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 reciprocal electron–nuclear polarization model produces retarded feedback and coupled modes, with energy-checked evolution of circular and petalled preparations.

Scope and missing physical input

This is a possible feedback extension of the already existing orbit-plus-precession construction, not the origin of its petals. The unperturbed h₀ below already supplies the selected exchange and clock; the induced response perturbs that prescribed motion. These calculations do not derive the precession lock from Coulomb binding or establish that nuclear feedback is necessary for it. In particular, this electric-dipole model does not calculate induced magnetic feedback; that requires a magnetic/current response and its reciprocal coupling.

This extends the actual-orbit coefficient dynamics with a responding nuclear electric dipole. It is a semiclassical mean-field model: c describes the electron's section amplitudes and p a collective nuclear polarization. It is not a microscopic proton calculation or a replacement by stationary orbitals. Magnetic/spin response, nuclear recoil, and radiative damping need distinct terms; the recoil and combined-source issues are treated separately.

The nucleus responds to the electron's local electric field, not necessarily to its far-field dipole. Let F_i be the Hermitian matrix for that local field in the full retained electron section space, so E_{e,i} = c†F_i c. Once the physical observer charge matrix R(x) is known, a quasistatic nonoverlapping source would give

Fi=14πϵ0∫R(x)(RN−x)i∣RN−x∣3 d3x. F_i=\frac1{4\pi\epsilon_0}\int R(x)\frac{(R_N-x)_i}{|R_N-x|^3}\,d^3x.

Finite nuclear size and overlapping charge require a regularized source calculation. Retardation requires the electromagnetic Green function and current, not this instantaneous Coulomb expression. The V2 current map still has to determine F_i; it cannot generally be inferred from a dipole matrix D_i alone. For the explicit diagnostic below we assume one channel F = κD and leave κ uncalibrated.

One reciprocal action

For one nuclear electric-polarization component, choose strength s_N > 0 and frequency Ω_N > 0. Its static polarizability is α_N(0) = s_N/Ω_N². With c†c = 1, the action is

S=∫dt[iℏc†c˙−c†h0c+p˙2−ΩN2p22sN+p c†Fc]. S=\int dt\left[i\hbar c^\dagger\dot c-c^\dagger h_0c +\frac{\dot p^2-\Omega_N^2p^2}{2s_N}+p\,c^\dagger Fc\right].

Both equations follow from the same interaction:

iℏc˙=(h0−pF)c,p¨+ΩN2p=sNc†Fc. i\hbar\dot c=(h_0-pF)c,\qquad \ddot p+\Omega_N^2p=s_Nc^\dagger Fc.

Thus the nucleus is driven by the electron, while its induced dipole enters the electron Hamiltonian as −pF. Keeping only the first effect would not be a backreaction model. The conserved energy is

E=c†h0c+p˙2+ΩN2p22sN−p c†Fc. E=c^\dagger h_0c+\frac{\dot p^2+\Omega_N^2p^2}{2s_N} -p\,c^\dagger Fc.

In the static mean-field limit, minimizing over p gives p = α_N(0)⟨F⟩ and energy shift −α_N(0)⟨F⟩²/2. The factor one half is required because energy is stored in the polarized nucleus. This is not the operator shift −α_N(0)F²/2 obtained in a suitable fast, quantized oscillator elimination: electron–nuclear correlations and fluctuations distinguish that quantum treatment from the present mean-field model.

Adding an internal damping term γ_N ṗ with γ_N ≥ 0 gives Ė = −γ_N ṗ²/s_N. Nuclear polarization does not provide an inexhaustible energy supply. This phenomenological damping is not the coherent radiation reaction of the complete atom, which must be derived from the combined source.

Induced charge and current

A nuclear polarization density P_N(x,t) = p(t)G_N(x−R_N) along the selected axis, with a normalized centered finite profile G_N, supplies ρ_pol = −∇·P_N and j_pol = ∂_tP_N. These satisfy continuity identically and carry zero net charge but dipole p. For a moving R_N, differentiating the full translated profile includes its motion. The nuclear net charge and center-of-mass current are additional sources, not part of this zero-net-charge polarization term. A magnetic response would add ∇×M_N to the current and require its own response function.

Retarded response and return field

With e^{-iωt} convention, the nuclear susceptibility is

αN(ω)=sNΩN2−ω2−iγNω. \alpha_N(\omega)=\frac{s_N}{\Omega_N^2-\omega^2-i\gamma_N\omega}.

For positive ω and nonnegative damping its imaginary part is nonnegative. A harmonic local field F_ω deposits average power ω Im(α_N)|F_ω|²/2 in this damped oscillator. Resonant enhancement therefore cannot be considered without phase, absorption, and backreaction.

