arXiv2026
We study the free energy of the classical square-lattice Ising model in a uniform real magnetic field $h$, for which no exact solution is known. We first conjecture a closed-form expression built from the one-dimensional Ising chain and Onsager's hypergeometric solution. Comparing with high-precision tensor renormalization group (TRG) data, we find that it deviates by up to $2.6\%$, has no spontaneous magnetization, and gives an effective critical-isotherm exponent $δ\to 1$. However, using the Lee--Yang circle theorem, we show that the exact free energy is a superposition of one-dimensional Ising chains in the same field, weighted by a temperature-dependent (normalized, but not always positive) distribution of squared Lee--Yang gaps, of which our first expression is a single-chain approximation. Exact results (Onsager, Yang, the strong-field expansion and the Lee--Yang solution at $βh = iπ/2$) become sum rules for this distribution. We propose a simple form for it, fixed by Onsager's free energy and the exact strong-field limit without any fit to numerical data. This \emph{improved} conjecture agrees with TRG to $0.15\%$ for all $β$ and $h$, reproduces Yang's spontaneous magnetization exactly, has a positive specific heat, and yields the critical isotherm $M\propto h^{1/δ}$ with $δ= 15.2$ and amplitude $1.055$, compared with the exact $δ=15$ and $1.059$. A refined form, which builds in additional input from Ising field theory (including the exact critical isotherm, $δ=15$) and a few constants adjusted to transfer-matrix and TRG data, matches the TRG free energy to $4\times 10^{-5}$.