Isometric Equivalence and Cyclic Shift Properties of Polygonal Residue Vectors in Fₚ for p ≡ 3 4

This paper investigates the structural identity between indicator vectors of quadratic residues and higher-order polygonal residues in finite fields Fₚ. For primes of the form $p = 4k+3$, we prove that polygonal residue vectors are isometrically equivalent to the fundamental Legendre sequence. By applying a modular completion of the square, a closed-form algebraic expression for the deterministic cyclic shift offset s is derived. Numerical verification and statistical uniformity are confirmed for all orbits up to $p < 100,000$, establishing a structural foundation for algorithmic optimization in Paley Type~I Hadamard matrix construction.