A Matrix-Degree Obstruction to Rational Generation of Boolean-Lattice Pseudo-Roots

For the neighborhood seed associated with the four-vertex path $P_4$, the diamond operations do not recover all Boolean-lattice pseudo-roots.

The corresponding question for unrestricted rational operations in the free skew field is subtler: the seed map has an invertible linearization and therefore a unique formal inverse near every generic scalar point.

We prove that this formal inverse is not free rational.

A symmetric one-parameter curve of $2 \times 2$ matrix outputs has a formal inverse whose coefficient field contains an element of degree three over $\mathbb{Q}(t)$.

An exact elimination in a quadratic Pauli algebra produces the irreducible cubic.

Its conjugate inverse branches are unramified, forcing the generic matrix degree of the seed map to be at least three in every size $n \ge 2$.

This contradicts the degree-one consequence of any free rational inverse.

The same matrix-degree argument, without specializing a hypothetical inverse, extends the obstruction to every graph containing an induced $P_4$.