Self-Contained Corrections and Resolutions for Three Revised Fibonacci–Lucas Problems

We give self-contained resolutions of three revised Fibonacci–Lucas problems via the companion matrix of the Fibonacci recurrence. For the corrected infinite product, we prove its exact Mahler-type form, bound it strictly between zero and one, and solve the full coefficient-selection problem on its maximal positive interval. For the reversed inequality, we establish sharp two-sided estimates with optimal constant and extend the result to arbitrary coefficients. For the nested radical sum, we provide exact finite-product evaluations for both natural Lucas-subscript readings and completely classify the inverse problem of prescribing arbitrary target sums. No external results are used.