IDOS Energy, Coherence, and Supporting Dynamics — Part 4

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This positive local operator has kernel exactly $k_j\ge0$, $\sum_jk_j\le N$. The one-coordinate terms first force $k_j\in\{0,\ldots,N\}$. If the sum exceeds $N$, some multi-index $a\le k$ with $|a|=N+1$ gives a nonzero term. Its principal symbol is $(\sum_j\xi_j^2)^{N+1}$, and its kernel has $D=\binom{N+d}{d}$ canonical pairs for one paired-real polarization. With $R$ polarization copies it has $RD$ pairs until an additional selector removes them. The cell is boundary data, not compactification. On whole ordinary space, the same constant-coefficient operator has no nonzero square-integrable kernel: its Fourier zeros form a finite set of measure zero.

For an actual coherent polynomial $P(x)=\sum_kv_ke^{ik\cdot x}$, uniform density requires every nonzero autocorrelation to vanish:

$$\sum_{k-l=q}v_l^*v_k=0\quad(q\ne0).$$

This condition allows cancellation between cross terms. It does not require every coefficient vector to be orthogonal. A concrete degree-two example homogenizes the simplex using $z_0=1$ and $m=d+1$ variables. If a unitary matrix $U$ satisfies $|U_{ii}|^2=2/(m+1)$ and $|U_{ij}|^2=1/(m+1)$ off the diagonal, then

$$P_i=\sqrt{(m+1)/2}\,z_i\sum_jU_{ij}z_j$$

has constant norm squared $m(m+1)/2$, and every quadratic monomial has coefficient norm one. Explicit finite matrices and a block construction give paired-real carrier costs $6,8,18,12,14,22$ at depths $2,3,4,5,6,10$. Thus many equally energized modes can form a genuinely coherent uniform-density field in a small carrier. These degree-two polarizations do not inherit a uniform degree-one prefix; they are not yet a common nested solution at the assigned large particle cutoffs.

There is an exact restriction on that stronger requirement. Write $P_{\le N}=\sum_{n=0}^NP_n$ with inherited homogeneous degree-$n$ polynomials. If every prefix $0,\ldots,M$ has constant norm, subtract consecutive norm identities and rotate all spatial variables by the same phase. The distinct positive frequencies $N-n$ force $P_n^*P_N=0$ pointwise for $n<N$, and each shell norm is constant. For nonzero shells this proves $R\ge M+1$. Equal-energy full simplex support also gives $R\ge d+2$ when $M\ge2$. Saturation $R=M+1$ requires $d\mid M(M+1)/2$: the square matrix of normalized shell columns is polynomial unitary, its determinant is a monomial, and each coordinate winding equals $\sum_{n=0}^M n/d$.

These bounds require every intermediate prefix. For only sparse selected prefixes, cross-shell cancellation survives. For example, coefficients $(1,0)$, $(1,-1)/\sqrt2$, $(0,1)$ at degrees $0,1,2$ have unit norms and uniform prefixes zero and two, but not one. The all-prefix bound must not be used to exclude the maintained sparse particle assignments.

One Kinetic Law for the Actual Carrier

Let $\rho=\psi^T\psi/2$, $\eta=-\psi^TJ\,d\psi/(2\rho)$, and choose either rotation sense $s=\pm1$:

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