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: Perturbing the actual orbit generator separates petal formation, observer-image closure, and recurrence of the full projective state.

Starting point: an actual orbit

This calculation continues the Hamiltonian trajectory and observer map in the 6D orbit demo. It concerns [c(t)] in P(V_L), not stationary oscillator states or conventional orbitals. The sector coordinate [Z] and Dirac field on physical space remain distinct objects as specified in the demo. Its unperturbed four-state generator, after subtracting the irrelevant common ωI/4, is (Lω σ_x) ⊕ diag(−ω,0).

Keep the same physical readout axes and initial amplitudes c(0) = (A,0,B/√2,B/√2), A,B > 0, A²+B² = 1. Test the Hermitian family

H=(ϵI2+gσx+δσz)⊕diag⁡(−ν,0),g,ν>0. H=(\epsilon I_2+g\sigma_x+\delta\sigma_z) \oplus\operatorname{diag}(-\nu,0),\qquad g,\nu>0.

Here g, δ, ε, ν are frequencies in units with ℏ = 1; g is not the scalar Hartree coupling. This is a perturbation diagnostic, not a claim that the projected V2 action produces every term. No amplitude-to-clock block mixing or leakage outside V_L is included. A basis rotation that removes δ would also rotate the specified observer readout, so it cannot silently remove the detuning in this question.

1. Exact orbit and loss of center crossings

Put Ω = √(g²+δ²). Direct exponentiation gives

c1=Ae−iϵt(cos⁡Ωt−iδΩsin⁡Ωt),c2=−iAe−iϵtgΩsin⁡Ωt, c_1=Ae^{-i\epsilon t} \left(\cos\Omega t-i\frac\delta\Omega\sin\Omega t\right),\quad c_2=-iAe^{-i\epsilon t}\frac g\Omega\sin\Omega t,

c3=B2eiνt,c4=B2. c_3=\frac B{\sqrt2}e^{i\nu t},\qquad c_4=\frac B{\sqrt2}.

The demo's amplitude fraction and aligned readout become

q(t)=1−g2Ω2sin⁡2Ωt,Γxy(t)=Rq(t)eiνt. q(t)=1-\frac{g^2}{\Omega^2}\sin^2\Omega t,\qquad \Gamma^{xy}(t)=R\sqrt{q(t)}e^{i\nu t}.

Thus q_min = δ²/Ω² and the minimum projected radius is R|δ|/Ω. For this preparation, exact center crossings require δ = 0. Small detuning retains radial modulation but opens a central hole in the traced plane curve. The demo's robustness under monotone detector response does not imply robustness of these crossings under a changed Hamiltonian. With f(0)=0 and strictly increasing f, positive q_min also implies positive minimum radius for the more general Rf(q) readout.

2. Preparation is a separate requirement

Even at δ = 0, a general pure normalized initial amplitude pair has a Bloch vector (s_x,s_y,s_z). Rotation about x gives

q(t)=12(1+szcos⁡2gt+sysin⁡2gt),qmin⁡=1−1−sx22. q(t)=\frac12\left(1+s_z\cos2gt+s_y\sin2gt\right),\qquad q_{\min}=\frac{1-\sqrt{1-s_x^2}}2.

Complete exchange requires s_x = 0. The demo's initial pair (1,0) satisfies it, as does its entire time-translated orbit. A generic nearby preparation does not. This is a condition on the physical orbit, not a choice of an orbital quantum number. Conservative evolution preserves s_x and does not relax an arbitrary initial preparation onto that orbit.

3. State recurrence is stronger than image closure

For ε = 0 and the stated preparation, all four eigenmodes are occupied. The occupied frequencies are Ω, −Ω, −ν, 0. A full projective recurrence therefore requires both ΩT and νT to be integer multiples of 2π. If Ω/ν = p/q in lowest positive integer terms, its period is 2πq/ν. An irrational ratio gives no exact recurrence.

The plane readout forgets the sign of the amplitude pair. It closes after one clock turn whenever 2Ω/ν is an integer, even if Ω/ν is a half-integer. For example, g/ν = 3/2, δ = ε = 0 produces a three-maximum plane image after one turn, while the full state ray needs two turns. Thus the even 2L sequence is tied to the full-state integer matching used in the demo, not to closure of the visible curve alone.

There is another invisible phase: ε shifts the whole amplitude block relative to the clock block. Neither q nor c₃ conjugate(c₄), nor the demo's depth coordinate, detects it. Yet full-state recurrence requires

(ϵ+Ω)T,(ϵ−Ω)T,νT ∈ 2πZ. (\epsilon+\Omega)T,\quad(\epsilon-\Omega)T,\quad\nu T \ \in\ 2\pi\mathbb Z.

Taking Ω/ν = L but ε/ν irrational preserves the entire displayed orbit while eliminating exact recurrence of the coefficient ray. Unlike adding a multiple of I₄, this is a physical relative phase in the four-state model, not a removable global phase. A closed observer image alone cannot prove closure of the actual state orbit.

4. Sharpened dynamical target

The ideal demo satisfies δ = ε = 0, g/ν = L and s_x = 0. Its selected representation gives the correct weights exactly; the problem is not to replace those weights with fitted frequencies. The projected action must select and preserve that representation and prepared orbit, fix the nuclear frame, and control the relative phases not visible in the readout. If environmental relaxation is invoked, it must be derived separately: unitary Hamiltonian flow alone has no attracting preferred trajectory. The physical rate and finite-width observer response remain additional closure calculations. This identifies testable operator conditions for actual orbits without claiming to have derived their selection.

Reproduction

orbit_exchange_checks.py verifies the differential solution symbolically, checks the ideal curves and their center crossings, and tests the detuning, preparation, and recurrence counterexamples. The visualization page now includes this calculation; the rendering code was subsequently extended to show detuning, relative-phase recurrence, and exact alignment presets while retaining the ideal petals. The renderer regression test checks the actual page JavaScript with canvas/DOM mocks. No connected browser was available for a visual screenshot check.