Spatial oscillator point contact

Summary: A point contact on actual flat spatial oscillator waves has a position-dependent reproducing strength, whose central value is a weighted Hockey-Stick count rather than the projective simplex dimension.

The physical spatial evaluation operator

Fix ordinary spatial coordinates $x\in\mathbb R^d$, Lebesgue measure, oscillator width $w>0$, and the cumulative normalized Hermite band $\mathcal B_{N,d}$ of oscillator modes. Let $P_N$ be its orthogonal projector. Point evaluation at $X$ defines a finite positive quadratic form

$$\langle f,A_N(X)f\rangle=|f(X)|^2, \qquad A_N(X)=P_N\delta_XP_N. \tag{1}$$

No untruncated bounded delta multiplication operator is assumed. In the real normalized Hermite basis, the finite matrix is $A_{\alpha\beta}=h_\alpha^w(X)h_\beta^w(X)$. It has rank one and its sole nonzero eigenvalue is

$$\boxed{K_{N,d}^w(X,X)=\sum_{|\alpha|\le N}|h_\alpha^w(X)|^2.} \tag{2}$$

This strength has physical units of inverse spatial volume and depends on position and width. Integrating (2) over $X$ gives the dimension $D_d(N)=\binom{N+d}{d}$, because each basis mode is normalized. The dimension is therefore the integrated trace density, not its value at an arbitrary spatial point.

Mehler's formula and the central count

Apply Mehler's formula to each spatial coordinate, including the Gaussian factors in the normalized modes. The shell generating function is

$$\sum_{\alpha\ge0}|h_\alpha^w(X)|^2t^{|\alpha|} =\left(\frac w\pi\right)^{d/2}(1-t^2)^{-d/2} \exp\left[-w|X|^2\frac{1-t}{1+t}\right]. \tag{3}$$

Multiplication by $(1-t)^{-1}$ sums all shells through $N$:

$$\boxed{\sum_{N=0}^{\infty}K_{N,d}^w(X,X)t^N =\frac{(w/\pi)^{d/2}}{(1-t)(1-t^2)^{d/2}} \exp\left[-w|X|^2\frac{1-t}{1+t}\right],\quad |t|<1.} \tag{4}$$

This depends on $X$ only through $|X|$, as required by physical rotation covariance of the complete isotropic band. The evaluation vector itself retains all spatial mode amplitudes.

At the oscillator center, odd occupations vanish. For $M=\lfloor N/2\rfloor$, the coefficient in (4) becomes

$$\boxed{K_{N,d}^w(0,0)=\left(\frac w\pi\right)^{d/2} \sum_{r=0}^{M}\frac{(d/2)_r}{r!} =\left(\frac w\pi\right)^{d/2}\binom{M+d/2}{M}.} \tag{5}$$

This is the weighted Hockey-Stick identity derived in oscillator source marginals. At fixed $d,w$ and large $N$, the gamma-function ratio gives

$$K_{N,d}^w(0,0)\sim \frac{(w/\pi)^{d/2}}{\Gamma(d/2+1)} \left(\frac N2\right)^{d/2}, \qquad D_d(N)\sim\frac{N^d}{d!}. \tag{6}$$

For example, at $d=6$ and the same width,

$$\frac{K_{34,6}^w(0,0)}{K_{12,6}^w(0,0)} =\frac{\binom{20}{17}}{\binom{9}{6}} =\frac{95}{7}\simeq13.5714, \qquad \frac{D_6(34)}{D_6(12)}\simeq206.7647. \tag{7}$$

These are operator comparisons, not predicted particle mass ratios. An independently derived width $w_N$ changes (5); choosing it to fit a desired power is additional structure. Even restoring a leading power would not prove the exact finite-band simplex law.

Fourier duality with the normal marginal

For $q$ normal coordinates, define the restriction $(Ef)(x)=f(x,0)$ and the scaled unitary Fourier transform

$$(\mathcal F_w f)(y)=\left(\frac w{2\pi}\right)^{q/2} \int e^{-iw y\cdot y'}f(y')dy'.$$

Direct Fourier transformation of the Hermite generating function gives $\mathcal F_w h_\beta^w=(-i)^{|\beta|}h_\beta^w$. At zero normal position,

$$h_{2m}^w(0)=\left(\frac w\pi\right)^{1/4} (-1)^m\sqrt{\frac{\binom{2m}{m}}{4^m}}, \qquad h_{2m+1}^w(0)=0.$$

Evaluation of the Fourier integral at zero therefore proves, on the finite band,

$$\boxed{R=\left(\frac{2\pi}{w}\right)^{q/2}E\mathcal F_w.} \tag{8}$$

The Fourier transform preserves total oscillator degree. Consequently, the restriction and marginal have the same singular spectrum after their displayed dimensional factors are removed. Their even-mode phases differ. Equality of singular strengths does not make them the same response to a fixed complex spatial preparation.

Relation to projective ray contact

The projective contact calculation uses $H^0(\mathbb{CP}^d,\mathcal O(N))$ with a homogeneous unit-volume Fubini–Study measure. Its point contact has constant strength $D_d(N)$. The point in that construction is a mode ray; the point $X$ in (1) is a position in an actual flat spatial chart. An equality of finite dimensions does not identify the two evaluation operators or measures.

There is an elementary vector test. Under the usual normalized homogenization of occupations, the spatial vector at $X=0$ has a nonzero ground coefficient, zero degree-one coefficients and nonzero degree-two coefficients whenever $N\ge2$. A Veronese coherent vector $z^{\otimes N}$ with nonzero ground coefficient and vanishing all degree-one coefficients must have $z_1=\cdots=z_d=0$, which also eliminates every degree-two coefficient. Thus central spatial evaluation is not a projective coherent ray in that identification.

The unfinished coordinate-to-ray kernel bridge must specify the actual transform, measure, field-dependent data and kinetic normalization. Equations (1)–(8) give a fully specified spatial comparison against which such a bridge can be tested. They do not replace the proposed compact physical geometry of an IDWT particle.

Verification

spatial_oscillator_projection_checks.py checks direct Hermite evaluation against (3)–(5), integrates the trace density, verifies physical rotation covariance and Fourier phases, and constructs the full marginal and restriction matrices independently.