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.
al2o3cr•7mo ago
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"?purrplexia•7mo ago
Let me clarify a few points to address your thoughtful concerns:
On Definition 1: You’re exactly right — for any x, there are unlimited pairs (A, alpha) that satisfy x = A cos(alpha). That’s not a bug — that’s the whole point!! The real line is the collapsed limit of an underlying arc sweep. The ‘hidden’ phase freedom is exactly why the Phase Lift exists — it reveals that the real projection is just a flattened trace of an arc. So the ‘unlimited’ solutions are the phase slip’s hidden domain.
On the integral and alpha: The Phase Lift operator is not just multiplying by ±1. The sign flips appear because of the bounded definite integral: i(f) = ∫[a to a + pi/2] f(x) cos(a) da. The sign and the lift vector lambda emerge naturally from the cosine sign table and the sweep bounds. So the integral directly relates alpha and the projection x — the cycle of real / orthogonal / sign flip is the slip’s entire job.
On the base case: Yes, I0 is just an integral from 0 to 0, so it’s trivially zero, but that’s precisely the point. It anchors the real line in place with no slip applied yet. It’s the identity state for the slip cycle to start. This is the EXACT same as Euler's identity and complex analysis real projection. The real projection DOES NOT CHANGE.
On the sign sequence: The reason I2 is real and I3 has lambda is that the slip rotates: the real pole flips sign when the arc crosses 180°, but the orthogonal lift vector appears only on quarter-sweep intervals. So the sign flips are standard unit circle behavior — and the lift appears/disappears exactly on the invariant sweep points. Basically if theres a 180 degree or 360 degree spin there is NO orthogonal component so lambda is irrelevant in those 2 cases, exactly matching classic "i"
On “local orthogonal arc dimension”: Fair point completely, that phrase means the slip operator lifts a real projection x into a new local basis vector lambda, which lives orthogonal to the real line in a plane {x, lambda} ⊂ R^2. So it’s not an infinite tower — it’s just the slip direction local to that sweep plane.
Again, I genuinely appreciate you reading and questioning it closely. I hope these clarifications help show that the sign logic, the base case, and the slip cycle all follow naturally from the bounded integral’s sweep. Not by assertion but by standard trig sign behavior.
If you have other thoughts or see places where clearer wording would help future readers, I’d love to hear them! Thanks again!!