What the companion examples and the thought experiment establish

The companion items contain three conditional examples and one hypothetical scenario in declared finite-mode or countable-mode models:

• The two-site bond model: exact canonical weights, mean energy, entropy and free energy of one particle on two sites solves the one-particle two-site bond model exactly: the tunnelling-split eigenbasis, the canonical weights and the exact mean energy, entropy and free energy;
• Finite-mode non-interacting thermal ensembles: two fermions in three modes and two bosons in two modes evaluates the canonical partition sums and occupation means of two fermions in three modes and two bosons in two modes, with the total-occupation identity and ground energies checked against the page's diagonalisation theorem;
• Free-mode statistics at low occupation: exact dilute identities, a finite arithmetic check and two convergence examples verifies the exact Bose and Fermi dilute identities, evaluated in exact rational arithmetic at $q=1\mathrm{/}10$, and contrasts the finite product partition of linearly spaced mode energies with the divergent product of constant energies;
• A site-localised preparation of the two-site bond model never becomes canonical, and its readouts need not take their thermal means shows, in one declared hypothetical preparation of the bond model, that the unitary evolution never produces the canonical preparation, that the site-occupation readout takes the canonical value $1\mathrm{/}2$ at some times while differing from it at others, and that the energy readout never takes its canonical value.

None of these items contains measured data, a measured uncertainty, a calibration, a sampling model or a detector model; all numbers are exact model values in the declared finite-mode or countable-mode idealizations, and the assignment of a physical system to any of these models is a tested identification under P7 (Preparation and apparatus postulate P7: declared preparations, isolation and sampling assumptions). In particular:

• the bond model is not any particular molecule, and its splitting $2t$ is an assigned parameter, not a measured spectroscopic value;
• the finite-mode examples are not ideal or interacting gases; they make no statement about a thermodynamic limit, pressure, condensation or a continuum of modes;
• the comparison of readout means with canonical values in the thought experiment is a statement about the declared preparation and the exact unitary flow; it is not evidence for or against thermalisation in nature, and the canonical ensemble remains the adopted assumption of The canonical equilibrium-ensemble assumption: the Gibbs preparation of a declared model at assigned temperature;
• the thought experiment uses the no-unitary-production conclusion (c) of Unitary evolution preserves entropy and cannot produce the canonical preparation from a different one. The separate entropy-conservation conclusion (a) of that theorem concerns the same finite closed-system class but is not a premise of the readout calculation; neither conclusion forbids or asserts open-system or environment-induced behaviour.

The cited treatments supply the model formulas used here at the recorded locators; they do not supply measured values for these particular finite-mode or countable-mode models, and no source is used as an empirical premise on this page. Where a comparison with data would be attempted, the preparation, isolation, detection and sampling models plus the uncertainty of every assigned parameter would have to be declared and carried through, as P7 requires.