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Summary: Auxiliary model only: A pure quadratic six-dimensional orbital can supply a two-plane spin source, but the complete orbital-spin operator does not freeze that orbital automatically.

Interpretation correction

This calculation concerns an auxiliary field or local spin model, not the actual electron orbit constructed in the 6D demo. Its stationary states, angular expectations, and stability results have not been mapped to that trajectory or its observer readout. They must not be used as a replacement for the existing orbit-shape derivation. See Actual orbit research correction.

1. Replace a supplied tensor with an orbital expectation

For the central quadratic potential V=k r^2/2 in the local Dirac model, the leading spin-orbit operator is k A/(4m^2), where

A=∑a<bΣab⊗Lab,Lab=−i(xa∂b−xb∂a). A=\sum_{a<b}\Sigma_{ab}\otimes L_{ab},\qquad L_{ab}=-i(x_a\partial_b-x_b\partial_a).

Given a normalized orbital phi, projecting onto that fixed orbital yields H_spin proportional to sum Sigma_ab <phi|L_ab|phi>. The expectation tensor need not be simple even though the frozen classical symbol x wedge p is. The question is whether a pure wave, not an imposed statistical mixture, can produce the two-plane source found in the preceding audit.

These orbitals are scalar harmonics of the flat R6 coordinate model.

2. Linear orbitals cannot provide two independent planes

A normalized degree-one orbital is a complex vector v in C6. Its angular expectation has components

⟨Lab⟩=2Im⁡(vˉavb). \langle L_{ab}\rangle=2\operatorname{Im}(\bar v_a v_b).

Writing v=u+iw gives 2(u wedge w), of rank at most two. Therefore every pure degree-one orbital leaves the frozen spinor contraction doubly degenerate when nonzero. Dropping coherence between different linear orbitals can change this rank, but then one has changed the state to a mixture.

3. An explicit pure quadratic source

Let z1=(e1+i e4)/sqrt(2) and z2=(e2+i e5)/sqrt(2), regarded as orthonormal complex vectors. In Sym^2(C6), take the normalized pure state

ϕ=2 z1⊗z1+z2⊗z23. \phi=\frac{\sqrt2\,z_1\otimes z_1+z_2\otimes z_2}{\sqrt3}.

Its symmetric tensor is traceless, hence represents a genuine degree-two harmonic. Equivalently the polynomial is a normalized combination of (x1+i x4)^2 and (x2+i x5)^2 with the same relative amplitudes.

The one-body angular generator acts on either tensor slot. Its cross matrix elements between the two squared modes vanish by orthogonality, without discarding the coherence of the full state. Consequently

⟨L⟩ϕ=43e1∧e4+23e2∧e5. \langle L\rangle_\phi= \frac43e_1\wedge e_4+\frac23e_2\wedge e_5.

This full bivector has rank four; the mixed block has rank two. The projected spin operator has spectrum {-2,-2/3,2/3,2}, with a unique locked ground ray in the chirality convention of the preceding audit. This is a constructive source from a specified pure orbital, rather than an arbitrary coefficient matrix. Its particular orbital state is still an input.

4. Restoring orbital dynamics changes the problem

On vector harmonics tensor a chiral Spin(6) spinor, the complete operator A has eigenvalues -5 with multiplicity 4 and +1 with multiplicity 20. On degree-two traceless harmonics tensor the spinor, its eigenvalues are -6 with multiplicity 20 and +2 with multiplicity 60. The calculation is performed explicitly in dimensions 24 and 84, with the four scalar-trace spinors removed from the latter to isolate the 80-dimensional ell=2 space.

For positive k/(4m^2), the fixed-orbital product ground expectation -2 is therefore not the coupled ell=2 ground eigenvalue -6. The orbital cannot be frozen merely because that projection yields the desired spin ray. If a separate orbital Hamiltonian selects phi with a sufficiently large gap, projection could become controlled; that gap and selection must be derived.

The SO(6)-invariant coupled problem also has a nontrivial ground multiplet, not a unique absolute orientation. This does not exclude a relational lock in a driven, symmetry-broken, or otherwise selected state. It locates the remaining dynamical input instead of replacing it with the chosen phi.

Verification and next step

orbital_source_projection_checks.py builds the normalized symmetric tensor representation, verifies harmonicity, evaluates all angular moments, and diagonalizes both full coupled operators. The source rank and projection identities are analytic; numerical eigenspectra are finite checks of the stated matrices, not a V2 master-spectrum computation.

The next useful calculation is a controlled orbital projection with an explicit source-selecting gap, or an actual cross-depth operator that supplies the correlated orbital state. The existence of the polynomial above does not establish either condition.

  • Spin-orbit source audit — derives the local derivative vertex and explains why the full two-form, not only its mixed block, matters.
  • Research status — current proof boundaries and correction routes.