M-theory Cosmology, Octonions, Error Correcting Codes

We study M-theory compactified on twisted 7-tori with $G_2$-holonomy. The effective 4d supergravity has 7 chiral multiplets, each with a unit logarithmic K"ahler potential. We propose octonion, Fano plane based superpotentials, codifying the error correcting Hamming (7,4) code. The corresponding 7-moduli models have Minkowski vacua with one flat direction. We also propose superpotentials based on octonions/error correcting codes for Minkowski vacua models with two flat directions. We update phenomenological $\alpha$-attractor models of inflation with $3\alpha =7,6,5,4,3,1$, based on inflation along these flat directions. These inflationary models reproduce the benchmark targets for detecting B-modes, predicting 7 different values of $r=12\alpha \mathrm{/}{N}_e^2$ in the range ${10}^{-2}\gtrsim r\gtrsim {10}^{-3}$, to be explored by future cosmological observations.