A Synthetic Angular-Scattering Comparison for Two Sheet-Response Models

A finite scattering table can distinguish two declared models without identifying which physical structure produced it. This synthetic example compares a wavevector-dependent sheet response with a constant-admittance local-sheet comparator. They agree at normal incidence by construction and differ at three sampled oblique angles.

The distinction is deliberately narrow. The coefficients were selected during AI-assisted work solely to define the example; they were not fitted to measurements or chosen from a material source. The result is a dimensionless, single-frequency calculation, not evidence that a particular sheet exists or that angular scattering uniquely reveals spatial nonlocality.

The declared comparison

The model uses identical vacuum ports, transverse-electric plane waves, and the $e^{-i\omega t}$ convention at one fixed frequency $\omega_0$. Write $\kappa=k_{\parallel}/k_0=\sin\theta$, restrict the aperture to $|\kappa|\leq\sqrt{3}/2$, and normalise the TE port admittance as $y_p=\cos\theta=\sqrt{1-\kappa^2}$.

The wavevector-dependent response is $y_s^{\mathrm{NL}}(\kappa)=\frac15+\frac{3}{10}\frac{\kappa^2}{1+\kappa^2}+\frac{i}{10}$. Its comparator is the angle-independent local sheet $y_s^{\mathrm{L}}=\frac15+\frac{i}{10}$. The two have the same response at $\kappa=0$; the real part of the first response increases with $\kappa^2$ over the declared aperture.

For unit incident amplitude, tangential-field continuity gives $t=1+r$. The sheet-current boundary condition is $y_p(1-r)-y_pt=y_st$. Solving yields $r=-\frac{y_s}{2y_p+y_s}$ and $t=\frac{2y_p}{2y_p+y_s}$. With $y_s=g+ib$, let $D=(2y_p+g)^2+b^2$. Then reflected power is $R=\frac{g^2+b^2}{D}$, transmitted power is $T=\frac{4y_p^2}{D}$, and absorbed fraction is $A=\frac{4y_pg}{D}$. Since $g>0$ in both declared models, the absorption is nonnegative for these propagating modes. For the identical ports and this scalar response, the two-port scattering matrix is symmetric and its symmetric/antisymmetric channel eigenvalues satisfy the stated passivity check. This setup-specific symmetry is not a general same-wavevector reciprocity rule for arbitrary spatially dispersive interfaces.

Four sampled angles

The values distinguish this chosen response from this constant-admittance comparator at the three listed oblique angles. They do not establish a unique response: infinitely many functions can agree at four samples, and conventional multilayers or patterned resonators could reproduce the same finite table. Absorption changing with angle is not by itself a nonlocality test, since the local comparator also changes with incidence angle through $y_p$.

What a physical test would still need

A physical discriminator would need a named material and geometry, a frequency-dependent constitutive law, appropriate reciprocal, passive and causal conditions, calibration of the ordinary structures allowed as alternatives, uncertainty limits, complex measurements from both sides, and held-out angle or frequency conditions fixed in advance. This example supplies none of those measurements or calibrations. It does not demonstrate a material, establish broadband causality, prove a theorem about all nonlocal interfaces, claim novelty, or support a hidden-layer interpretation.

The bounded derivation and reproduction instructions are recorded in version 0.1.1 of the public Research Artifact Ledger entry. Its coefficients and finite comparison were AI-selected to define this toy example. The reported independent check was performed by an AI reviewer; it was not human specialist review, peer review, a proof-assistant check, or physical validation. No systematic prior-art or novelty search was performed. The Ledger artifact has no open licence; this article links to it rather than copying its reproduction script.

AI assisted with model formulation, derivation, calculation and drafting; a separate AI reviewer checked the equations and finite results. The author remains responsible for checking the source and claims. The accompanying illustration is AI-generated conceptual artwork, not a device depiction, experimental result or data. No material, experimental dataset or measurement was analysed for this article.