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: Auxiliary model only: An exactly solved orbital-gap model quantifies when a projected spin source survives orbital backreaction.

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.

Supplied selector, not a derived interaction

Let A = Σ σ_ab ⊗ L_ab on the harmonic degree-two orbital space tensored with one chiral spinor. Its spectrum is −6 (multiplicity 20), +2 (multiplicity 60). Choose the normalized two-plane quadratic orbital φ in the projection source test. Let P = I₄ ⊗ |φ⟩⟨φ| and Q = I − P. Study the deliberately supplied selector

H=ϵA+ΔQ,ϵ>0,Δ>0. \mathcal H=\epsilon A+\Delta Q,\qquad \epsilon>0,\quad\Delta>0.

Δ is an orbital gap assumption. This projector penalty is not claimed to be a local potential, a result of the scalar moment functional, or part of the already derived V2 action. Its role is to turn the frozen-orbital approximation into a solvable diagnostic.

Exact reduction

The compressed spin operator H₀ = ⟨φ|A|φ⟩ has eigenvalues h = −2, −2/3, 2/3, 2. On the harmonic orbital space, A² + 4A = 12I. For v_h = s_h ⊗ φ, define c_h = 12 − 4h − h². If c_h > 0, w_h = (A − h)v_h / √c_h is orthogonal to P and the two-dimensional span is invariant. In its orthonormal basis,

Hh=(ϵhϵchϵchΔ+ϵ(−4−h)), \mathcal H_h=\begin{pmatrix} \epsilon h&\epsilon\sqrt{c_h}\\ \epsilon\sqrt{c_h}&\Delta+\epsilon(-4-h) \end{pmatrix},

E±(h)=Δ−4ϵ±Δ2−4ϵΔ(2+h)+64ϵ22. E_\pm(h)=\frac{\Delta-4\epsilon \pm\sqrt{\Delta^2-4\epsilon\Delta(2+h)+64\epsilon^2}}{2}.

The h = 2 case has c_h = 0 and only the actual vector v_h with energy 2ε. The second formal root is not a second vector in this collapsed two-dimensional construction. Other harmonic states can independently have the dark energies Δ − 6ε and Δ + 2ε.

Selection and its error

For every positive Δ and ε the unique ground branch is h = −2:

E0=Δ−4ϵ−Δ2+64ϵ22=−2ϵ−16ϵ2Δ+O(ϵ4/Δ3). E_0=\frac{\Delta-4\epsilon-\sqrt{\Delta^2+64\epsilon^2}}2 =-2\epsilon-\frac{16\epsilon^2}{\Delta} +O(\epsilon^4/\Delta^3).

It is below the dark energy Δ − 6ε. At Δ = 0 the original 20-fold ground degeneracy returns; the gap begins as Δ/6 and approaches 4ε/3 as Δ/ε grows. In this particular orbital example, tracing out the orbital gives fidelity with the projected ground spin ray

F=12(1+ΔΔ2+64ϵ2)=1−16ϵ2Δ2+O(ϵ4/Δ4). F=\frac12\left(1+\frac{\Delta} {\sqrt{\Delta^2+64\epsilon^2}}\right) =1-\frac{16\epsilon^2}{\Delta^2} +O(\epsilon^4/\Delta^4).

At Δ/ε = 1, 10, 100 the fidelities are approximately 0.5620, 0.8904, 0.9984. Existence of a unique ground is therefore much weaker than accuracy of a frozen pure-spin state. The second-order effective operator is εH₀ − (ε²/Δ)(12I − 4H₀ − H₀²), preserving its spin eigenvectors in this model but not eliminating orbital entanglement.

Reproduction and boundary

orbital_gap_checks.py checks invariant blocks, full eigenvalues, and reduced-spin fidelities at five gap strengths. Its 84-dimensional implementation retains the scalar trace sector as four decoupled states of energy Δ; this does not alter the harmonic ground. What remains is to derive an orbital-selecting operator, its strength, its sign, and the appropriate field-to-spatial-coordinate relation.