kernel -k/|q| as a control. It is not silently identified with the Green function of an unrestricted six-dimensional field, nor with the unresolved interaction of a flat nucleus and a depth-six electron. A radial potential -k/r^a has circular effective-potential curvature a k (2-a)/r^(a+2). Thus a=1 has stable radial circular motion, whereas the unrestricted six-dimensional inverse-fourth-power potential does not in this flat point-particle model. Kernel and geometry selection remain explicit.

The responding source is now a real skew 6 by 6 angular-momentum matrix C on a fixed coadjoint orbit. This is a new source phase-space premise, motivated by angular-momentum-bearing response, not a relabeling of the previous singlet-plus-gapped-amplitude source. No equivalence between the two source models or identification with a measured nuclear current is claimed. In particular C=0 remains zero: this source can reorient a prepared angular momentum but cannot induce its magnitude from an initially zero source. The petal preparations require nonzero source angular momentum as input.

One autonomous Hamiltonian

Define L=sum_i(q_i p_i^T-p_i q_i^T), pairing <A,B>=-tr(AB)/2, and

$$H_0=\sum_i\left(\frac{p_i^2}{2m}-\frac{k}{|q_i|}\right) +\sum_{i<j}\frac{\gamma}{|q_i-q_j|},\qquad H=H_0+\lambda\langle C,L\rangle+\frac{\langle C,C\rangle}{2I}.$$

Use canonical brackets for q,p and the angular-momentum Lie-Poisson bracket for C. The equations are

$$\dot q_i=p_i/m-\lambda Cq_i,\qquad \dot p_i=F_i(q)-\lambda Cp_i,\qquad \dot C=\lambda[C,L].$$

Here F is the full attraction-plus-pair-repulsion force from H0. The source energy is a Casimir and adds no orientation torque. The source is not frozen: C and L generally precess, with dot L=-dot C. Total J=C+L, energy, the spectrum of C, and the spectrum of total orbital L are conserved. The source and orbit exchange angular momentum.

There are no prescribed time-dependent fields in these equations. They describe classical actual trajectories. The two-electron version includes direct repulsion, but is not an antisymmetric quantum many-body state and does not replace the separate Pauli-compatible carrier tests. The canonical p is not m dot q when the angular coupling is present.

Exact reduction to Coulomb motion plus reciprocal precession

Let Q_i(t),P_i(t) solve the uncoupled rotationally invariant H0 problem, with conserved total L0 and the same initial Cartesian data. Then

$$R(t)=e^{-\lambda Jt}e^{\lambda L_0t},\qquad q_i(t)=R(t)Q_i(t),\quad p_i(t)=R(t)P_i(t), \qquad C(t)=e^{-\lambda Jt}C_0e^{\lambda Jt}$$

solve the coupled equations. Indeed dot R R^T=-lambda C(t), and the central and pair forces are rotation-covariant. This is an exact autonomous backreaction solution, not an imposed precession schedule. It also shows a limitation: this interaction only rotates the ordinary Coulomb configuration; it does not repair its radial or pair instabilities.

Prepared signed-rose family

Set radius and mass to one, k=1, lambda=0.2. For N=1 choose q=e1,p=ell e2 with ell=1. For N=2 take an antipodal pair with opposite momenta and gamma=0.4. Both remain separated by distance two, and their canonical circular momenta obey ell=sqrt(k-gamma/4)=sqrt(0.9).

In either case define beta=ell/m-N lambda ell. Choose conserved total angular momentum J=j(E14+E25), where Eab is the unit skew generator; indices here run from one to six. These planes mix the first two visible axes with the fourth and fifth axes. Set lambda j=l beta and C0=J-L0. Then the first electron has the exact Cartesian path

$$q(t)=(\cos(l\theta)\cos\theta,\ \cos(l\theta)\sin\theta,\ 0, \ \sin(l\theta)\cos\theta,\ \sin(l\theta)\sin\theta,\ 0), \qquad\theta=\beta t.$$

Its full radius is one, so projected center crossings are not collisions with the nucleus. The source backreaction is nonzero. The second electron, when present, is its antipode. The computed clocks are beta=0.8 for N=1 and beta=0.569209978830 for N=2. Neither clock is imposed in the RHS.

However, the integer ratio is selected by the initial conserved J, not dynamically attracted or quantized by this Hamiltonian. The source coadjoint spectrum changes when this preparation changes. No shell index or nuclear response spectrum has been derived from choosing integer l.

Relation to the visualization

The direct planar projection is the smooth signed rose w=cos(l theta) exp(i theta). For even integer l it has 2l distinct petals and the same planar image as the demo's |cos(l theta)| exp(i theta). Thus l=2 and l=4 yield actual four- and eight-petal projected paths. Their time parametrization differs from the modulus-folded demo at center crossings; the demo's additional height readout is not derived here.

