Once first answers in any dimension to the Green-Griffiths and Kobayashi conjectures for generic algebraic hypersurfaces $mathbb{X}^{n-1} subset mathbb{P}^n(mathbb{C})$ have been reached, the principal goal is to decrease (to improve) the degree bounds, knowing that the `celestial horizon lies near $d geqslant 2n$. For Green-Griffiths algebraic degeneracy of entire holomorphic curves, we obtain: [ d ,geqslant, big(sqrt{n},{sf log},nbig)^n, ] and for Kobayashi-hyperbolicity (constancy of entire curves), we obtain: [ d ,geqslant, big(n,{sf log},nbig)^n. ] The latter improves $d geqslant n^{2n}$ obtained by Merker in arxiv.org/1807/11309/. Admitting a certain technical conjecture $I_0 geqslant widetilde{I}_0$, the method employed (Diverio-Merker-Rousseau, Berczi, Darondeau) conducts to constant power $n$, namely to: [ d ,geqslant, 2^{5n} qquad text{and, respectively, to:} qquad d ,geqslant, 4^{5n}. ] In Spring 2019, a forthcoming prepublication based on intensive computer explorations will present several subconjectures supporting the belief that $I_0 geqslant widetilde{I}_0$, a conjecture which will be established up to dimension $n = 50$.
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