Kim's own formulations in 'Supervenience and Mind' leave the existence of inter-world isomorphisms as a substantive metaphysical assumption, not a logical guarantee, breaking the entailment.
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A belief about what fundamentally exists or what reality is really made of that you're taking for granted without proving it.
supervenience(Philosophy of mind and reduction; contrasted with full reduction)
A relation in which mental (or higher-level) states are dependent on physiological (or lower-level) states such that any two cases with identical lower-level bases are identical in their higher-level states; a necessary but not sufficient condition for reduction.