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\\mathbb{P}^n(\\mathbb{C})$ have been reached, the principal goal is to decrease (to improve) the degree bounds, knowing that the `celestial' horizon lies near $d \\geqslant 2n$.\nFor Green-Griffiths algebraic degeneracy of entire holomorphic curves, we obtain: \\[ d \\,\\geqslant\\, \\big(\\sqrt{n}\\,{\\sf log}\\,n\\big)^n, \\] and for Kobayashi-hyperbolicity (constancy of entire curves), we obtain: \\[ d \\,\\geqslant\\, \\big(n\\,{\\sf log}\\,n\\big)^n. \\] The latter 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