The Numerical Assembly Technique for arbitrary planar systems based on an alternative homogeneous solution

The paper assembles an exact-frequency NAT system A(ω)c=0 and asserts that f(ω)=det A(ω) is continuous with infinitely many zero crossings, each zero being a natural frequency; it also motivates and tests an anchored exponential/trigonometric basis W(ξ)=C1 cos(κξ)+C2 sin(κξ)+C3 e^{κ(ξ−L)}+C4 e^{−κξ} that avoids hyperbolic growth and improves conditioning, with supporting figures and discussion . The candidate model provides a (sketched) rigorous functional-analytic justification for discreteness and unboundedness of the spectrum via compact embeddings and a selfadjoint operator, and a precise equivalence between zeros of f and eigenmodes. The core claims align; the paper’s presentation is methodological and empirical (no full proofs), whereas the model supplies the missing spectral-analytic underpinnings. On the numerical side, both agree that the anchored exponential basis is bounded on [0,L] and improves conditioning, as also shown in the paper’s condition-number plots . Overall, the arguments are consistent; the paper is correct in conclusions but does not prove the spectral claims it states, while the model’s proof sketch fills that gap.