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variational free energy per spin. 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explicit edge witnesses for the ideal uniform-site lazy Metropolis kernel; log π(s)P(s,s′) equals its reverse. 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variational free energy per spin. 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Finite random resolution is an implementation approximation. Fixed laziness 1/2 prevents the all-accepted parity cycle at J=h=0; the ideal random kernel is irreducible and aperiodic at T>0. One sweep is N random-site attempts with replacement. Samples are post-sweep after burn; burn is not evidence of equilibrium. Centered population moments and short-lag descriptive correlations are not independent-sample error estimates; constant trajectories have undefined correlation. 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variational free energy per spin. Roots solved in finite effective-field coordinate y, m=tanh(y), so low-temperature saturation can round m to ±1 without losing a root. Curvature F″=T cosh²(y)-K. χ=sech²(y)/(T-K sech²(y)) applies to a stable local branch. A null value marks critical/unstable response. Near-degenerate minima within 1e-10 are candidates; largest m chosen for a displayed branch, not finite-volume equilibrium magnetization."},"squareInfinite":{"applicable":true,"Tc":2.269185314213022,"spontaneousMagnetization":0.911319377877496,"scope":"Onsager critical temperature and Yang spontaneous magnetization: infinite ferromagnetic square lattice, nearest-neighbor coupling, h→0+ after N→∞. 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variational free energy per spin. 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explicit edge witnesses for the ideal uniform-site lazy Metropolis kernel; log π(s)P(s,s′) equals its reverse. This does not turn a deterministic PRNG trajectory into independent exact equilibrium draws. The proof for all states is algebraic in the lesson."},"scope":"Finite microscopic Ising models, exact density-of-states sums, finite transfer matrices, deterministic lazy Metropolis traces and explicitly distinguished mean-field/infinite-square references. T>0, J≥0, kB=1; no critical-exponent fitting or automatic equilibration claim."}}]}