Oscillator source marginals

Summary: Spatial oscillator marginals have exact weighted Hockey-Stick singular strengths, while physical bilinear currents require their full coefficient contractions before projection.

A specified spatial projection

In a flat product comparison, let a containing physical spatial sector be $\mathbb R^{d+q}$ and a lower sector be $\mathbb R^d$. Write coordinates $(x,y)$ relative to this inclusion, with $q$ perpendicular directions. Every coordinate is an ordinary macroscopic spatial coordinate of the same kind. This calculation choice does not give the containing particle an inherent division into two spaces. Curved geometry, transport of spinor indices and the actual particle profiles require their own reduction.

At a common width $w>0$, use the normalized functions

$$h_n^w(y)=\left(\frac w\pi\right)^{1/4} \frac{H_n(\sqrt w y)e^{-wy^2/2}}{\sqrt{2^n n!}}, \qquad h_\alpha^w(x)=\prod_i h_{\alpha_i}^w(x_i). \tag{1}$$

Their cumulative band $\mathcal B_{N,d}$ contains $|\alpha|\le N$ and has dimension $D_d(N)=\binom{N+d}{d}$. The normalized oscillator construction fixes this spatial measure and basis.

Splitting occupations proves the exact decomposition

$$\mathcal B_{N,d+q} =\bigoplus_{j=0}^N \mathcal B_{N-j,d}\otimes \operatorname{Sym}^{j}\mathbb C^q, \qquad D_{d+q}(N)=\sum_{j=0}^N \binom{j+q-1}{q-1}D_d(N-j). \tag{2}$$

For $q=1$ this is the ordinary Hockey-Stick identity. It counts the available spatial modes. To obtain a coupling map one must also calculate their amplitudes in the specified source pairing.

Marginal of a linear scalar source

First regard the higher band as a space of linear scalar sources $j$, with arbitrary complex coefficients. Define

$$(Rj)(x)=\int_{\mathbb R^q}j(x,y)\,dy. \tag{3}$$

Integrating the Hermite generating function against $e^{-y^2/2}$ gives

$$c_{2m+1}=0,\qquad c_{2m}=\int h_{2m}^w(y)dy =c_0\sqrt{\frac{\binom{2m}{m}}{4^m}}, \qquad c_0=\sqrt2\,\pi^{1/4}w^{-1/4}. \tag{4}$$

Thus the complete matrix, before squaring any amplitude, is

$$\boxed{R_{\alpha;\alpha',\beta} =\delta_{\alpha\alpha'}\prod_{i=1}^q c_{\beta_i}, \qquad |\alpha'|+|\beta|\le N.} \tag{5}$$

Every even normal occupation contributes. A normal ground-state bra would give a different operator. The constant cylindrical dual in (3) need not belong to the untruncated $L^2$ space. On these finite source bands, $R^\dagger$ is well defined and represents its projection back into the band, consistently with nested reciprocity.

Divide $R$ by $c_0^q$ only to express its entries in ground-integral units. For a shared occupation of degree $k=|\alpha|$, put $M=\lfloor(N-k)/2\rfloor$. Different rows have disjoint support, and

$$\frac{(RR^\dagger)_{\alpha\alpha}}{c_0^{2q}} =\sum_{|m|\le M}\prod_{i=1}^q\frac{\binom{2m_i}{m_i}}{4^{m_i}}. \tag{6}$$

The one-coordinate generating function is $(1-t)^{-1/2}$. Its $q$th power shows that the shell of paired degree $r$ has weight $w_q(r)=(q/2)_r/r!$, where $(a)_r$ is the rising factorial. A final cumulative sum multiplies the generating function by $(1-t)^{-1}$:

$$\boxed{\sum_{r=0}^{M}\binom{r+q/2-1}{r} =\binom{M+q/2}{M}.} \tag{7}$$

Generalized binomials here mean the rising-factorial expressions; all weights are positive. Equation (7) is the squared singular strength of $R/c_0^q$, with multiplicity $\binom{k+d-1}{d-1}$ at shared degree $k$. For even $q$ it is an ordinary integer binomial sum.

For $q=4$, the sum is $1+2+\cdots+(M+1)=\binom{M+2}{2}$. A ten-to-six spatial inclusion has this codimension. Identifying these particular scalar source bands with occupied tau and electron patterns would require their actual field and bundle construction.

Both responses and spectral weights

For the declared scalar pairing $U=g\operatorname{Re}\langle a,KRb\rangle$, real differentials give the reciprocal source forces $gKRb$ and $gR^\dagger K^\dagger a$. The convention is $\delta U=\operatorname{Re}\langle\delta a,F_a\rangle +\operatorname{Re}\langle\delta b,F_b\rangle$; a complex Hamiltonian using Wirtinger derivatives must retain its corresponding factor of two. Field-dependent sources or geometry require additional derivatives.

