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As the necessary background to construct from the aspect of Grothendieck'sAlgebraic Geometry dynamical fermionic D3-branes along the line ofRamond-Neveu-Schwarz superstrings in string theory, three pieces of thebuilding blocks are given in the current notes: (1) basic$C^\infty$-algebrogeometric foundations of $d=4$, $N=1$ supersymmetry and$d=4$, $N=1$ superspace in physics, with emphasis on the partial$C^\infty$-ring structure on the function ring of the superspace, (2) thenotion of SUSY-rep compatible hybrid connections on bundles over the superspaceto address connections on the Chan-Paton bundle on the world-volume of afermionic D3-brane, (3) the notion of $\widehat{D}$-chiral maps$\widehat{\varphi}$ from a $d=4$ $N=1$ Azumaya/matrix superspace with afundamental module with a SUSY-rep compatible hybrid connection$\widehat{\nabla}$ to a complex manifold $Y$ as a model for a dynamicalD3-branes moving in a target space(-time). Some test computations related tothe construction of a supersymmetric action functional for SUSY-rep compatible$(\widehat{\nabla}, \widehat{\varphi})$ are given in the end, whose furtherstudy is the focus of a separate work. The current work is a sequel toD(11.4.1) (arXiv:1709.08927 [math.DG]) and D(11.2) (arXiv:1412.0771 [hep-th])and is the first step in the supersymmetric generalization, in the case ofD3-branes, of the standard action functional for D-branes constructed inD(13.3) (arXiv:1704.03237 [hep-th]).

$N=1$ fermionic D3-branes in RNS formulation I.  $C^\infty$-Algebrogeometric foundations of $d=4$, $N=1$ supersymmetry,  SUSY-rep compatible hybrid connections, and $\widehat{D}$-chiral maps from a  $d=4$ $N=1$ Azumaya/matrix superspace