Angular Mode Kernels Select Shapes and Their Small Oscillations

The angular addition theorem and the directed-state construction can be combined into an explicit energy. Its minima determine direction sets, and its second variation determines small-oscillation frequencies. This supplies more than an allowed bond angle: it gives a testable response spectrum under one stated effective interaction.

Here $k$ denotes the number of spatial directions resolved by the angular construction. A molecular observer uses $k=3$. Identifying its modes with the reduction of a depth-six electron remains the separate physical map in the bond-geometry module. An observer's angular space does not replace the particle's spatial sector.

1. The Complete Scalar–Vector Band Fixes the Overlap

On the unit $S^{k-1}$ with probability measure,

$$\int n_a n_b\,d\nu=\frac{\delta_{ab}}k.$$

Thus the scalar function $1$ and vector harmonics $\sqrt{k}\,n_a$ form an orthonormal band of dimension $k+1$. Its normalized evaluation state is

$$u(n)=\frac{(1,\sqrt{k}\,n)}{\sqrt{k+1}},\qquad u(n)^\dagger u(m)=K_1(n\cdot m) =\frac{1+k\,n\cdot m}{k+1}.$$

The scalar and vector weights $1:(k)$ follow from this complete band and its measure. The overlap is the same $s$-plus-$p$ structure used in the directed-state theorem.

For $k+1$ unit directions, specify the positive pair energy

$$E_1=\kappa\sum_{i<j}|K_1(n_i\cdot n_j)|^2,\qquad \kappa>0.$$

Each term is nonnegative. A regular simplex attains zero because its off-diagonal dot products are $-1/k$. Conversely, zero energy requires every off-diagonal dot product to equal $-1/k$. Its Gram matrix has rank $k$, fixing the configuration up to an orthogonal spatial transformation.

Therefore this energy selects the regular simplex as its global minimum:

$$k=2:\text{ three directions at }120^\circ,\qquad k=3:\text{ four directions at }\arccos(-1/3).$$

The energy and the angular-band choice are explicit physical hypotheses. The minimum and its uniqueness follow mathematically. Minimization here keeps the norm and scalar–vector relative phase fixed within the displayed real directed family; the field equation must establish that family's physical role. A generic orthogonal change of occupied wavefunctions is not enough to produce this result; the evaluation states specify actual directions.

2. A Conservative Shape Dynamics Has a Fixed Spectrum

For a common positive angular inertia $I$, take the effective constrained Lagrangian

$$\mathcal L_{\rm ang}=\frac I2\sum_i|\dot n_i|^2-E_1,\qquad |n_i|=1.$$

At a simplex minimum, a tangent displacement obeys $n_i\cdot\delta n_i=0$. The first variation of the pair overlap is

$$\delta K_{ij}=\frac{k}{k+1} (n_j\cdot\delta n_i+n_i\cdot\delta n_j).$$

Because every initial overlap is zero, the Hessian is the positive operator defined by

$$\delta^2E_1=2\kappa\left(\frac{k}{k+1}\right)^2 \sum_{i<j}(n_j\cdot\delta n_i+n_i\cdot\delta n_j)^2.$$

Its complete spectrum on the $(k+1)(k-1)$ tangent coordinates is

The second positive family is absent at $k=2$. All nonrotational shape directions have positive stiffness.

For $k=3$, the physical angular frequencies in this model are

$$\omega_{\rm triplet}^2=\frac{2\kappa}{I},\qquad \omega_{\rm doublet}^2=\frac{3\kappa}{I},\qquad \frac{\omega_{\rm doublet}}{\omega_{\rm triplet}}=\sqrt{\frac32}.$$

The same four-state construction therefore determines both the tetrahedral minimum and a triplet–doublet shape spectrum. One common scale $\kappa/I$ remains; the ratio needs no separate adjustment of the two families. This is an ideal angular prediction, not a claim to have computed all molecular vibrations, bond lengths or actual electron trajectories.

The conservative dynamics oscillates near its minimum. It does not by itself relax from arbitrary initial conditions; damping or energy transfer to additional modes would require their derived response.

3. Derivation of the All-Dimensional Hessian

Let $\mathsf N$ have the simplex directions as its $k+1$ rows. Then

$$\mathsf N^T\mathsf N=cI,\qquad \mathsf N^T\mathbf1=0,\qquad c=\frac{k+1}{k}.$$

Every displacement matrix has the unique form

$$\delta\mathsf N=\mathsf N A+\mathbf1b^T.$$

The skew part of $A$ gives common rotations. For the symmetric part $S$, the tangent constraints give $\operatorname{tr}S=0$ and

$$n_i^TSn_i+n_i\cdot b=0.$$

The traceless symmetric space splits into a $k$-dimensional subspace with

$$|b|^2=\frac{k-1}{k}\|S\|_F^2$$

and its orthogonal complement with $b=0$, of dimension $(k+1)(k-2)/2$. To see the split, use the matrices $n_in_i^T-I/k$: their Gram matrix has diagonal $(k-1)/k$ and off-diagonal $-(k-1)/k^2$, hence rank $k$.

