Quadratic Dynamics over Hyperbolic Numbers

The paper proves that MH equals the square S in the (a,b)-plane and classifies KH(fc) into four topological types via characteristic coordinates X=x−y, Y=x+y, which decouple the map into the real quadratics X↦X^2+c1 and Y↦Y^2+c2 with c1=a−b, c2=a+b. It states THEOREM 1 (MH=S) and THEOREM 2 (four cases for KH), with the key Lemma T(KH(fc))=KR(fc1)⊕j KR(fc2) and the real one-dimensional trichotomy for KR(fc) (interval on [−2,1/4], Cantor for c<−2, empty for c>1/4) . The candidate solution reproduces the same argument: the characteristic-coordinate decoupling, MH=S via c1,c2∈[−2,1/4], and KH(fc)=KR(c1)×KR(c2) yielding the same four cases. It supplies more explicit details for the real quadratic lemma. No contradictions were found; both are correct and essentially the same proof strategy.