How spatial preparation sets the spectrum

5. Spatial preparation and kernel-mass heterogeneity

5.1 A conditional Gaussian-exchange realization

Fix distinct sites $x_i\in\mathbb R^2$. A free-space heat mediator with diffusivity $D$ has kernel $H(\tau,x)=(4\pi D\tau)^{-1}\exp[-|x|^2/(4D\tau)]$. At preparation time $\tau$, sample its normalized influence:

$$G_{ij}=H(\tau,x_i-x_j)/H(\tau,0)=e^{-|x_i-x_j|^2/(2\sigma^2)},\qquad \sigma^2=2D\tau.$$

Hold the coefficients fixed during response. With $s=G\mathbf1$ and $L=\operatorname{diag}(s)-G$, diagonal self terms cancel. This is the complete Gaussian graph in the spatial analysis. Setting $\sigma$ to the median pair distance chooses $D\tau$ rather than deriving a universal diffusion time.

For complex amplitudes, set $F=\kappa z^*Lz/2$ with $\kappa>0$ and scalar dissipative and reactive mobilities $\gamma>0$, $\alpha>0$. The real Hessian commutes with the complex structure, giving

$$\dot z=-(\gamma+i\alpha)\kappa Lz,\qquad K_d=\gamma\kappa L,\qquad C=\alpha\kappa L.\tag{27}$$

This realizes both CSCF faces under the declared assumptions, with $h(\lambda)=\alpha\lambda/\gamma$ on the retained spectrum and a bounded nondecreasing extension. The limit $\alpha=0$ is purely dissipative. Prepare $\sum_i z_i=0$ to remove the uniform stationary sector. If amplitudes have units A, $\kappa$ has energy/A² and the mobilities have A²/(energy × second). Amplitude decay is $\gamma\kappa\lambda$, phase frequency is $\alpha\kappa\lambda$, and energy decay is twice amplitude decay. Eliminating a live mediator requires its memory law; freezing the sampled network does not perform that elimination.

5.2 A finite degree-to-spectrum theorem

Theorem 5.1 (Gaussian kernel mass). Order $s_{(1)}\le\cdots\le s_{(N)}$ and the Laplacian eigenvalues $0=\lambda_1<\lambda_2\le\cdots\le\lambda_N$. Then

$$\lambda_i\le s_{(i)},\qquad\sum_{i=1}^N(s_{(i)}-\lambda_i)=N.$$

The equal-weight measures on $\lambda_2/N,\ldots,\lambda_N/N$ and $s_{(2)}/N,\ldots,s_{(N)}/N$ have Wasserstein distance

$$W_1=\frac{N-s_{(1)}}{N(N-1)}\le\frac1{N-1}.\tag{28}$$

Proof. The Fourier representation makes $G$ positive semidefinite: its quadratic form integrates $|\sum_jc_je^{ik\cdot x_j}|^2$ against a nonnegative Gaussian density. Thus $L\le\operatorname{diag}(s)$ and min-max gives the ordered inequalities. Since $\operatorname{tr}G=N$, trace subtraction gives the sum. Positive off-diagonal weights make the kernel exactly the constants. Removing the zero eigenvalue subtracts $s_{(1)}$ from the deficit. Sorted coupling is optimal for one-dimensional $W_1$.

Kernel mass at a site controls the retained bulk. To control sharp windows, pair $a_i=\lambda_{i+1}$ with $b_i=s_{(i+1)}$ and set $e_i=|\log a_i-\log b_i|$. A pair's log-ratio error is at most $e_i+e_j$. Membership changes only if the predicted log-ratio lies within that distance of an endpoint. Counting boundary pairs bounds each incidence error. A small global distance alone controls neither discontinuous windows nor nearly zero modes. Saved-centroid calculations verify these finite inequalities for retained spatial views.

5.3 A preparation law generates a ratio distribution

Prepare iid sites $X\sim N(0,a^2I_2)$ by harmonic confinement and quench them before measuring. Gaussian convolution gives

$$q(x)=\mathbb E e^{-|x-X|^2/(2\sigma^2)} =\frac{\sigma^2}{\sigma^2+a^2}e^{-|x|^2/[2(\sigma^2+a^2)]}.$$

The population median pair distance has $\sigma^2=4a^2\log2$. Since $|X|^2/(2a^2)$ is unit exponential,

$$\log q(X)=\text{constant}-\beta E,\qquad \beta=(1+4\log2)^{-1},\quad E\sim\operatorname{Exp}(1).\tag{29}$$

Independent differences have density $e^{-|d|/\beta}/(2\beta)$. Exponentiating gives

$$f_R(r)=\frac{1}{2\beta r}e^{-|\log r|/\beta},\qquad r>0.\tag{30}$$

Integrate this density over the original windows to predict limiting incidences without fitting an eigenvalue density.

