Determinants, Adjugate Decompositions, and Multi-Wave Mersenne Perturbations of Shifted k-Step Fibonacci Lo Shu Matrices and Pisot-Weighted Sextics

We investigate the algebraic, spectral, and combinatorial structure of Pisot-weighted sextic polynomials and 3x3 matrices formed by populating an arbitrarily shifted Lo Shu magic square I(m) = L + m J_3 (m in Z, central index c = m + 5) with terms of the k-generalized Fibonacci (k-bonacci) sequence F_n⁽ᵏ⁾ (k >= 2) and its companion Lucas sequence L_n. First, we establish exact biquadratic-to-cubic reductions and Vieta root-product identities for sextic polynomials weighted by powers of the Tribonacci constant alpha_3 and the Golden Ratio phi = alpha_2, showing that dual phi± 6-weighted sextics embed the Tribonacci minimal polynomial and recover the squared minimal polynomial derivative (P_2'(phi))^2 = 5. Second, for the shifted Fibonacci and Lucas Lo Shu matrices M_F(m) and M_L(m), we derive closed-form cubic determinants, quadratic adjugate decompositions, column-shift operators, and diagonal projections in Z[F_c, L_c] linked by a Galois-Binet ring duality D. Third, for the Tribonacci Lo Shu matrix M_T(m), we derive its three-parameter centered matrix pencil, homogeneous cubic determinant, and six-monomial adjugate, proving that M_T(m) has full rank 3 for all m in Z except at the unique rank-drop singularity m = -6 (rank M_T(-6) = 2). Fourth, for arbitrary step order k >= 2, we prove that the central (2,2) cofactor C₂₂⁽ᵏ⁾(m) is the unique cofactor of M_k(m) whose leading O(alpha_k²ᶜ) Pisot mode cancels identically, derive the universal Laurent determinant kernel K(x) = -(x^6 - x⁻⁶)(x^3 - x⁻³), establish the dyadic freeze law det(M_k(m)) = -257,985 * 2³ᵐ for k >= m + 8, and prove a universal diagonal-sum theorem. Finally, we derive an all-orders multi-wave combinatorial inclusion-exclusion expansion for F_n⁽ᵏ⁾, prove its Lagrange-Burmann equivalence to Dresden's dominant Binet mode, and obtain exact multi-wave Mersenne perturbation formulas for C₂₂⁽ᵏ⁾(m) and det(M_k(m)).