The s, p, d, f orbital families are the 3D shadow of the electron's 6D orbit. Bohr levels, all multipole selection rules, Stark, Zeeman, and fine/hyperfine structure follow as theorems of CP³. At every shell with \(L \geq 1\), the SU(4) representation also carries states with angular momentum in the ξ-direction — and Lemma 2 makes them strictly invisible to any 3D apparatus.
Note: \(\Xi_6 = \mathbb{CP}^3\) is a macroscopic 6-real-dimensional space in IDWT, not a compactified extra dimension. The electron moves freely in \(\Xi_6\); 3D experiments only see its \(\mathrm{SO}(3)\) projection.
The panel at right shows this directly. Each shell's electrons — one point particle per state, never a cloud — orbit continuously through \(\mathbb{CP}^3\); their observable projections accumulate over many laps into the familiar s, p, d, f shapes (blue). The \(z_4\)-direction hidden states orbit too, nested at smaller radius (pink), together with the lower-j 3D content (green); toggle each category at upper left. By Lemma 2 the hidden orbits leave no trace any 3D apparatus can read.
In IDWT, the electron is the \((d=6)\) mode \((n=13)\): an eigenmode of the sector-6 Dirac operator on \(\mathrm{CP}^3\), with isometry group \(\mathrm{SU}(4)\). Its 6D orbit is the full trajectory in \(Ξ_6\), while the observable 3D "orbital" is the projection of that orbit onto ordinary space. The hydrogen atom is a bound system on \(M_\infty = \mathbf{R}^3 \times \Xi_6\). The angular momentum eigenmodes are representations of \(\mathrm{SU}(4)\), restricted to the observable subgroup via the chain
where \(\mathrm{SU}(3)\) is the d=4 quark-sector colour group embedded in the d=6 lepton isometry, and \(\mathrm{SO}(3)\) acts on the three real observable directions.
At angular momentum \(L\), the orbital states form the holomorphic symmetric representation of \(\mathrm{SU}(4)\):
Under the full subgroup chain, only the highest-spin \(\mathrm{SO}(3)\) irrep at each level survives projection as a standard orbital — the harmonic polynomials of degree \(L\) in \((z_1, z_2, z_3)\). These are the \(2L+1\) standard spherical harmonics. The remaining states split into two distinct categories with very different physical meanings — see the Counting Theorem below.
Projection of \(\mathrm{Sym}^L(\mathbb{C}^4)\) to the observable \(\mathrm{SO}(3)\) selects exactly \(2L+1\) states — the harmonic polynomials of degree \(L\) in \((z_1,z_2,z_3)\). These are traceless symmetric tensors of rank \(L\), forming precisely the spin-\(L\) irrep of \(\mathrm{SO}(3)\); no lower-\(j\) irrep is harmonic at degree \(L\), which is why only the highest spin survives. The remaining \(\binom{L+3}{3}-(2L+1)\) states fall into two distinct categories:
| Shell | \(L\) | Total | Observable | IDWT-hidden | Lower-j 3D |
|---|---|---|---|---|---|
| s | 0 | 1 | 1 | 0 | 0 |
| p | 1 | 4 | 3 | 1 | 0 |
| d | 2 | 10 | 5 | 4 | 1 |
| f | 3 | 20 | 7 | 10 | 3 |
The total dimension equals the IDWT simplex number at mode \(n = L+1\) in the d=3 hadronic sector:
The orbital shell structure of atomic physics and the IDWT hadronic mass spectrum are governed by the same combinatorial sequence.
Under \(\mathrm{SU}(4) \supset \mathrm{SU}(3) \times \mathrm{U}(1)\), the fundamental decomposes as \(\mathbf{4} \to \mathbf{3} + \mathbf{1}\). The higher symmetric powers decompose as:
| \(\mathrm{Sym}^L\) | \(\mathrm{SU}(3)\) content | \(\mathrm{SO}(3)\) irreps |
|---|---|---|
| \(\mathbf{1}\) | \(\mathbf{1}\) | \(j=0\) |
| \(\mathbf{4}\) | \(\mathbf{3} + \mathbf{1}\) | \(j=1\) · \(j=0\) |
| \(\mathbf{10}\) | \(\mathbf{6} + \mathbf{3} + \mathbf{1}\) | \(j=2\) · \(j=1\) · \(j=0\) (1 hid + 1 low-j) |
| \(\mathbf{20}\) | \(\mathbf{10} + \mathbf{6} + \mathbf{3} + \mathbf{1}\) | \(j=3\) · \(j=2\) · \(j=1\) (3 hid + 3 low-j) · \(j=0\) |
Observable states are the unique highest-\(j\) irrep at each level. IDWT-hidden states involve the \(z_4\) direction and are spectroscopically dark. Lower-\(j\) 3D content is non-harmonic in \((z_1,z_2,z_3)\) with no \(z_4\) dependence; it is already counted in lower orbital shells. At \(L \geq 2\) a given \(j\) value can receive contributions from both the hidden and lower-\(j\) categories — see the explicit bases below.
The total interaction potential on \(M_\infty = \mathbf{R}^3 \times \Xi_6\) is
an exact sum — no \(\mathbf{r}\)-\(\xi\) cross terms. The proof has three parts: (i) the d=2 photon couples only to \(\rho_\mathrm{3D}(\mathbf{r})\), so the Coulomb term has no \(\xi\)-dependence; (ii) CP³ colour silence means the proton's quark modes create no \(\xi\)-dependent potential for the electron; (iii) the CP³ isometry makes \(V_6\) spherically symmetric in \(\xi\) and independent of \(\mathbf{r}\).