The displacement norm and Gram variation satisfy

$$\|\delta\mathsf N\|_F^2=c\|S\|_F^2+(k+1)|b|^2,$$

$$\|\delta(\mathsf N\mathsf N^T)\|_F^2 =4c^2\|S\|_F^2+2(k+1)c|b|^2.$$

Substituting the two cases into the second variation gives $2\kappa$ and $4\kappa/c$, respectively. This proves the spectrum without a dimension-by-dimension fit.

4. A Vector–Quadratic Kernel Selects Antipodal Orthogonal Axes

A second normalized feature state is

$$w(n)=\frac1{\sqrt2}(n,\ n\otimes n),\qquad K_2(n\cdot m)=\langle w(n),w(m)\rangle =\frac{x+x^2}{2},\quad x=n\cdot m.$$

It uses an equally weighted vector and symmetric quadratic spatial tensor. Decomposing the latter into trace and traceless parts gives the angular weights

$$K_2(x)=\frac1{2k}+\frac12x+ \frac{k-1}{2k}\frac{kx^2-1}{k-1}.$$

At $k=3$ these are the $L=0,1,2$ weights $1/6,1/2,1/3$. They follow from the stated feature norm, which remains a choice for the physical interaction. They are not the weights of an unweighted complete $L\leq2$ spherical band.

For $2k$ directions, specify

$$E_2=\kappa\sum_{i<j}|K_2(n_i\cdot n_j)|^2.$$

Zero energy requires every pair to be orthogonal or antipodal, since the roots are $0$ and $-1$. There can be at most $k$ mutually orthogonal lines and two directions on each. A zero-energy set of $2k$ distinct directions is therefore exactly a rotated cross-polytope:

$$\{+e_1,-e_1,\ldots,+e_k,-e_k\}.$$

At $k=3$ this is the octahedron. The same rule produces a square at $k=2$ and the higher-dimensional cross-polytopes.

With the same angular kinetic convention, each coordinate-plane block of its Hessian has eigenvalues $0,\kappa,\kappa,2\kappa$. Thus

At three dimensions the six lower shape modes and three higher shape modes have frequency ratio $\sqrt2$. Tangent variations of an antipodal pair have zero first-order dot-product change; the orthogonal pairs supply the displayed quadratic stiffness. The coordinate-plane blocks account for all $2k(k-1)$ tangent coordinates.

5. How This Fits V2

Both examples follow the same sequence:

$$\text{spatial mode feature} \longrightarrow\text{normalized overlap} \longrightarrow\text{reciprocal energy} \longrightarrow\text{shape and response spectrum}.$$

The scalar–vector feature reuses the proposed $s\oplus p$ reduction of the electron's spatial field. The vector–quadratic feature reuses the angular contact tensors already present in coupling calculations. Neither construction fits an angle directly. Both retain one common interaction coefficient and one common angular inertia within the stated model.

For a physical application, derive that feature band, the fixed-length approximation and the coefficients from the electron–nucleus interaction. Add the radial, multi-center and spin-dependent terms under the same action before comparing a complete molecular spectrum. For particle-sector use, compare the actual physical angular carrier at every depth rather than substituting the observer's three directions universally.

6. Temporal Phase Can Split the Shape Response

If the directed source carries a separable finite sawtooth, temporal averaging multiplies each pair energy by the positive density overlap $W_{ij}$. Every zero-energy simplex shape survives, while its Hessian becomes $2\kappa\sum_{i<j}W_{ij}|\delta K_{ij}|^2$. Equal weights preserve the original frequency ratios; unequal relative phases split them.

For the limiting sawtooth overlap, opposite phases have one sixth of the aligned weight. A tetrahedron prepared with phases $(0,\pi,0,\pi)$ consequently has squared shape frequencies

$$0^{(3)},\quad (1/3)^{(2)},\quad (1/2)^{(1)},\quad 2^{(1)},\quad (13/6)^{(1)}$$

in units $\kappa W(0)/I$. Its geometric angles remain unchanged. The sawtooth derivation gives the exact generalized Hessian, finite-cutoff corrections and all-dimensional positive-weight checks. These are conditional shape spectra at specified phases; the zero-energy angular term alone does not select those phases or determine their inertia.

Verification

The shared calculation checks the simplex minima and shape spectra across $k=2,\ldots,12$. It also computes the tetrahedral and octahedral generalized Hessians exactly. In units $\kappa=I=1$, their characteristic polynomials are

$$p_{\rm tet}(\lambda)=\lambda^3(\lambda-2)^3(\lambda-3)^2, \qquad p_{\rm oct}(\lambda)=\lambda^3(\lambda-1)^6(\lambda-2)^3.$$

These are finite geometric calculations, with no particle time-evolution simulation.