What the one-dimensional potentials examples establish

The three computations. Tunnelling through a rectangular barrier: an exactly solvable scaling and its thick-barrier comparison treats the rectangular barrier of Rectangular barrier: scattering solution, flux balance and transmission at the exactly solvable scaling $U=2E$ with $\kappa a=\ln 10$: it obtains $T=400\mathrm{/}10201$ and $R=9801\mathrm{/}10201$ as exact rationals, brackets them by $3921\mathrm{/}100000<T<3922\mathrm{/}100000$, compares the exact value with the thick-barrier value $1\mathrm{/}25$ (the difference is exactly $-201\mathrm{/}255025$) and verifies the theorem's error bound $201\mathrm{/}125000=0.001608$; the computation is a finite check of the model formulas, including the positivity of transmission through a classically forbidden barrier. Finite square well: an exactly solvable bound state and exact bound-state counts exhibits a declared well depth at which the unique bound state is exactly solvable (even state, $k=\kappa =\pi \mathrm{/}(4a)$, binding energy $E=-U\mathrm{/}2=-{\mathrm{\hslash }}^2{\pi }^2\mathrm{/}(32m{a}^2)$), verifies the even root condition with the special value $\tan (\pi \mathrm{/}4)=1$, and verifies the count and branch criterion at declared depths: exactly three bound states with types even, odd, even when $2aK\mathrm{/}\pi =5\mathrm{/}2$, exactly one when $2aK\mathrm{/}\pi =1$, exactly two when $2aK\mathrm{/}\pi =2$, and at least one for every $a,U>0$. The Dirichlet box: exact spectrum, nodes, constant shift and gaps records the exact free Dirichlet box eigenpairs ${\mathcal{E}}_n={\mathrm{\hslash }}^2{\pi }^2{n}^2\mathrm{/}(2m{L}^2)$ with sine eigenfunctions, their explicit $n-1$ nodal points $kL\mathrm{/}n$, the exact spectrum ${\mathcal{E}}_n+c$ of a constant potential with the min-max bound attained, and the positive gaps ${\mathrm{\hslash }}^2{\pi }^2(2n+1)\mathrm{/}(2m{L}^2)$.

What is and is not compared. The comparisons are internal: the exact barrier formula and its thick-barrier approximation, the exact well root condition and the exact branch count, and the exact box spectrum with the min-max bound. They verify the cited theorems at declared parameter points by exact rational and algebraic checks; they are not numerical approximations of generic transcendental roots, they contain no simulation presented as data, and they report no measured value, no uncertainty and no significance statement. The displayed decimals are rational bracket endpoints or exact terminating decimal expansions of rational values, including $201\mathrm{/}125000=0.001608$; each is justified in the verifications.

Evidence status. All three items are conditional computations in the declared line, well and box models of the pair's A page, under the same P1, P2, P4, P5 assumptions and the same cited mathematics; the empirical qualifications of the A page carry over unchanged. Nothing on this page is an observation, an apparatus model or a statistical inference, and agreement of the displayed identities with any measurement would be evidence about the combined model and apparatus assumptions rather than a proof of the model.