Pseudorandomly Weighted Lucas Sums, Lucas Bridge Theorems, and Lower-Bound Conjectures

This paper uses the companion-matrix viewpoint to separate and classify two obstructions to lower-bound principles for Lucas-type periods. The organizing shift is from isolated scalar coordinates to full Lucas state vectors: for MP,Q=pmatrixP&-Q\\1&0pmatrix, the first-kind state period modulo an odd prime p Q is exactly the order of MP,Q in GL₂( Fₚ). Thus the matrix is not only a computational device but the primary period object. The first obstruction occurs for multiplicatively weighted Lucas partial sums Sₘ=∑ₙ₌₁ᵐ (aₙ p)Uₙ(P,Q), aₙ≡ kⁿ⁻¹ t. The upper-bound theory is governed by the reduced weight period λₚ, not merely by the ambient order λ=ₜ(k). We prove the complementary lower-bound obstruction: the visible pair (π(p),λ) cannot force growth of the sum period τ or a fixed positive proportion of (π(p),λ). Explicitly, taking $(P,Q)=(2,1)$, choosing $t=pq$ with q≡2 p, imposing k≡1 p, and making k primitive modulo q, gives π(p)=p, λ=q-1, λₚ=1, τ=p, so τ/(π(p),λ)=1/(q-1)→0. The second obstruction is recurrence-level torsion. For fixed $(P,Q)$ with Q≠0, the periods πP,Q(p) are bounded over admissible odd primes if and only if the integer companion matrix is torsion; in companion form this gives exactly (P,Q)∈\(0,-1),(-1,1),(0,1),(1,1)\. The same bridge records the split/nonsplit discriminant bounds, rank-period relations, a GRH-conditional Artin benchmark for the classical Fibonacci matrix, and higher-order analogues for order- recurrences. The unified conclusion is that lower-bound conjectures fail for two structurally different reasons—weight periods can disappear after reduction, and recurrence parameters can specialize to torsion—while the modular period theory itself is governed by integer companion matrices reduced modulo primes. --- Keywords Lucas sequence; pseudorandom weighted sums; bridge theorem; companion matrix; modular periodicity; lower-bound conjectures; period collapse; torsion classification; reduced weight period --- MSC Classification 11B39, 11B50, 11T06, 37B15