Complementary Symmetry and the Reconstruction of Modular Trajectories in a Family of Moduli

We study the family of integers n(g,a) = g^a - 1 + g^(a+g) = g^a(1 + g^g) - 1, where g >= 2 and a >= 1. For the least nonnegative residue r_k = [g^(a+g+k)]_n, we prove the congruence n - r_k + g^k ≡ g^(a+k) (mod n). An exact integer identity is also derived using the quotient from Euclidean division. The form of the modulus implies that n is odd and that g is a unit modulo n. Consequently, the powers of g generate a periodic orbit whose period is the multiplicative order ord_n(g). We also study the transformation x -> [g^k - x]_n, showing that it is an affine involution with a unique fixed point. Hence, for each fixed k, the residue classes modulo n decompose into one fixed point and (n-1)/2 pairs. The paper examines the relationship between this complementary symmetry and modular trajectory reconstruction, while distinguishing the proven algebraic properties from possible computational applications. The example g = 2, a = 3, n = 39 is developed in detail, including its complete modular orbit, numerical table, and graphical representations. The results establish a general modular identity and describe the associated arithmetic and symmetry properties of the family.