The graph y=x³+x² separates two and three states

Exact statement

No nontrivial rational two-variable homogeneous linear loop remains on $y=x^3+x^2$ , while the orbit $x(n)=2^n$ , $z(n)=-4^n$ , $y(n)=8^n+4^n$ gives a three-state witness.

StatusBoth the lower bound and witness are proved

External reviewNo documented external or specialist review of this FPRD exposition is recorded.

Context

The source states the separation; the FPRD page supplies the missing direct proof.

Proof or evidence

The polynomial has two nonlinear monomials, so FPRD-LL-T03 excludes two states. In state order (x,z,y), the initial state (1,-1,2) and update x'=2x, z'=4z, y'=8y+4z give x(n)=2^n, z(n)=-4^n, and y(n)=8^n+4^n.

A strict two-versus-three separation →

Verification notes

The signs, n=0 boundary, initial state, update matrix, invariant equation, and infinite-orbit condition were checked directly through 21 steps in the independent audit.

Limitations

This single example does not imply an unbounded hidden-dimension hierarchy.

Open work

Seek independent review of the exact classification proof that supplies the two-state lower bound.