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: An explicit quadratic nuclear-plane kernel reproduces the precession flow, while inverse-rate reconstruction exposes nonuniqueness and the oriented source required for the full petal generator.

What is being reverse engineered

The target is first the fixed-nucleus sector precession just derived, and then the stronger target of the visualization's selected rotation generator. These are distinct inverse problems. Constructing a kernel that yields a specified motion proves compatibility, not independent physical selection. No measured masses, fields, or conventional atomic-orbital densities are used in the construction below.

Retain J in SO(6)/U(3), the nuclear three-plane projector P, and the symplectic convention dot J = [grad V,J]/I of the precession note. The scale I is an action normalization. The common 6-vector frame and projectors are the binding module's geometric premises, not new coordinates added to the particle.

1. An explicit nuclear-plane interaction kernel

Let u denote a nuclear configuration, P(u) its rank-three projector in the shared frame, and gamma(u) a scalar coefficient. Define

KN6orient(u,J)=γ(u)2∥P(u)JP(u)∥F2=−γ(u)2tr⁡(P(u)JP(u)J). \boxed{ K_{N6}^{\rm orient}(u,J) =\frac{\gamma(u)}2\|P(u)JP(u)\|_F^2 =-\frac{\gamma(u)}2\operatorname{tr}(P(u)JP(u)J). }

This is a smooth, quadratic orientation interaction. Under a simultaneous SO(6) rotation of both arguments it is invariant. It measures how much the electron complex structure rotates nuclear-plane directions within that plane instead of into its complement. It does not model the nucleus as a point or a thin charged sheet.

For a normalized nuclear density with a common projector P(u)=P0,

VN(J)=∫ρN(u)KN6orient(u,J)dμN(u)=γˉs(J;P0),γˉ=∫ρN(u)γ(u)dμN(u). V_N(J)=\int\rho_N(u)K_{N6}^{\rm orient}(u,J)d\mu_N(u) =\bar\gamma s(J;P_0),\qquad \bar\gamma=\int\rho_N(u)\gamma(u)d\mu_N(u).

It gives precisely the previous V(s)=kappa s with kappa=mean gamma. Thus the previously unspecified orientation kernel now has an explicit candidate. For varying P(u), the integral remains well defined but generally does not collapse to one scalar times s; its anisotropy must be retained. The coefficients, density, and identification of the shared frames are still physical inputs.

This kernel is the orientation channel only. It does not supply radial Coulomb binding, spatial falloff, retardation, a field propagator, or a definition of physical separation on the sector. Adding a J-independent term leaves the reduced J flow unchanged, but cannot be advertised as solving the separate relative-position dynamics.

2. Reciprocity is available from the same energy

For fixed u and the skew pairing −tr(XY)/2,

∇JK=2γPJP. \nabla_J K=2\gamma PJP.

For projector variations, with the ordinary symmetric trace pairing,

δPK=tr⁡(GPδP),GP=−γJPJ. \delta_P K=\operatorname{tr}(G_P\delta P),\qquad G_P=-\gamma JPJ.

The latter expression is an ambient gradient restricted to allowed tangent variations delta P = [B,P], B skew; it is not by itself a nuclear velocity. Simultaneous rotation of both arguments gives

δJK∣δJ=[B,J]+δPK∣δP=[B,P]=0. \delta_J K\big|_{\delta J=[B,J]} +\delta_P K\big|_{\delta P=[B,P]}=0.

Hence a joint dynamical model can use this single interaction to obtain reciprocal orientational forces. A nuclear kinetic or symplectic action is still needed for the response equation; no mechanical inertia or magnetic susceptibility is silently supplied by this identity.

3. Invert an entire precession-rate function

For 0<s<1 the previous flow gives the signed rate

ωp(s)=2V′(s)sI. \omega_{\rm p}(s)=\frac{2V'(s)\sqrt{s}}{\mathcal I}.

Therefore a prescribed rate function determines the reduced potential up to a constant on an interval:

V(s)=C+I2∫sωp(v)v dv. \boxed{ V(s)=C+\frac{\mathcal I}{2} \int^s\frac{\omega_{\rm p}(v)}{\sqrt v}\,dv. }

For example, omega_p(s)=a sqrt(s)+b s^(3/2) gives V(s)=C+I a s/2+I b s²/4. Substituting s(J;P(u)) into this expression gives a higher-order orientation-kernel candidate, for a fixed common plane.

A single trajectory at conserved s0 gives much less information: it fixes only V'(s0)=I omega_p(s0)/(2 sqrt(s0)). Adding lambda(s−s0)² changes the kernel away from that trajectory without changing its motion. Likewise, any delta K with zero nuclear marginal leaves V unchanged. Thus neither one precessing orbit nor an entire reduced V identifies a unique microscopic kernel.