In a positive same-field response with diagonal stiffness $\ell_{k+2r}>0$, the particular matrix $RL^{-1}R^\dagger$ has entries

$$G_k=c_0^{2q}\sum_{r=0}^{\lfloor(N-k)/2\rfloor} \frac{\binom{r+q/2-1}{r}}{\ell_{k+2r}}. \tag{8}$$

Only constant stiffness on these channels permits the denominator to leave the sum. The full common field response also retains direct terms and the actual source maps. Its retarded version needs the physical frequencies and boundary conditions. Canonical normalization acts on the field's kinetic metric; dividing by $c_0^q$ does not perform that step.

The $\beta=0$ columns prove that $R$ has full row rank $D_d(N)$ and nullity $D_{d+q}(N)-D_d(N)$. The map admits both visible and invisible source combinations at every codimension. There is no special depth-five zero in (7).

Phases can change a source cancellation. In one normal direction, $(h_0-\sqrt2e^{-i\varphi}h_2)/\sqrt3$ has squared integral $c_0^2(4/3)\sin^2(\varphi/2)$. Free oscillator evolution changes $\varphi$ at the gap $4w$. This is source beating, not a reciprocal encounter calculation or a conserved electric-neutral sector.

Construct a bilinear current before using the sum

For an actual spatial wave, let each $c_{\alpha\beta}$ be a spinor coefficient and write

$$\Psi(x,y)=\sum_{|\alpha|+|\beta|\le N} c_{\alpha\beta}h_\alpha^w(x)h_\beta^w(y), \qquad j=\Psi^\dagger\Gamma\Psi, \tag{9}$$

with a specified constant Hermitian vertex $\Gamma$. The positive density has $\Gamma=I$; the Dirac scalar has $\Gamma=\beta_{\rm D}$. Spatially varying vertices and frame transport must remain inside the integral. Orthonormality in the normal coordinates gives the exact result

$$\boxed{(Rj)(x)=\sum_{\alpha,\alpha',\beta} c_{\alpha'\beta}^{\dagger}\Gamma c_{\alpha\beta} h_{\alpha'}^w(x)h_\alpha^w(x).} \tag{10}$$

The sums obey both cumulative bounds. In particular, the normal occupation is contracted with the same normal occupation, rather than multiplied by the one-field integral $c_\beta$ in (4). For $\Gamma=I$ this is the partial trace of the full mode density matrix. It keeps shared-coordinate coherence and all spinor components.

There is no contradiction with (5). The bilinear in (9) is a Gaussian times a polynomial of degree at most $2N$, so it lies in the scalar source band $\mathcal B_{2N,d+q}$ at width $2w$. First forming its coefficients there, and then applying (5) at that width and cutoff, reproduces (10). Applying $R$ to $\Psi$ and squaring would instead introduce interference between orthogonal normal modes.

For a normalized separated positive density $|\psi(x)|^2|f(y)|^2$, (10) simply gives $|\psi(x)|^2$ because $\int|f|^2dy=1$. A normal wave whose linear integral vanishes still has a positive unit density integral. Thus the source singular strengths are not universal multipliers of normalized particle charge or contact strength.

For the declared density interaction $U=g\int \rho_a(x)(R\rho_b)(x)dx$, variation gives $V_a=gR\rho_b$ and $V_b(x,y)=g\rho_a(x)$, followed by the respective band compressions. Both reciprocal responses follow from the same energy. This particular cylindrical response does not by itself transfer normal oscillator occupation: its higher operator is block diagonal in $\beta$. More general transported vertices and spatial kernels can change that property.

Covariance and verification

The isotropic oscillator band is preserved by common rotations of the physical coordinates. Rotate the containing field, lower inclusion and normal frame together in (3); a change of integration variables gives the same marginal in the rotated lower coordinates. The result does not assign a preferred physical split. Holding the inclusion fixed while rotating only one pattern describes a different preparation.

spatial_oscillator_projection_checks.py verifies exact occupation sums, independent Hermite quadrature, full finite singular spectra, complex reciprocal variation, chain composition, rotated inclusions, nonuniform response weights and the independent bilinear-current construction (9)–(10).

The same paired-occupation weights also appear in spatial point contact.

Spectral weights in oscillator response derives the cutoff convergence criterion for (8), with exact affine-spectrum limits and the remaining shared-mode dependence in codimension four. The criterion retains the source map and response spectrum together.