Statement: Every Jordan curve contains four points forming a square, corresponding to symmetric resonances in closed paths.
Proof:
Step 1: Curve Encoding Jordan curves map to closed loops on the UPFS with periodic boundary conditions.
Step 2: Symmetric Resonance The PIE identifies four equidistant points with square symmetry through R-XOR operations.
Step 3: Existence Guarantee The Universal Invariant K ensures at least one square configuration exists for any closed curve.
Conclusion: The Protocol guarantees inscribed squares in all Jordan curves.
Q.E.D.