For odd l the signed curve instead has l distinct petals. In particular l=1 is an offset circle and l=3 is a three-petal rose, not the demo's two- and six-petal images. The absolute-value detector map cannot be silently substituted for this linear spatial projection. Matching selected even-l images is a concrete partial match, not reproduction of the entire display.

Perturbation and two-electron instability

Independent integration agrees with the exact prepared paths over a full clock turn for N=1,2 and l=1,2,3,4. For one electron, perturbed bound Kepler motion stays radially bounded because R is orthogonal. Small changes can alter the clock ratios and lose exact recurrence; the periodic petal path is not an attractor.

For the l=2 test, changing one unobserved position component and one canonical tangential momentum component by 0.0001 or 0.001 gives the following maximum position deviations over two intended clock turns:

The repelling pair has an analytic exponential instability, not merely a poor numerical trajectory. Its common-displacement mode has no first-order change in total orbital angular momentum and therefore is not stabilized by the reciprocal rigid rotation. In the ordinary co-rotating Coulomb frame its 4 by 4 Jacobian is [-W,I; diag(2k,-k),-W], with W=[[0,-Omega],[Omega,0]] and Omega²=k-gamma/4. In the tested units,

$$\sigma^2=\frac{k-2\Omega^2+\sqrt{9k^2-8k\Omega^2}}2>0.$$

The growth rate is 0.520404067288 in the test's time units, predicting amplification 64.2789706 over eight units. The checker computes this eigenvalue and tests its predicted growth in a full coupled integration from a 1e-7 perturbation. Consequently the exact two-electron family is an existence result, not a stable atomic model.

Magnetic quadratic completion must not be omitted

The linear angular coupling can stand as a phenomenological gyroscopic Hamiltonian, but calling it a complete magnetic minimal coupling requires the quadratic term. For a spatially uniform linear connection, the completed kinetic Hamiltonian is

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

This adds m lambda² sum |Cq|²/2. The source equation must change as well: dot C=[C,lambda L-m lambda² sum(q wedge Cq)]. Both versions are checked against their own energy gradients, with conserved total angular momentum. This is a consistency test for this uniform-connection interpretation, not a uniquely derived nuclear electromagnetic field.

On the prepared trajectories, the initial quadratic/uncoupled-kinetic energy ratios are:

The term is not perturbatively small for the complex petal examples. Adding it reciprocally changes the old l=2 preparation: over one intended turn the one-electron position error reaches 1.86827 with radius change 0.595951; the pair position error reaches 2.53216 with radius change 0.527079. These are tests of the same initial data, not an exclusion of other solutions of the completed equations. The subsequent completed relative-equilibrium calculation finds different exact solutions of the completed Hamiltonian, including one-electron four- and eight-petal branches without resolved exponential linear modes. Thus this initial-data failure is not a no-petal conclusion.

Conclusion

Actual autonomous petal-shaped projections with reciprocal source motion now exist in an explicit Coulomb-plus-angular-response Hamiltonian, including an exact repelling two-electron family. The source, coupling, and preparation are explicit, and the even-l planar matches are checked, not drawn by an assigned clock.

The remaining problems are specific: the pair is unstable; the integer frequency ratio is prepared rather than selected; odd-l and height readouts do not match the visualization; and the magnetic quadratic completion is large and changes these trajectories. That next calculation has now found bound relative equilibria of the completed Hamiltonian; see Completed Coulomb petals. Source induction, integer selection, and stable many-electron motion remain unfinished.

Reproduction and related notes

Final normal and optimized runs completed 9,812 evaluations across 28 conditions. Independent integration matched the exact prepared motion within 1.88e-9; full-turn energy drift was at most 6.02e-9. Perturbed trajectory energy drift was 3.94e-9, and the quadratic-completion energy drift was 1.01e-8. Predicted unstable-mode amplification agreed with the coupled integration to relative error 1.66e-8. Reversing source backreaction, omitting the quadratic force, or reversing pair repulsion each failed an independent gradient or angular-momentum conservation check. All 50 diagnostic scripts and source checks passed in the regression run; the final zero-source control also passed the separate normal and optimized reruns.

• coulomb_angular_response_checks.py — actual six-coordinate integrations, exact solution, reciprocal gradients, pair instability, and quadratic completion.
• Full response extension — preceding response-only model; its negative result is not overwritten.
• Actual orbit research correction — provenance of the Coulomb-plus-precession target.
• Two-electron shared response — separate antisymmetric carrier model, not this classical Cartesian calculation.
• Research status — current proof boundaries.