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 W₂(μ,δ₁)²=Var_μ(s) for every unit-mean probability measure with finite second moment, identifying the CPFT coordinate $g=0$ exactly with the Wasserstein boundary point δ₁. The Ising application unexpectedly leads to a stronger finite-volume result. Writing the L× L periodic square lattice as the toroidal graph CL CL, the Ising energy is determined by the cut size b of the spin partition, E=-2JL²+2Jb. We give a constructive classification of all possible cut sizes for every L≥3. If L is odd, b∈\0\∪\4,6,8,…,2L(L-1)\, whereas if L is even, b∈\0\∪\4,6,8,…,2L²-4\∪\2L²\. Consequently the number of distinct energies is exactly $L(L-1)$ for odd L and L²-1 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≥3, Pc,LPF=(1-qL)δ1/(1+qL)+qLδ2/(1+qL). At the exact Ising critical point we prove limL→∞-1/L²log qL =δ =12log2+2G/π-log(1+√2) =0.048321811322…, where G is Catalan's constant. Hence Pc,LPF→δ₁ in W₂, with limL→∞-1/L²log W₂(Pc,LPF,δ₁)=δ/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.