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 finite-inertia rank-three nuclear plane coupled reciprocally to a 6D electron preserves its projector constraints, energy, visible momentum, and combined rotational charge while undergoing nontrivial orientation response.

Purpose

The plane-response audit showed that a nuclear three-plane needs conjugate momentum. This note constructs that missing finite-inertia system before any molecular periodic orbit is sought. It is a consistency model for reciprocal orientation dynamics, not a microscopic identification of the nuclear kernel.

Hamiltonian variables and interaction

Let q,p∈R6q,p\in\mathbb R^6 describe the electron, let R,PR∈R3R,P_R\in\mathbb R^3 describe the lower-depth nuclear centre, and let

PT=P,P2=P,tr⁡P=3 P^T=P,\qquad P^2=P,\qquad\operatorname{tr}P=3

be its oriented three-plane. A skew matrix Ω∈so(6)\Omega\in\mathfrak{so}(6) is the plane's angular velocity and carries kinetic energy I⟨Ω,Ω⟩/2I\langle\Omega,\Omega\rangle/2, with ⟨A,B⟩=−tr⁡(AB)/2\langle A,B\rangle=-\operatorname{tr}(AB)/2.

For the selected bound-within/free-without ontology use the cylindrical distance

s=dTPd,d=q−(R,0), s=d^TPd, \qquad d=q-(R,0),

and V=−Zs−1/2V=-Zs^{-1/2}. Coordinates perpendicular to the nuclear plane do not increase separation from its cylindrical support. The numerical consistency run uses dimensionless MN=12M_N=12, I=2.3I=2.3, and Z=0.7Z=0.7. These values are not fitted to an atom. An earlier η=0.35\eta=0.35 interpolation gave the nucleus finite unobserved localization and has been removed as inconsistent with this ontology.

Reciprocal equations

Write G=∂V/∂PG=\partial V/\partial P. Variation under δP=[K,P]\delta P=[K,P] fixes the plane equations

P˙=[Ω,P],IΩ˙=2[P,G]. \dot P=[\Omega,P], \qquad I\dot\Omega=2[P,G].

The electron receives −∇qV-\nabla_qV. The three-coordinate nuclear centre receives the opposite visible component +(∇qV)1:3+(\nabla_qV)_{1:3}. This respects the depth mismatch: the nucleus is not assigned unobserved translational momentum.

The torque factor and sign are fixed by differentiating the common energy. They are also required for the visible rotational Noether charge

JO=qO∧pO+R∧PR−IΩOO \mathcal J_O =q_O\wedge p_O+R\wedge P_R-I\Omega_{OO}

to be conserved. The minus sign reflects the angular-velocity convention in P˙=[Ω,P]\dot P=[\Omega,P].

Nonlinear verification

Sixteen independent random states check the differential conservation of energy, visible total momentum, and JO\mathcal J_O. A nonlinear integration over 18 model time units gives:

quantity maximum error
total energy 2.57×10−102.57\times10^{-10}
projector idempotence 1.01×10−111.01\times10^{-11}
visible total momentum 2.05×10−152.05\times10^{-15}
visible rotational charge 5.12×10−115.12\times10^{-11}

Symmetry of PP, skewness of Ω\Omega, and tr⁡P=3\operatorname{tr}P=3 remain preserved. The final projector differs from its initial value by more than 0.10.1 in norm, so conservation is not a consequence of a frozen plane.

What this changes

The project now has an energy-conserving mechanism by which a flexible lower-depth nucleus can store orientation momentum and return it to a 6D electron. This closes the phase-space consistency gap identified in the finite-response note.

It does not yet establish gyroscopic stabilization of the trigonal saddle. The next calculation must duplicate this rotor for all symmetry-related nuclei, regularize the supported-plane Coulomb singularity using the derived finite nuclear profile, and compute the full monodromy matrix. The physical kernel must eventually determine II, ZZ, and the finite-size profile rather than accepting diagnostic values.

Reproduction

nuclear_plane_rotor_checks.py checks the differential identities, nonlinear evolution, projector constraints, Noether charges, and nontrivial plane response.

  • Finite nuclear-plane response — prerequisite: proves why conjugate orientation momentum is required and gives the local gyroscopic stability target.
  • Reciprocal cylindrical molecule — continuation: combines multiple rotors with moving nuclear centres, the finite profile, and a relative electron source.