Critical Partition Function Theory VIII

Critical partition function theory (CPFT) treats critical spacing data as probability measures rather than assuming from the outset a smooth spacing density. This permits atomic critical measures, a possibility that becomes essential for the finite periodic square-lattice Ising model. We first prove the model-independent identity $W_2(\mu,\delta_1)^2=Var_\mu(s)$ for every unit-mean probability measure with finite second moment, identifying the CPFT coordinate $g=0$ exactly with the Wasserstein boundary point $\delta_1$. The Ising application unexpectedly leads to a stronger finite-volume result. Writing the $L\times L$ periodic square lattice as the toroidal graph $C_L\square C_L$, the Ising energy is determined by the cut size $b$ of the spin partition, $E=-2JL^2+2Jb.$ We give a constructive classification of all possible cut sizes for every $L\ge3$. If $L$ is odd, $b\in\{0\}\cup\{4,6,8,\ldots,2L(L-1)\},$ whereas if $L$ is even, $b\in\{0\}\cup\{4,6,8,\ldots,2L^2-4\}\cup\{2L^2\}.$ Consequently the number of distinct energies is exactly $L(L-1)$ for odd $L$ and $L^2-1$ for even $L$, and adjacent distinct-energy gaps are only $4J$ and $8J$: the lower edge is always $8J$, and an additional upper-edge $8J$ gap occurs exactly for even $L$. It follows that the partition-function-weighted critical gap measure is an exact two-atom law for all $L\ge3$, $P_{c,L}^{\rm PF}=(1-q_L)\delta_{1/(1+q_L)}+q_L\delta_{2/(1+q_L)}.$ At the exact Ising critical point we prove $\lim_{L\to\infty}-\frac{1}{L^2}\log q_L$ $=\delta$ $=\frac12\log2+\frac{2G}{\pi}-\log(1+\sqrt2)$ $=0.048321811322\ldots,$ where $G$ is Catalan's constant. Hence $P_{c,L}^{\rm PF}\to\delta_1$ in $W_2$, with $\lim_{L\to\infty}-\frac{1}{L^2}\log W_2(P_{c,L}^{\rm PF},\delta_1)=\frac{\delta}{2}.$ The result was found by following an atomic CPFT critical measure back to the finite Ising spectrum. CPFT itself does not require randomness, a symmetry class, or spatial dimension as inputs of its distributional sector; nevertheless its finite-size application exposed an exact discrete support structure of a standard critical model. To the best of our knowledge, the parity-resolved all-$L$ support classification and the resulting exact atomic CPFT boundary theorem have not previously been stated.