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For locality we surely mean observer. And I'd agree, take the human observer who is aware of their surroundings. Their perception is strictly local with time and their experience of it --- but our experience of time arises because of a gene regulation inside our heads, the so-called, suprachiasmatic nucleus. Beleiving for a moment that time exists objectively (outside the human perception) time is then part of the understanding of the manifold which requires a non-linear geometry. That would be most important, not the perception of it.