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A monocyclic, approximately planar, continuously conjugated pi system with $4n+2$ pi electrons has a closed-shell aromatic configuration in the elementary Hückel model, where $n$ is a nonnegative integer. Fused systems and heterocycles require a specified orbital and resonance description rather than counting every drawn double bond in each polygon.

Rewrite the real-energy probability as $T=[1+U_0^2{\sin }^2(\zeta a)/(4E(E+U_0)){]}^{-1}$. Thus $\boxed{T_{\max }=1}$ at $\zeta a=j\pi$, namely $E_j={\hslash }^2{j}^2{\pi }^2/(2m{a}^2)-U_0\ge 0$, with a strictly positive resonance energy in the ordinary scattering problem. Between resonances it is less than one and its oscillations approach one as energy increases. Generically $T(0^{+})=0$; a well tuned to a zero-energy resonance instead has $T(0^{+})=1$. The original transmission plot in S1 illustrates these maxima for a chosen well, not a universal specified depth.

On the physical sheet the bound-state amplitude poles lie at $k=i{\kappa }_j$, ${\kappa }_j>0$, and map to real negative $E_j=-{\hslash }^2{\kappa }_j^2/(2m)$. The square-root map has two sheets meeting at the continuum threshold. Analytic continuation can also have virtual-state poles on the negative imaginary $k$ axis and resonance poles in the lower half-plane; these are not normalizable bound states. In the energy plane the resonances belong to the nonphysical sheet, rather than additional negative-axis bound levels.

For the algebraically continued probability, include the mirror poles contributed by $D(-k)$ as explained in part d; the two $k$ signs map to the same numerical energy but need not impose the same boundary condition. A sketch must label the sheet or the amplitude being plotted, rather than treating every singularity as a bound state.

A coordinate transformation considered solely as a relabeling of events has no independent observable outcome: changing labels changes neither a detector record nor a clock comparison. But the Lorentz transformations also express substantive relations between operationally realized inertial coordinates. With clocks and rulers specified, their velocity-dependent time dilation, length relations and velocity composition make testable claims about how material systems and electromagnetic signals behave.

The distinction matters for synchronization. One-way coordinate light speed depends on how distant clocks are synchronized; a conventional alteration changes the displayed transformation law while leaving two-way timing observations unchanged. It would therefore be incorrect to regard every coefficient in an unspecified coordinate formula as a separately measurable physical fact. Testable content comes from the transformation together with the protocol and dynamical behavior of the systems implementing the coordinates.

Tests can compare clock proper times, decay lifetimes or resonance behavior across relative motion and orientation. Agreement supports Lorentz-invariant kinematics and the associated matter dynamics over the range tested; it does not prove exact invariance at every possible scale. A rival theory with an undetectable preferred frame can reproduce the same observations by changing its explanatory commitments. Empirical equivalence of that rival does not make the observable clock relations untestable.

Thus the answer is qualified: transformations as passive mathematical changes are not experiments, while Lorentzian relations among specified measuring procedures have substantial empirical consequences. Separating these roles avoids both treating coordinate convention as a measured force and declaring the entire relativistic structure immune to evidence.

Acta Crystallographica C46, pp. 1725–1727. Structural planarity and nonuniform bond geometry; aromaticity is discussed as a resonance model rather than a directly measured binary label.