4. A nonzero universal rate has a regularity cost

Demanding omega_p(s)=omega0 at every 0<s<1 gives

V(s)=C+Iω0s. V(s)=C+\mathcal I\omega_0\sqrt{s}.

This is not smooth at the aligned configuration. Along the smooth signed-coordinate curve A=epsilon K, Q=diag(sqrt(1−epsilon²), sqrt(1−epsilon²),1), C=−epsilon K, it becomes proportional to |epsilon|. The two directional derivatives disagree at zero. This is a property of the inverse construction, not a prohibition on precession.

A smooth alternative is

Vη(s)=Iω0(s+η2−η),ωp,η(s)=ω0ss+η2,η>0. V_\eta(s)=\mathcal I\omega_0(\sqrt{s+\eta^2}-\eta),\qquad \omega_{{\rm p},\eta}(s)=\omega_0\sqrt{\frac{s}{s+\eta^2}},\quad\eta>0.

It approaches the desired rate away from alignment and tends to zero near alignment. Eta is an additional model parameter, not a derived nuclear width and not the visualization's imbalance parameter. Matching one selected s0 to an orbital clock instead only calibrates a coefficient; it does not establish a dynamically attracting frequency lock.

5. Reverse engineering the full chosen rotation

For a prescribed skew generator B_star with units of inverse time, the linear potential

Vtarget(J)=I⟨B⋆,J⟩=−I2tr⁡(B⋆J) V_{\rm target}(J)=\mathcal I\langle B_\star,J\rangle =-\frac{\mathcal I}{2}\operatorname{tr}(B_\star J)

gives exactly dot J=[B_star,J] and J(t)=exp(t B_star)J0 exp(−t B_star). Choose the 1:1:3 vector-plane generator corresponding to the existing magnetic construction and this supplies its base rotation. Its specified spin/section lift can then be tested against the existing coefficient flow; the physical observer map and preparation are still required for petals.

An inverse source kernel is consequently

Kdrive(u,J)=I⟨B(u),J⟩,∫ρN(u)B(u)dμN(u)=B⋆. K_{\rm drive}(u,J)=\mathcal I\langle B(u),J\rangle,\qquad \int\rho_N(u)B(u)d\mu_N(u)=B_\star.

This identifies the source tensor that would be needed, rather than deriving that tensor from a scalar nuclear density. It is a separate candidate from the quadratic plane kernel; summing them generally changes the target flow. No claim is made that the sum preserves the desired petals automatically.

6. What a bare nuclear plane can and cannot select

A kernel depending only on the unoriented plane P and J must respect the stabilizer SO(3)_O times SO(3)_H. No nonzero fixed skew matrix B_star commutes with this whole group: each diagonal skew 3x3 block must commute with all SO(3) rotations and is therefore zero; the off-diagonal block cannot be invariant under independent rotations of O and H either.

Thus a bare plane cannot select that particular linear drive. This does not contradict the quadratic kernel's precession: its effective generator 2 mean(gamma) PJP/I also depends on the electron's prepared J. The result does not exclude a joint symmetry-breaking electron–nucleus solution. It says that an oriented nuclear current, a relational electron–nucleus tensor, or another dynamically selected structure must supply the missing orientation if the full fixed generator is to follow from the source. Flatness already provides the split; it does not provide every axis and relative rotation rate inside that split.

Status and next test

We have constructed a smooth quadratic inverse kernel, derived both orientation variations, and inverted general precession-rate functions. We have also identified an exact linear source kernel for the stronger target and the extra source information it requires. None is a unique first-principles interaction or a completed electron–nucleus binding model.

Next, use the actual projected nuclear/electron currents to calculate the allowed source tensors and their energy cost. Test whether the quadratic orientation channel and a selected linear drive emerge together with the same orbital clock, rather than fitting their sum to the displayed curve.

Verification

inverse_nuclear_kernel_checks.py tests covariance, both energy variations, reciprocal rotation balance, nuclear marginalization, synthetic inverse recovery, the constant-rate cusp, smooth rounding, one-trajectory nonuniqueness, exact linear-drive motion, and the full-rank bare-plane centralizer constraints. Explicit runtime conditions remain active under Python -O; synthetic targets are not experimental data and no literal PASS is used as a physical verdict. Normal and optimized runs each complete 188 numerical evaluations across 18 conditions. The largest reported residual is approximately 2.2e-10. An in-memory sign reversal of the electron gradient fails its independent finite-difference check in both modes, with nonzero exit status; no mutated source is saved.