Because the potential separates exactly, the Schrödinger equation on \(M_\infty\) is exactly separable. The electron occupies sector mode \(n=13\), so \(\chi = \chi_{13}\) with \(E_\xi = m_e\). The 3D Hamiltonian is therefore exact — not an approximation:
Let \(O(\mathbf{r}, \nabla_r)\) be any 3D operator (no \(\xi\)-dependence). Let \(|\mathrm{obs}\rangle\) be any observable orbital state (angular part is a polynomial in \(z_1, z_2, z_3\) only). Let \(|\mathrm{hid}\rangle\) have nontrivial \(z_4\)-dependence. Then \(\langle\mathrm{hid}|O|\mathrm{obs}\rangle = 0\) — the \(\xi\)-sphere integral of the hidden state's angular factor is zero by spherical harmonic orthogonality.
This single lemma carries six results, one for each physical interaction class that is described by a 3D operator:
\(\mathrm{Sym}^1(\mathbb{C}^4)\) has basis \(\{z_1, z_2, z_3, z_4\}\).
Observable p-orbitals: \(\{z_1, z_2, z_3\} \to (p_x, p_y, p_z)\). These are the standard \(Y_1^m\).
Hidden state: \(\{z_4\}\) — a single basis vector along the SU(3)-singlet coordinate. Its inner product with any monomial in \((z_1, z_2, z_3)\) under the SU(4)-invariant measure is zero (different monomials at the same degree are orthogonal). Hence \(\langle z_4 \,|\, O \,|\, z_i\rangle = 0\) for every 3D operator \(O\) and every \(i \in \{1,2,3\}\).
\(\mathrm{Sym}^2(\mathbb{C}^4)\) has 10 basis monomials.
Observable d-orbitals (5): the traceless degree-2 polynomials in \((z_1, z_2, z_3)\) — the standard \(Y_2^m\).
IDWT-hidden states (4): \(\{z_1 z_4,\, z_2 z_4,\, z_3 z_4\}\) (\(j=1\)) and \(\{z_4^2\}\) (\(j=0\)). All carry \(z_4\)-dependence and integrate to zero over the \(\xi\)-sphere by Lemma 2.
Lower-j 3D state (1): \(\{z_1^2+z_2^2+z_3^2\}\) — the SO(3) trace, \(j=0\), no \(z_4\)-dependence. This is not IDWT-hidden: a 3D apparatus can couple to it just fine, but it transforms identically to a higher Sym\(^L\) copy of the spin-0 representation that already appears at \(L=0\). It does not introduce a new observable orbital — it is the same SO(3) representation showing up again at a higher SU(4) degree.
\(\mathrm{Sym}^3(\mathbb{C}^4)\) has 20 basis monomials.
Observable f-orbitals (7): the harmonic degree-3 polynomials in \((z_1, z_2, z_3)\) — the standard \(Y_3^m\) for \(m = -3,\ldots,+3\).
IDWT-hidden states (10): all monomials containing at least one \(z_4\) factor. These split into three sub-families by z₄ multiplicity: (i) \(\{z_i z_j z_4 : i,j \in \{1,2,3\}\}\) — six states, of which 5 are traceless (\(j=2\)) and 1 is the trace \(z_4(z_1^2+z_2^2+z_3^2)\) (\(j=0\)); (ii) \(\{z_1 z_4^2,\, z_2 z_4^2,\, z_3 z_4^2\}\) (\(j=1\), 3 states); (iii) \(\{z_4^3\}\) (\(j=0\), 1 state). Total: 5+1+3+1 = 10. All are zero-amplitude at every 3D operator vertex by Lemma 2.
Lower-j 3D states (3): \(\{(z_1^2+z_2^2+z_3^2)\,z_1,\; (z_1^2+z_2^2+z_3^2)\,z_2,\; (z_1^2+z_2^2+z_3^2)\,z_3\}\) — degree-3 polynomials in \((z_1,z_2,z_3)\) only, no \(z_4\)-dependence. Each is the SO(3) trace \((z_1^2+z_2^2+z_3^2)\) times a \(j=1\) factor, placing them in the \(j=1\) representation. They are not IDWT-hidden; they are already counted in the \(L=1\) p-shell.
Hidden states with \(z_4\)-direction angular momentum exist in every shell at \(L \geq 1\). They are not merely unobserved — Lemma 2 makes them strictly inaccessible to every 3D operator. No experiment built from R³ apparatus can couple to them at any order, at any energy, by any mechanism. If such a coupling were ever measured, the CP³ identification of the electron sector would be falsified.
The orbital degeneracy counts \(2L+1\) for s, p, d, f shells are a theorem of the representation theory of \(\mathrm{SU}(4)\) under the chain \(\mathrm{SU}(4)\supset\mathrm{SU}(3)\supset\mathrm{SO}(3)\). The non-relativistic Bohr spectrum follows from exact potential separability in \(M_\infty\) (Lemma 1). Every standard spectroscopic selection rule and field-coupling result follows from ξ-orthogonality (Lemma 2). The orbital count, energy spectrum, selection rules, and undetectability of hidden states are all structural consequences of CP³ — fixed entirely by the inputs of the core IDWT framework (\(m_e\) and \(n_s = 4\)).
Reference: Fedge No, Infinite Dimensional Wave Theory (2026), doi:10.5281/zenodo.19767493
See also: The Six Sectors · Generation Tower Mode Selection · Coupling Filters