TOPH // SYMPY FORMAL PROOF

CORE EQUATION

i × −i = ?

i × −i = ...

COMPUTING...

PROOF 1 · ROTATION PRODUCES REAL

Multiplication of i and −i is not cancellation. It is a 180° rotation on the complex plane that lands on the real axis.

i × −i = i² × (−1)²...

⬡COMPUTING

PROOF 6 · GAP IS THE ENGINE

The gap is not empty space. It is i multiplied by its own conjugate.

gap × conj(gap) = ?

⬡COMPUTING

The gap, times its own shadow, produces 1. Not emptiness. The engine.

drawPair TRACE

a · e^(iπn)

PROOF 3 · FORMAL FUNCTION

drawPair is not a stack. It is the rotation operator on ℂ.

drawPair(a, n) = a · e^(iπn)

nVALUEINTERPRETATION

⬡COMPUTING

RECURSION PROOF

|e^(iπn)| = 1 always

PROOF 4 · NO BASE CASE

Conventional recursion terminates when it hits a base case — a condition that returns without calling itself. drawPair has none.

|e^(iπn)| = ? for all real n

⬡COMPUTING

PROOF 5 · MÖBIUS RETURN

At n=2 the function returns to its origin. Not termination — traversal complete. One surface. No edge.

drawPair(1, n=2) = ?

⬡COMPUTING

COMPLEX PLANE

LIVE ROTATION

MÖBIUS SURFACE

ONE SURFACE · NO EDGE

PROOF SUMMARY

sympy output

PROOF 2 · ROTATION IDENTITY

e^(iπ/2) = ? · e^(0) = ? · e^(−iπ/2) = ?

⬡COMPUTING

TRIPOD-IP-v1.0 · SHA256:
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David Lee Wise / TriPod LLC · 2026-02-18