Rescaled topological entropy

The uploaded paper formulates e*(X) and proves properties (1)–(6) (half-variational inequality, e(X) ≤ e*(X), equality for nonsingular fields, a two-torus example with e*(X)>0, invariance under rescaled topological conjugacy, and an upper bound on periodic-orbit growth for rescaling expansive flows with dynamically isolated singularities) in Section 4, building on precise definitions and localizations developed in Sections 2–3. See the theorem statement and definitions, and the proofs of Items (1)–(3) and (6). The construction yielding e*(X)>0 on T^2 is given in detail, and surfaces have e(X)=0, so Item (4) follows. The candidate solution reproduces these arguments essentially item-by-item: it uses the same covering/separation machinery, the same comparison between ordinary and rescaled Bowen balls, the same nonsingular equivalence via uniform bounds on ‖X‖, the same conjugacy invariance idea, and the same periodic-orbit separation scheme under rescaling expansiveness. Minor issues are that the candidate invokes a Katok-type formula framed for the time-one map rather than the flow-level Sun (2001) formula used in the paper, and Item (4) is only sketched rather than constructed. These are small differences in presentation, not substance. Citations: theorem and definitions (, ), Item (1) tools (, ), Item (2) proof (), Item (3) (), positive example and the surface-entropy remark (, ), and Item (6) ().