On the unperturbed actual orbit, the electron's response to −pF is

χFF(t,s)=iℏc0†[F(t),F(s)]c0,δ⟨F⟩=χFF∗p. \chi_{FF}(t,s)=\frac{i}{\hbar}c_0^\dagger[F(t),F(s)]c_0, \qquad \delta\langle F\rangle=\chi_{FF}*p.

Here the star denotes the causal time integral. Hence p = α_N*(F₀+χ_FF*p). The companion expression δd_i = χ_{D_iF}*p gives the change in the electron's measured dipole. For a nonstationary orbit these are generally two-time kernels. A scalar frequency denominator is justified only for a time-translation-invariant channel, a justified orbit-phase average, or a Floquet generalization.

In the scalar stationary channel F = κD, write the electron susceptibility as χ_e. Solving the two response equations gives

p=αNκ1−αNκ2χe d0,dtotal=1+αNκ1−αNκ2χe d0. p=\frac{\alpha_N\kappa}{1-\alpha_N\kappa^2\chi_e}\,d_0, \qquad d_{\rm total}=\frac{1+\alpha_N\kappa}{1-\alpha_N\kappa^2\chi_e}\,d_0.

The denominator determines feedback and coupled poles. The numerator determines electron–nucleus electric-dipole interference. They are distinct: changing κ to −κ leaves the feedback denominator unchanged but changes the radiation amplitude. Any use of this formula also needs the physical conversion between local field and dipole represented by κ.

Coupled frequencies and stability are calculable, not automatic

For a positive-response oscillator approximation χ_e = s_e/(Ω_e²−ω²), s_e > 0, and zero damping, the coupled frequencies obey

ω±2=Ωe2+ΩN2±(Ωe2−ΩN2)2+4κ2sesN2. \omega_\pm^2=\frac{\Omega_e^2+\Omega_N^2 \pm\sqrt{(\Omega_e^2-\Omega_N^2)^2+4\kappa^2s_es_N}}2.

Both squared frequencies are positive precisely when κ²s_es_N < Ω_e²Ω_N². Near resonance the modes mix; at stronger coupling this particular quadratic model becomes unstable. Conservative mode mixing does not by itself attract preparations into a preferred locked orbit. The previously calculated electron response can have either sign depending on its prepared occupations, so positivity of s_e must not be assumed for every orbit channel.

Direct reciprocal evolution on the actual section model

The executable diagnostic uses the full section spaces at L = 1, 2, 3, not just the four initially occupied modes. Its h₀ = ℏνĜ_L and D = (T_34+T_43)/2 in chosen dipole units are the earlier geometric candidate. It also initializes the three-state ring embedded at degree 2. No field frequencies are fitted to experimental lines.

Illustrative dimensionless parameters are ℏ = ν = 1, Ω_N = 1.3, s_N = 0.8, κ = ±0.7, γ_N = 0. The initial polarization and its velocity are zero. These parameters are deliberately visible-response test values, not estimates of the proton's response. Evolution runs to t = 20.

The integrator alternates exact half-steps of the free electron/nucleus flow and an exact coupling step. During the latter p is fixed and ⟨F⟩ is conserved, so c → exp(iΔt pF/ℏ)c and ṗ → ṗ+s_NΔt⟨F⟩. This preserves coefficient norm; the second-order splitting's energy error is checked by halving the step from 0.02 to 0.01.

preparation maximum nuclear dipole magnitude final 1−fidelity to unforced orbit maximum energy error
two petals 0.102915 0.0003331 5.2×10⁻⁸
four petals 0.122709 0.0016087 7.4×10⁻⁸
six petals 0.129525 0.0023806 8.3×10⁻⁸
ring embedded at degree 2 0.122709 0.0016087 7.4×10⁻⁸

The degree-2 ring and four-petal preparations give the same active clock response for this particular single-channel probe; it does not distinguish their spectator amplitude occupations. That is a property of the chosen coupling, not an identification of their actual orbits.

Changing κ's sign gives the same evolved electron state and opposite nuclear dipole. Postprocessing the combined dipole acceleration changes the radiation diagnostic, as expected. Both signs give nonzero radiation diagnostics in these transient runs; no cancellation or attracting stable atom is found. Radiation is not fed back into this conservative integrator, so its excellent energy conservation must not be presented as proof of radiative stability.

Outcome

This provides a reciprocal, energy-audited electric-polarization feedback model. It can shift and mix modes and change the combined radiation. Whether it selects V2's observed orbit families depends on the actual nuclear susceptibility, local-field matrices, and radiation reaction. The companion benchmark tests the scale of ordinary proton polarization without claiming it bounds every multidimensional source map.

nuclear_feedback_checks.py checks the feedback algebra, passive nuclear response, coupled-mode threshold, four orbit integrations, norm/energy convergence, sign reversal, and total-dipole diagnostics. No microscopic nuclear response is derived by these tests.