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We predict the low energy signatures of a Flipped $SU(5) \times U(1)_{\chi}$effective local model , constructed within the framework of F$-$theory based on$ A_{4} $ symmetry. The Flipped SU(5) model from F Theory in the field ofparticle physics is prominent due to its ability to construct realisticfour$-$dimensional theories from higher$-$dimensional compactifications whichnecessitates a unified description of the fundamental forces and particles ofnature, used for exploring various extensions of the Standard Model. We studyFlipped $SU(5) \times U(1)_{\chi}$ Grand Unified Theories (GUTs) with $ A_{4} $modular symmetry. In our model due to different modular weights assignments,the fermion mass hierarchy exists with different weighton fields. Theconstraints on the Dirac neutrino Yukawa matrix allows a good tuning to quarkand charged lepton masses and mixings for each weighton field, with theneutrino masses and lepton mixing well determined by the type I seesawmechanism which occurs at the expense of some tuning which manifests itself incharged lepton flavour violating decays which we explore here. The minimalFlipped $SU(5$) model is supplemented with an extra right$-$handed type and itscomplex conjugate electron state, $ E_{c} + \bar{E_{c}} $, as well as neutralsinglet fields. The $ E_{c} + \bar{E_{c}} $ pair gets masses of the order ofTeV which solves the $ g_{\mu}- 2$ discrepancy. The predictions of the modelfor charged lepton flavour violation decay rate and proton decay could betested in near future experiments. Also we detect in our model the existence ofneutralino, its charge mass and spin via direct and indirect detection.

FCNCs, Proton Stability, $ g_{\mu}-2$ Discrepancy, Neutralino cold Dark  Matter in Flipped $SU(5) \times U(1)_{\chi}$ from $F$ Theory with $ A_{4} $  Symmetry