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.

Geometric lift, not a substituted orbital

In the nuclear-relative spin frame the geometric homogeneous lift is C4=C2 tensor C2. For normalized Z let rho=ZZ†, and define real expectation values

ai=tr⁡(ρ σi⊗I),bj=tr⁡(ρ I⊗σj),Tij=tr⁡(ρ σi⊗σj). a_i=\operatorname{tr}(\rho\,\sigma_i\otimes I),\quad b_j=\operatorname{tr}(\rho\,I\otimes\sigma_j),\quad T_{ij}=\operatorname{tr}(\rho\,\sigma_i\otimes\sigma_j).

With [a]_cross the usual cross-product matrix, the convention

J(Z)=([a]×−TTT[b]×) J(Z)=\begin{pmatrix}[a]_\times&-T\\T^T&[b]_\times\end{pmatrix}

satisfies J²=−I for every pure normalized ray. The mixed block is Q=−T. For the canonical Schmidt representative Z=cos(eta)|01>−sin(eta)|10>, a_z=−b_z=cos(2eta), Q=diag(sin(2eta),sin(2eta),1). Local SU(2) rotations give the corresponding independent SO(3) frame rotations and cover the pure-ray family. This reproduces the binding module's canonical J(s).

This map expresses the orientation matrix in terms of homogeneous coefficient data.

The explicit mixed-plane operator

The response energy −g tr(X^T Q) is exactly the expectation of

hX=g∑i,jXij σi⊗σj. h_X=g\sum_{i,j}X_{ij}\,\sigma_i\otimes\sigma_j.

Thus the required nine-component operator map is explicit. Under the projective action i hbar Z† dot Z minus Z†h_XZ, direct Pauli commutators give

J˙=−2gℏ[BX,J]. \dot J=-\frac{2g}{\hbar}[B_X,J].

Comparing with the matrix convention dot J=−g[B_X,J]/I_e fixes

Ie=ℏ/2 \mathcal I_e=\hbar/2

when the geometric projective action is normalized with hbar. This fixes a relative factor, not an independently measured action scale. If the master projection supplies a different coefficient for that action, the same calculation replaces hbar by that coefficient.

The operator construction does not derive a physical source for X or the coupling g. It explains precisely which nonfactorizing geometric-spin operator the proposed response would enter. A scalar identity vertex is not silently being relabelled as this tensor interaction.

Checks

source_geometry_maps_checks.py checks the complex-structure identity for generic rays, the energy expectation, the commutator normalization, and independent frame changes. The microscopic field/current identification remains open.