low_temp_limit
plain-language theorem explainer
The theorem shows that for positive energy gap the ground state probability in the J-cost Boltzmann distribution tends to 1 as inverse temperature beta tends to infinity. Workers deriving statistical mechanics from recognition cost functionals cite it to confirm ground-state dominance in the low-temperature regime. The proof reduces the claim to the standard real-analysis limit of exp(-beta * gap) to zero, followed by continuity of addition and division.
Claim. Let $gap > 0$. Then the ground-state probability $P_0(beta, gap) = 1 / (1 + exp(-beta * gap))$ satisfies $lim_{beta to +infty} P_0(beta, gap) = 1$.
background
The module derives the Boltzmann distribution from the J-cost functional of Recognition Science, where each state carries a recognition cost J(x) and the most probable occupation maximizes microstate count subject to fixed total cost. The ground-state probability is the normalized weight of the lowest level in a gapped two-level system, with beta emerging as the Lagrange multiplier conjugate to that cost constraint. Upstream cost-algebra results establish that J-automorphisms compose multiplicatively, preserving the exponential form under the derived cost map.
proof idea
The tactic proof first unfolds groundStateProb to the explicit ratio. It applies Filter.Tendsto.atTop_mul_const to show beta * gap tends to infinity, then composes with Real.tendsto_exp_neg_atTop_nhds_zero to obtain the exponential term tending to zero. Const_add and div are applied in sequence, with a positivity check ensuring the denominator never vanishes, yielding convergence to 1.
why it matters
The result completes the low-temperature analysis inside the J-cost derivation of statistical mechanics, confirming that cost minimization concentrates probability on the ground state. It supports the module's claim that temperature arises as partial derivative of J-cost with respect to entropy and aligns with the Recognition Science forcing chain in which unique minimal-cost states are selected. The declaration feeds no downstream theorems in the current graph but underpins the paper proposition on statistical mechanics from Recognition Science.
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