Reimagining the imaginary unit I – new proof (PGP signed)
1 purrplexia 1 7/1/2025, 1:02:37 PM
Hi HN,
I’m sharing a short paper that proposes a new interpretation of the imaginary unit i as a definite arc integral — showing it as a continuous phase lift that naturally recovers Euler’s identity and the classical i roots of unity.
It’s fully cryptographically signed (PGP) and timestamped. Free to download, share, verify, or break. ~50% of readers so far have downloaded it — very unusual for a pure math post.
Please verify the signature, read it critically, and try to find any flaw in the logic or trust chain.
Zenodo DOI (PDF, .asc, public key): https://doi.org/10.5281/zenodo.15783356
Repo: https://github.com/Purrplexia/Maths
Would love to hear your thoughts — good, bad, skeptical, or corrections.
Trust, but verify.
Definition 1 is meaningless, as for any x there are an unlimited number of values of A and alpha that satisfy x = A cos(alpha)
The phase-shift operator defined in Lemma 1 is literally just multiplying by +/-1. The integral doesn't show any relationship between alpha and x.
At the top of page 5, your "I_0 = base case" writes out an integral that is identically zero, but then simply labels it "grounded to the real line".
Just below that, the results for I_2 and I_3 suggest that there's something peculiar going on with your definition of integration. Why is one's result just "-1" and the other "-1*lambda"?
Overall there appears to be a lot of nonstandard terminology that is used but not defined in the paper; for instance, consider Definition 3:
What is a "local orthogonal arc dimension"?