The transfer from preparation to spectrum is quantitative. At fixed bandwidth, conditional Hoeffding bounds and a union bound give, with probability at least $1-\delta$,

$$\max_i|s_i/N-q(X_i)|\le N^{-1}+\sqrt{\log(2N/\delta)/[2(N-1)]}.$$

The law of large numbers and Theorem 5.1 transfer the degree-law limit to $L/N$. Pair U-statistic consistency transfers the empirical distance median because the population density there is continuous and positive. The kernel derivative with respect to $\log\sigma$ is bounded by $2/e$, so using the empirical bandwidth changes all normalized degrees by a vanishing amount. Since $q(X)>0$ almost surely, continuous mapping gives the log-spectrum and product weak convergence gives the pair law. Removed diagonal mass vanishes and endpoints have zero limiting mass. Finite eigenvalues and overlapping pairs need not be independent.

The isotropic curve matches H30's 3% slot ordering while differing in amplitudes. That is a mechanism for the ordering under this preparation, not an identification of Gaussian preparation in hydrogen or photographs. With $X\sim N(0,C)$,

$$q(x)=\det(I+C/\sigma^2)^{-1/2}\exp[-\tfrac12x^*(\sigma^2I+C)^{-1}x].$$

If $c_j$ are covariance eigenvalues and $b_j=c_j/(\sigma^2+c_j)$, then $\log q(X)=\text{constant}-\frac12\sum_j b_jZ_j^2$. Its variance is $\frac12\sum_j b_j^2$. The independent difference characteristic function is $\prod_j(1+b_j^2t^2)^{-1/2}$. Equal $b_j$ in two dimensions give the Laplace law. The three frozen 512-site examples use covariance pairs $(1,1)$, $(4,1/4)$ and $(16,1/16)$. They change amplitudes while retaining high rank agreement, as reported in Table 3. They are synthetic preparation checks.

5.4 Extraction uncertainty and source-image scope

Proposition 5.2 (Matched-site displacement). If each labelled site moves by at most $\epsilon$ and the original median bandwidth satisfies $\sigma>2\epsilon$, every edge changes by at most

$$\delta=\min\{1,2\epsilon e^{-1/2}/\sigma+(2/e)|\log(1-2\epsilon/\sigma)|\}.$$

Consequently $\|\Delta L\|\le 2(N-1)\delta$, which also bounds the change in each ordered eigenvalue.

Proof. Each pair distance changes by at most $2\epsilon$, as does the median by matched order statistics. The radial derivative at fixed bandwidth is at most $e^{-1/2}/\sigma$ and the derivative in $\log\sigma$ is at most $2/e$. Split the changes and use $\sigma'\ge\sigma-2\epsilon$. A signed edge-Laplacian perturbation has norm at most twice its largest absolute row-weight sum by its quadratic form. Apply Weyl's inequality.

If $k$ sites are deleted at fixed bandwidth with surviving block $S$, then

$$L_{SS}=L_{\rm deleted}+\operatorname{diag}\left(\sum_{j\notin S}W_{ij}\right), \qquad 0\le L_{SS}-L_{\rm deleted}\le kI.$$

Interlacing and Weyl give $\lambda_j-k\le\mu_j\le\lambda_{j+k}$. Re-estimating bandwidth adds at most $2(N-k-1)(2/e)|\log(\sigma'/\sigma)|$ to the operator bound. Keep the changed dimension and pair exposure. A negative lower endpoint supplies no positive-ratio guarantee.

Nine nominated displacement checks and three deletion checks pass for saved embryo, galaxy and leaf coordinates. Physical robustness additionally needs independent bounds on coordinate error and site disagreement. The matched-site bound does not cover births, mergers or unbounded component changes induced by thresholding.

The established extraction uses EXIF orientation, at most 512 pixels on the longest side without upsampling, fixed stored-sRGB luminance, strict dark thresholds, eight-connected components, inclusive area limits and a two-pixel separation rule. Saved centroids support graph-level replay. The photographs are absent, six stable acquisition mappings remain unresolved, and redistribution permissions were not supplied. Pixel-level replay and image formation are separate. The small dark-embryo graph has different kernel mass and weaker hydrogen correlation. A uniform ring has a circulant graph whose Fourier spectrum concentrates near its common degree, making the nominated away-from-one incidences vanish with size. Preparation can therefore generate recurrence or its absence under the same kernel. Neither result establishes image origin, authenticity or semantic content.