Finite ray resolution

Summary: A normalized coherent symbol of specified finite width has an exact full Toeplitz spectrum and a controlled point-contact error, with distinct fixed-width and fixed-band limits.

Specify a width independently of the retained band

On the auxiliary mode-ray space $\mathbb{CP}^d$, use unit FS measure, $\mathcal H_N=H^0(\mathbb{CP}^d,\mathcal O(N))$ and $D_N=\binom{N+d}{d}$. These are the mode fibres of the projective contact comparison, not additional physical spatial dimensions.

For a specified ray $z$ and integer $p\ge0$, define the positive symbol

$$f_{p,z}(u)=D_p|u^\dagger z|^{2p},\qquad \int f_{p,z}d\nu=1, \qquad T_{N,p}=P_N f_{p,z}P_N. \tag{1}$$

Here $p$ fixes the source width independently of the band $N$. $p=0$ is uniform; increasing $p$ concentrates the source about $z$. It is a prescribed comparison symbol. Its width and source variation would need to follow from the master action in a physical application.

Exact full spectrum

Choose $z=e_0$. Phase integration makes $T_{N,p}$ diagonal on normalized monomials. If their zeroth occupation is $k$, the FS moments are

$$\int\prod_{r=0}^{d}|u_r|^{2m_r}d\nu =\frac{d!\prod_r m_r!}{(d+\sum_r m_r)!}.$$

These follow from iterated Euler beta integrals on the unit simplex. Substitution into (1) gives

$$\boxed{\lambda_k(N,p) =D_p\frac{(N+d)!}{(N+p+d)!}\frac{(p+k)!}{k!}, \qquad m_k=\binom{N-k+d-1}{d-1}.} \tag{2}$$

For $p>0$ these eigenvalues increase with $k$. In particular,

$$\boxed{\lambda_{\rm top}(N,p)=\frac{D_ND_p}{D_{N+p}}, \qquad \frac{\lambda_{N-j}}{\lambda_{\rm top}} =\frac{N(N-1)\cdots(N-j+1)} {(N+p)(N+p-1)\cdots(N+p-j+1)}.} \tag{3}$$

Every eigenvalue is positive at finite $p$, and $\sum_k m_k\lambda_k=D_N$. At $p=0$, (2) gives the identity operator. At $p=N$, (3) reproduces the smooth coherent-density coefficient $D_N^2/D_{2N}$ used in the complete quartic action.

Exact distance from point contact

At fixed $N\ge1$, $p\to\infty$ gives $T_{N,p}\to D_N|z;N\rangle\langle z;N|$. The coherent line is its top eigenvector for every $p$. The sum of all complementary eigenvalues is $D_N-\lambda_{\rm top}$, so no complementary eigenvalue exceeds that deficit. Therefore

$$\boxed{\frac{\|T_{N,p}-D_N|z;N\rangle\langle z;N|\|}{D_N} =\varepsilon_{N,p} =1-\frac{D_p}{D_{N+p}} =1-\prod_{i=1}^d\frac{p+i}{p+N+i}.} \tag{4}$$

Writing each factor as $1-N/(p+N+i)$, the product bound $\prod(1-a_i)\ge1-\sum a_i$ and $1-a_i\le e^{-a_i}$ give

$$\boxed{1-\exp\left[-\frac{dN}{p+N+d}\right] \le\varepsilon_{N,p}\le\frac{dN}{p+N+1}.} \tag{5}$$

The upper bound may exceed one; (4) gives the exact value. These formulas state quantitatively how narrow a specified symbol must be for the point operator to be a useful approximation.

For two bands $N_2>N_1$ with the same $p$,

$$\frac{\lambda_{\rm top}(N_2,p)}{\lambda_{\rm top}(N_1,p)} =\frac{D_{N_2}}{D_{N_1}} \prod_{i=1}^d\frac{p+N_1+i}{p+N_2+i}. \tag{6}$$

Its relative correction to the point-contact ratio is independent of an overall common scale and remains nonzero at every finite $p$. This is an operator comparison; extracting particle frequencies still requires the complete normalized spatial field equation.

Limits with different physical inputs

At fixed $p$ and $N\to\infty$, $\lambda_{\rm top}(N,p)\to D_p=\|f_{p,z}\|_\infty$. It stays bounded while $D_N$ grows. More generally, $\|P_N fP_N\|\le\|f\|_\infty$ for any fixed bounded multiplier. A bounded symbol of fixed amplitude and width cannot produce an unbounded simplex eigenvalue merely by increasing the cutoff.

If instead $p/N\to a>0$, then

$$\frac{\lambda_{\rm top}(N,p)}{D_N} \longrightarrow\left(\frac a{1+a}\right)^d. \tag{7}$$

The leading simplex power can therefore coexist with a shrinking source width, but (7) is asymptotic. It does not make the finite-band coefficient exactly independent of $N$. A prescribed $N$-dependent width must be explained by the same action and normalization as the other field data.

There is also an exact extremal statement. For any positive unit measure $\mu$ on the ray space, define

$$T_\mu=D_N\int|u;N\rangle\langle u;N|\,d\mu(u).$$

Then $\|T_\mu\|\le D_N$. Equality for a normalized vector $v$ requires $|\langle v,u;N\rangle|=1$ for $\mu$-almost every $u$. For $N\ge1$, distinct projective rays give distinct coherent lines; thus equality holds exactly when $\mu$ is a point measure. This statement uses the positive unit-measure convention. Signed symbols, different total source strength or additional operator terms change its hypotheses.

Consistent band contacts addresses the separate requirement that a proposed family arise from one common bare operator or a specified effective reduction.

Verification

coherent_kernel_consistency_checks.py checks (2) by independent beta quadrature and complete CP1 matrices, the exact trace and full error spectrum, both bounds in (5), finite band ratios, and the distinct limits in (3) and (7).