Spectral Pisot–Galois Discrepancy, Extended Stride-s Hankel Operator Squares, and Complete 2-Adic and Cyclotomic Valuation Laws in Generalized k-Step Fibonacci Sequences

We present a unified spectral, Pade-determinantal, and p-adic arithmetic treatise on generalized k-step Fibonacci sequences {F_n^(k)}_{n in Z} (k >= 2). First, we connect the unimodular SL(2, Z) Mobius collapse at stride s = k, where P_k^(k)(y) = [y^{k+1} - (2y - 1)^k] / (y - 1), and the all-orders Binet-Galois discrepancy functional R_n^(k) = (F_n^(k))^2 - alpha_k^{-2} (F_{n+1}^(k))^2 to the monogenic stability of Pisot-Vijayaraghavan (PV) units (Z[alpha_k^k] = Z[alpha_k]), Parry invariant measures on symbolic beta-shifts Sigma_{alpha_k}, and non-reciprocality obstructions for Salem numbers. Second, we show how the (k-1) x (k-1) Cauchy-Binet time-reversal operator square H_{k-1}^(k,s)(m) = (-1)^{binom(k,2) + (k-1)n_0(k-1)} [Pi_s^(k)(B)]^2 F_{(k-2)s - m}^(k) and top-down Dodgson condensation involution provide an O(1) closed-form bypass for sub-maximal Hankel determinants and Pade approximant pole/singularity analysis without matrix inversion. Third, we resolve four fundamental open problems in the hierarchy:(1) We establish the complete stride-s 2-adic valuation law for the k x k Vandermonde quotient J_s^(k) across all k >= 2 and s >= 1 via root-cluster ramification over Q_2-bar, proving the exact Tribonacci Thue-Vandermonde identity J_s^(3) = Delta^(1)(T_s, T_{s+1}) and resolving multi-cluster F_4 Hurwitz resonances for mixed odd orders.(2) We characterize all shared odd prime divisors of the boundary quotients J_{k-1}^(k) and J_{k+2}^(k) via half-degree Sylvester-Bezout resultants and the universal quadratic lock x^2 - 2x + 4 = 0 (mod p), proving that same-pair shared primes occur if and only if k = 1 (mod 3) and p divides M_k = [2^{k+1} - (-1)^{(k-1)/3}] / 3 (yielding 11 at k = 4, 85 = 5 * 17 at k = 7, and 683 at k = 10).(3) We prove the Fundamental Pairwise Sum-to-Product Bridge Identity lambda_i + lambda_j = 2 + (1/2)(p_{ij} - p_{ij}^{-k}), collapsing the outer-shell strides s in {2, k-2, k+3} into exact 3-tap and 4-tap shift-operator squares on H^(k,k).(4) We derive the universal 2-adic difference operators Delta_{k+1} = 2 D_1 and Delta_{2k+2} = 4 B D_1, establish the binom(k,2) Universal Zero Triangle T_k up to m = k^2 + k - 1, and prove via Strassman's theorem that the 3 x 3 Tetranacci Hankel determinant H_3^(4,1)(m) vanishes across Z if and only if m in {0, 1, 2, 5, 6, 10}. MSC 2020 Classification: Primary 11B39, 15B05, 11R06; Secondary 37B10, 41A21, 15A15, 11D59, 12D10.DOI: 10.6084/m9.figshare.34809483