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The derivative of transverse betatron tune $Q$ with respect to relative momentum error $\delta$. For a thin-lens FODO cell with fixed quadrupole fields, natural cell chromaticity is $-\tan (\mu /2)/\pi$ on the stable phase-advance branch. Zero corrected chromaticity requires additional momentum-dependent focusing.

Let $D(d)=\left(\begin{array}{cc}1& d\\ 0& 1\end{array}\right)$ be the drift beam transfer matrix. Start immediately before the first lens and end immediately before its counterpart in the next FODO cell. Matrices act on column vectors from right to left. With $A=1/f_1$ and $B=1/f_2$, where both focal lengths are signed in the chosen plane,The two drifts account for both gaps. Direct multiplication gives $\det M=1$ andFor the equal-strength focusing-defocusing case, $f_1=f>0$ and $f_2=-f$:Choosing a different start within the cell gives a conjugate matrix, with the same trace and eigenvalues. If $f_2$ is specified as a positive magnitude of a defocusing lens, its signed value in these formulas is $-f_2$.

For a real uncoupled, lossless transverse map, $\det M=1$. Bounded nondegenerate oscillatory motion requires complex-conjugate eigenvalues $e^{\pm i\mu }$, equivalentlyIts Courant-Snyder parameters giveThe positive quadratic form ${\gamma }_T{x}^2+2{\alpha }_{T}x{x}^{\prime }+{\beta }_T{x}^{\prime 2}$ is invariant, so repeating the cell cannot produce secular growth. At trace $\pm 2$ a nontrivial Jordan block instead produces linear growth; only the exceptional maps $M=\pm I$ remain bounded at those exact boundaries. The criterion must hold in both transverse planes. For the symmetric FODO cell, both traces are $2-d^2/f^2$, giving

A suitable accelerator-science example is strong focusing. The apparent technical contradiction is that increasing a quadrupole magnet's focusing strength improves confinement in one transverse plane while worsening it in the other. Abandoning that magnet entirely loses its strong focusing action.

An alternating-gradient lattice separates the two actions in space: a focusing quadrupole in one plane is followed, after a drift, by one with the opposite sign, and the pattern repeats. In the other plane their roles reverse. The symmetric thin-lens FODO cell has the same trace $2-d^2/f^2$ in both planes, so both are stable for $0<d/|f|<2$, despite each individual quadrupole defocusing one plane. The noncommuting sequence of drift and lens maps is what makes net confinement possible.

In a TRIZ interpretation, this resolves the contradiction by spatial separation and periodic alternation of opposing functions, rather than demanding that one static element satisfy both. The useful result is smaller beam envelopes and magnet apertures for a suitable lattice. This is a retrospective design analysis; it does not claim that the historical inventors actually followed TRIZ. The matrix calculation supplies a test of the proposed resolution rather than just attaching a slogan to an invention.