= Solution
A <Peierls transition> is a lattice distortion coupled to a gap in an initially partly filled quasi-one-dimensional electronic band. A modulation at $2k_F$ couples the opposite Fermi points. At half filling with one orbital per atom, it doubles the cell and alternates strong and weak bonds; the <dimerized tight-binding chain> shows explicitly how the gap opens at the Fermi energy. In the ideal weak-coupling zero-temperature model, the <Peierls energy gain> has a negative term proportional to $-\delta^2\ln(t/|\delta|)/t$, which can outweigh a positive quadratic elastic cost.
Observation should connect electronic and structural evidence: cooling through the transition can reduce conductivity as the gap develops, while diffraction satellites identify the ordering wavevector. A softened phonon near $2k_F$ tests the precursor electron-lattice coupling. In <inelastic neutron scattering of TTF-TCNQ>, measurements from $60$ to $300\,\mathrm K$ found an increasing <Kohn anomaly> near $2k_F$ as the $54\,\mathrm K$ transition was approached. Those particular measurements were above the transition, so they probe the precursor dynamics rather than a static low-temperature diffraction pattern.<Phonon dispersion and Kohn anomaly in TTF-TCNQ by Gen Shirane and others>[Abstract: temperature interval and wavevector-dependent anomaly]{ref}
A real finite-temperature transition also depends on interchain coupling and fluctuations. The ideal one-dimensional energy argument by itself does not prove finite-temperature long-range order.
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