Breaking Pratt's Bound for Balanced Tripartition Tensors: R(P₃) ≤ 226, R(P₂) ≤ 28, and a Four-Cube Lemma

Pratt's balanced tripartition tensor P_n, the tensor of ordered partitions of a 3n-element set into three n-element blocks, is the bridge between Strassen's asymptotic rank conjecture and the Set Cover Conjecture: its asymptotic rank controls the running time of balanced tripartitioning and set cover algorithms. The previous upper bounds on its rank are Pratt's group construction R(P_n) ≤ 8ⁿ/2 and the partial-Fourier refinement of Flavi, Jelisiejew and Michałek, which gives 31 for P₂ and 247 for P₃. We prove R̲(P₃) ≤ 226 and R̲(P₂) ≤ 28. Both bounds come from explicit identities that write the cubic form of P_n as a sum of blocks ℓ(ℓw + pq), and every block has border rank four by a four-cube lemma: x(xw + pq) is the limit of four cubes. The identity for P₃ is invariant under the symmetric group S₇, uses 56 blocks indexed by the pairs and triples of a seven-element set plus one block of border rank two, and has rational coefficients 1/5, 2/5, 1/8, 1/200 and 1/2. Its proof reduces to 19 + 13 orbit checks that are listed in full. We further prove R̲(P₃) ≥ 150 and R(T₂,₂,₃) ≤ 55: the first from a Koszul–Young flattening whose 10584 × 10584 matrix has exact rank 10500, the second from a decomposition with entries 0, ±1 on two factors that beats the partial-Fourier bound 57; the B-part of the P₃ identity is itself a 56-term decomposition of T₂,₂,₃. On the side of group constructions we prove that no abelian group of order below 32 realises P₂ as a restriction of its group tensor, so Pratt's ℤ₂⁵ is optimal among abelian groups and the bound 28 lies beyond the reach of the method. A computational survey of the balanced type slices of W^⊗m for m = 4, …, 9 places every one of them at least 11.7 % below Pratt's bound and up to 15.6 %, including R(P₃) ≤ 240 and R̲(T₂,₂,₃) ≤ 54 numerically. Every exact statement comes with a verification script in rational or modular arithmetic; paper, scripts, certificates and numerical decompositions are included.