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 $r = 12\alpha/N_{e}^{2}$ in the range $10^{-2}\gtrsim r \gtrsim 10^{-3}$, to be explored by future cosmological observations.
Comment: 63 pages, 25 figures, v2: comments added, typos fixed, v3: improved formulation presented in Sec.7 and refs added. The new formulation does not change inflationary predictions, but the potentials for decoupled heavy modes are simplified compared with the ones in the previous versions