Lognormal spectrum optimization

Finite-sample realization of the Section 4 design objective from Path-Length Distributions, Inverse Geometry, and Designed Neighborhood Spectra (J. R. Landers, April 2026). 15 points in \(\mathbb{R}^2\) are evolved by Adam so each anchor's neighbor-distance distribution matches a target shifted lognormal.

\[ \mathcal{L}(Y)=\frac{1}{n-1}\sum_i \left\|\operatorname{sort}\left(\{\|y_i-y_j\|:j\ne i\}\right)-q^{\mathrm{LN}}\right\|_2^2, \qquad q^{\mathrm{LN}}_k=\exp\left(\mu+\sigma\Phi^{-1}\left(\frac{k-\tfrac12}{n-1}\right)\right). \]

Iteration

Mean spectrum mismatch

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Initial mismatch

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\(\mu\)0.60

\(\sigma\)0.45

\(\eta\)0.050

steps/frame3

Geometry — 15 points in \(\mathbb{R}^2\) with \(k\)-NN edges \((k=3)\)

Loss curve (log scale)

Anchor-wise distance distributions vs target lognormal (dashed)

Anchor 1

Anchor 5

Anchor 9

Anchor 13