Counting mode tools

Exact counting and spectral tools with explicit operator, normalization, and occupation assumptions.

All physical coordinates belong to one nested chain of ordinary macroscopic spatial submanifolds. The scalar models and coefficient spaces here are mathematical tools whose physical application requires a specified operator.

• Simplex band level moments — derives exact mean and variance for a uniform cumulative band.
• Weighted mode evaluation — states the necessary and sufficient point-evaluation bound.
• Simplex heat trace — bounds a specified positive spectral tower and its truncation error.
• Round sphere dirac counts — keeps eigenvalues and dimension-dependent spinor multiplicities explicit.
• Normalized oscillator modes — supplies normalized functions, shell convolutions, and weighted overlaps.
• One spatial band for counts, contact and transport — applies these counts and moments to actual evaluation operators, their kinetic dressing, connection and normalized overlaps.
• Angular kernels, shapes and small oscillations — uses the same finite-band normalization to obtain exact direction sets and their shape-frequency ratios.

Several frequencies from one field

The combined spectral construction makes the role of the moments explicit: the cumulative mean is the bright state's kinetic expectation, and the cumulative variance is its squared coupling to the kinetic complement. They therefore give a controlled correction to the contact frequency. For shell-weighted response, the exact identity $g_d(n-1)/S(n,d)=d/(n+d-1)$ also fixes the static tail and a convergent frequency-subtracted remainder. The same count enters operator strength, frequency corrections and error control.

A governing operator determines available frequencies; source coupling and nonlinear dynamics determine which excitations become occupied and persist. The motivating open question is how several frequencies that are not direct harmonics of one another arise within $\Psi_\infty$. Parallel multi-delay comb filters were an exploratory guess. Other resonance mechanisms remain possible within the field's dynamics.

For example, the conservative amplitude equation

$$\ddot q+\begin{pmatrix}2&-1\\-1&2\end{pmatrix}q=0$$

has squared frequencies $1,3$, with normalized eigenvectors $(1,1)/\sqrt2$ and $(1,-1)/\sqrt2$. Its frequencies are $1,\sqrt3$ in the chosen units. Such amplitudes describe modes of a field; they add no physical spatial dimensions. Connecting a resonance model to the occupied particle spectrum still requires the source, interaction, and state.