Zonal hybrid geometry

Directed angular functions

On the unit sphere of a chosen ordinary spatial three-plane, let $Y_{\ell m}$ be orthonormal with area measure. Define a normalized angular coefficient vector for unit direction $v$ by

$$|\ell,v\rangle=\sqrt{\frac{4\pi}{2\ell+1}} \sum_{m=-\ell}^{\ell}\overline{Y_{\ell m}(v)}|\ell,m\rangle.$$

The spherical-harmonic addition theorem gives

$$\langle\ell,v|\ell,w\rangle=P_\ell(v\cdot w), \qquad \|\,|\ell,v\rangle\|=1.$$

The normalization and addition formula are specified in DLMF 14.30.8–9. Different $\ell$ blocks are orthogonal. Thus a directed function $|h_v\rangle=\sum_\ell c_\ell|\ell,v\rangle$, with the same coefficients for each direction and $\sum|c_\ell|^2=1$, has overlap

$$\boxed{\langle h_v|h_w\rangle=\sum_\ell|c_\ell|^2P_\ell(v\cdot w).}$$

These are angular functions, not trajectories or additional physical spatial dimensions. The three-plane is selected relative to the system being described; the electron's six spatial directions remain equivalent.

The scalar-plus-vector case

Let $|h_i\rangle=a|s\rangle+b|p_{v_i}\rangle$, with $a,b>0$, $a^2+b^2=1$. Suppose the $n$ functions are orthonormal and their span contains $|s\rangle$. Completeness within that span implies

$$1=\sum_{i=1}^n|\langle h_i|s\rangle|^2=na^2.$$

Consequently $a^2=1/n$, $b^2=(n-1)/n$, and off-diagonal orthogonality requires

$$v_i\cdot v_j=-\frac1{n-1}\quad(i\ne j).$$

Their direction Gram matrix has one zero eigenvalue and $n-1$ eigenvalues $n/(n-1)$, so it can be realized in a three-plane only if $n\le4$. The sum of the directions is zero. This gives the regular simplex cases:

The simultaneous directed form, equal coefficients, orthonormality, and span condition are assumptions. Orthogonalizing arbitrary directed functions generally changes their form; it does not establish these assumptions or select nuclear equilibrium angles.

A six-direction construction

With squared shares $(|c_0|^2,|c_1|^2,|c_2|^2)=(1/6,1/2,1/3)$,

$$\frac16+\frac12x+\frac13P_2(x)=\frac{x(x+1)}2.$$

The six directions $\{\pm e_1,\pm e_2,\pm e_3\}$ have distinct pair cosines $0$ or $-1$, so their six directed functions are orthonormal. This proves an octahedral construction for the displayed shares. Selecting those shares or an actual molecular geometry requires dynamics.

An Explicit Energy and Its Shape Spectrum

The angular-kernel continuation turns these overlaps into the specified positive energy $E=\kappa\sum_{i<j}|K(v_i\cdot v_j)|^2$. It proves that the complete scalar–vector band selects a regular simplex among its directed evaluation states. The vector–quadratic feature selects antipodal orthogonal axes. Both are unique zero-energy shapes up to rotation at their stated numbers of directions.

The same energy, with a common angular inertia, gives an exact small-oscillation spectrum. At three resolved dimensions the tetrahedron has a triplet and doublet with ratio $\sqrt{3/2}$; the octahedron has six lower and three higher modes with ratio $\sqrt2$. The continuation proves the corresponding spectra for general resolved dimension and checks the three-dimensional Hessians exactly. The physical band, fixed-radius approximation, inertia and interaction coefficient remain premises to derive from the molecular reduction.

• Multicentre orbit axes — obtains related direction geometry from an actual prescribed route.

• Cylindrical molecular shape stability — tests full ligand forces beyond kinematic axes.

• Resonance and profile tools — overview and verification.

resonance_profile_checks.py checks finite examples by quadrature, matrix evolution, and independent algebra.