Block and Ramsey partition regularity of the sum-product formula

Classify the partitions of the variables x, y, z, t for which the formula (x ≠ y) ∧ (x + y = z) ∧ (x · y = t) is block partition regular and classify those for which it is Ramsey partition regular.

Background

The formula combining distinct inputs, pairwise addition, and pairwise multiplication is examined through tables listing known block and Ramsey partition-regularity statuses. Several variable-block configurations are marked as not previously considered or remain unresolved. The final problem asks for a complete classification across all partitions of the variables, both for ordinary block partition regularity and for the stronger Ramsey partition regularity.