We analyze the critical temperature of the one-dimensional Ising model with inverse-square interactions. In the Anderson-Yuval-Kosterlitz (AYK) renormalization-group framework the transition temperature depends sensitively on the bare domain-wall fugacity, which is controlled by the domain-wall core energy. We replace the standard Euler-Mascheroni approximation for the core energy by a brute-force microscopic evaluation of domain-flip energies, thereby fixing the bare fugacity consistently with the microscopic Hamiltonian used in Monte Carlo simulations. With this input, integrating the AYK equations yields $T_c^{\mathrm{RG}}=1.557\left(3\right)$ (in units of the coupling). We then perform Monte Carlo simulations using a long-range cluster algorithm for system sizes up to $L=7.3\times {10}^4$ and analyze the full magnetization distribution $P_{L,T}\left(m\right)$. We show that near criticality, finite-size bimodality above $T_c$ produces a systematic upward bias in $\langle m^2\rangle$ and drives moment-based pseudocritical estimators (including Thouless-jump matching) to underestimate $T_c$. Using the peak of the modulus susceptibility $Q=\langle m^2\rangle -{\langle \left|m\right|\rangle }^2$ and Kosterlitz-Thouless (KT)/AYK finite-size scaling, we obtain $T_c^{\mathrm{MC}}=1.544\left(3\right)$ and argue that the residual offset from $T_c^{\mathrm{RG}}$ is consistent with slow KT-like finite-size drift and distributional contamination that decays only logarithmically with $L$.

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• BKT transition
• Critical phenomena
• Order parameters
• Phase transitions
• Phase transitions by order
• Thermodynamics
• Transition temperature
• Monte Carlo methods
• Renormalization