You appear to have an error. You write:

  (ϵ + 1)(ϵ−1) = −1, or alternately (1 + ϵ)(1 − ϵ) = −1

That alternative should surely be:

    (1 + ϵ)(1 − ϵ) = 1

Not least, in a commutative system (1+x)(1-x) = 1-x^2. Thus

    (1 + ϵ)(1 − ϵ) = 1 - ϵ^2 = 1