Consistent band contacts

Summary: A fixed bare kernel must have compatible cumulative compressions, while coherent projective contact changes under the standard band inclusion and effective kernels obey a different Schur consistency law.

What a common bare operator requires

At fixed spatial depth $d$, let $\mathcal B_N=\bigoplus_{k=0}^N\operatorname{Sym}^k\mathbb C^d$ with normalized oscillator occupations $|\alpha\rangle$, $|\alpha|\le N$. Let $I_N:\mathcal B_N\hookrightarrow\mathcal B_{N+1}$ preserve each existing occupation. These are mode labels on the same specified spatial sector, not additional physical dimensions.

If a fixed bare operator or quadratic form $A$ is defined on the union of these spaces, its compressions $A_N=P_NAP_N$ satisfy

$$\boxed{I_N^\dagger A_{N+1}I_N=A_N.} \tag{1}$$

This follows by testing the two forms on any pair of vectors already in $\mathcal B_N$. Conversely, a Hermitian family satisfying (1) defines a unique form on finite occupation sums. Boundedness, closability and an infinite-space operator domain require additional control; finite consistency alone does not establish them.

The spatial point form at a fixed position and width satisfies (1): its matrix entries $\overline{h_\alpha(X)}h_\beta(X)$ remain the same when more modes are admitted. Its growing nonzero eigenvalue is consistent with an unchanged lower matrix block.

Test the normalized homogenized coherent family

Identify $\mathcal B_N$ with $\operatorname{Sym}^N(\mathbb C\oplus\mathbb C^d)$ using normalized occupation states. For a fixed unit vector $z=(z_0,z_1,\ldots,z_d)$, the coherent coefficients are

$$(v_N(z))_\alpha= \sqrt{\frac{N!}{(N-k)!\,\alpha!}}\, z_0^{N-k}z_1^{\alpha_1}\cdots z_d^{\alpha_d}, \qquad k=|\alpha|\le N. \tag{2}$$

The standard cumulative inclusion adds one unused, or zeroth, occupation. Direct comparison of (2) at two consecutive bands gives

$$I_N^\dagger v_{N+1}(z)=z_0 C_Nv_N(z), \qquad (C_N)_\alpha=\sqrt{\frac{N+1}{N+1-k}}. \tag{3}$$

For $\Lambda_N(z)=D_N|v_N(z)\rangle\langle v_N(z)|$ and $D_N=\binom{N+d}{d}$, the resulting compression is

$$\boxed{I_N^\dagger\Lambda_{N+1}(z)I_N =D_{N+1}|z_0|^2C_N|v_N(z)\rangle\langle v_N(z)|C_N.} \tag{4}$$

It is generally different from $\Lambda_N(z)$. At $z=e_0$, the same ground matrix entry is $D_{N+1}$ on the left and $D_N$ on the right. For $N\ge1$ and a ray with nonzero $z_0$ and a nonzero transverse coordinate, $C_N$ changes the relative ground and degree-one amplitudes, so even the compressed rank-one direction changes. A scalar rescaling alone cannot repair that directional difference. At $z_0=0$, the left side vanishes completely.

Consequently, the projective point-contact family is not the sequence of compressions of one fixed linear form under this fixed-ray, normalized cumulative identification. Each individual projective contact theorem remains exact. The family uses a different homogeneous line bundle and evaluation map at each $N$.

This is a test of a particular proposed operator bridge. If $N$ labels different physical branches rather than a common cutoff, if the master operator depends on the full field, or if the spatial embedding and kinetic metric depend on $N$, their actual relations must replace (1). Such data cannot be inferred from equality of dimensions alone.

Effective operators have Schur consistency

Compression and effective reduction are different operations. Let $D(\zeta)=A-\zeta I$ in a finite complete mode space, and split retained modes $P_N$ from the complement $Q_N$. When the indicated inverses exist, eliminating the complement gives

$$F_N(\zeta)=D_{NN}-D_{NQ}D_{QQ}^{-1}D_{QN}, \qquad \boxed{F_N(\zeta)^{-1}=P_ND(\zeta)^{-1}P_N.} \tag{5}$$

The identity follows by solving the complementary block equation and substituting it into the retained equation. Hence it is the inverse effective operator that has the compression relation

$$I_N^\dagger F_{N+1}(\zeta)^{-1}I_N=F_N(\zeta)^{-1}. \tag{6}$$

Equivalently, taking a Schur complement through an intermediate band gives the same result as direct elimination. For a positive static Hessian, the eliminated block also fixes the sign of the correction: $F_N=D_{NN}-B D_{QQ}^{-1}B^\dagger\le D_{NN}$.

Thus failure of (1) is not by itself a failure of an effective master reduction. A proposed effective family must instead supply the complementary modes, its frequency dependence and the consistency in (5)–(6). These are modes of the same field $\Psi_\infty$; no mediator particles have been introduced. The common field response also tracks its kinetic normalization and retarded boundary conditions.

A controlled local limit of the coherent evaluation map

There is a useful mathematical connection to an ordinary bosonic coherent coefficient sequence. In the affine chart set

$$z=\frac{(1,a/\sqrt N)}{\sqrt{1+|a|^2/N}},\qquad a\in\mathbb C^d.$$

For fixed occupation $\alpha$ of degree $k$, (2) becomes

$$(v_N)_\alpha= \sqrt{\frac{N(N-1)\cdots(N-k+1)}{N^k}} \left(1+\frac{|a|^2}{N}\right)^{-N/2} \frac{a^\alpha}{\sqrt{\alpha!}} \longrightarrow e^{-|a|^2/2}\frac{a^\alpha}{\sqrt{\alpha!}}. \tag{7}$$

The measure must transform as well. Unit-volume FS measure in this chart is

$$d\nu_{\rm FS}= \frac{d!}{\pi^dN^d}\left(1+\frac{|a|^2}{N}\right)^{-d-1} d^{2d}a.$$

Since $D_Nd!/N^d=\prod_{j=1}^d(1+j/N)$, its contact measure satisfies

$$\boxed{D_Nd\nu_{\rm FS}\longrightarrow\pi^{-d}d^{2d}a} \tag{8}$$

at fixed $a$. The coherent operator density then has a finite limit per unit $a$ volume. Equations (7)–(8) are a local large-band limit, not an exact finite-band identification. The coordinates $a$ describe mode amplitudes; they do not replace any physical spatial dimensions. Changing the measure in an action also transforms its delta functions and source coefficients. A normalization factor cannot simply be discarded while holding those other ingredients fixed.

Finite ray resolution separately quantifies how a specified smooth symbol approaches point contact at fixed $N$.

Verification

coherent_kernel_consistency_checks.py checks (3)–(4) in every retained occupation, fixed spatial point compression, direct and staged complex Schur elimination, compressed resolvents and the coefficient/measure limits in (7)–(8).