A Torus Ladder for Pratt's Tripartition Tensor: R(P₃) ≤ 216 and R(T₂,₂,₃) ≤ 54

Pratt's balanced tripartition tensor P_n connects Strassen's asymptotic rank conjecture with the Set Cover Conjecture through its rank. For P₃, a tensor of format 84 × 84 × 84, the upper bounds were Pratt's 256, the partial-Fourier bound 247 of Flavi, Jelisiejew and Michałek, and the border rank bound 226 of an S₇-invariant block identity (Foss 2026, 10.5281/zenodo.23333860); the lower bound is 150. We prove R̲(P₃) ≤ 216 and R̲(T₂,₂,₃) ≤ 54. The proof combines block decompositions Σ ℓ(ℓw + pq) of the cubic form of P₃, in which every block has border rank four, with a torus relaxation: along a one-parameter torus under which the cubic form has weight zero, every cubic that agrees with it on the monomials of non-negative weight degenerates to it. The monomials of negative weight, a quarter to a third of all coefficients, drop out of the equations, and the number of blocks is preserved. Distinguishing a growing set D of ground elements gives a ladder of symmetric search problems. The rungs |D| = 1, 2, 3 carry exact rational identities with 56, 55 and 54 blocks and give R̲(P₃) ≤ 224, 220 and 216; the B-part of the last identity is a 54-term border decomposition of T₂,₂,₃, below the rank bound 55. The rung |D| = 4 carries a 53-block solution with residual below 10⁻²¹² in 240-digit arithmetic, which gives R̲(P₃) ≤ 212 and R̲(T₂,₂,₃) ≤ 53 numerically. We further prove a simplex lemma, x(xw + Q) is a limit of rank Q + 2 cubes, which prices blocks that share a linear form. Every exact statement is verified in rational arithmetic and cross-checked by an exact evaluation of the limit; paper, LaTeX source, certificates, verification scripts and the numerical solution are included.