Growth of generalized greatest common divisors along orbits of self-rational maps on projective varieties

The paper proves the limit using a refined r-jet/linear-system method on the blow-up and the iteratively finite locus, culminating in a uniform bound for h_{f^{-n}(Y)} relative to h_H and a block-growth inequality for h_H along the orbit. The candidate solution hinges on an unsubstantiated uniform inequality h_Z(P) ≲ deg_H(Z)^{1/c} h_H(P) derived from a naive Hilbert-polynomial count and Bertini; the paper explicitly notes that such a clean general upper bound for subscheme heights is not known and is highly nontrivial for positive-dimensional Y. The model also blurs scheme-theoretic inverse images f^{-n}(Y) (used in functoriality) with numerical cycle pullbacks (f^n)^*Y, and assumes degree growth control that the paper carefully avoids.