Exact error thresholds and relaxed-host measurements for elastic mode discrimination

Abstract

Released and clamped stiffnesses can agree for elastic assemblies with different internal support modes. We study whether finite support sweeps distinguish them when an associating nematic host relaxes during force measurements. Classical Stieltjes and parametric Loewner theory give the ideal response structure. An exact joint bounded-error criterion specializes rational interval fitting to positive one-mode responses, including closed sample intervals and strict physical constraints. For a two-rail spring design, the exact four-setting distance to the one-mode class is (611 − √366193)/7920. A determinant test with shared sample errors reaches that threshold. We then derive observation bounds for both physical hypotheses. The competitor is a bounded constitutive family of population-dependent one-rail springs, which includes the sharp ideal attainer. At zero terminal strain, fixed host fields give a support-independent relaxation offset. Explicit derivative estimates bound finite-force bias, sensor and displacement errors, support uncertainty, and finite preparation and settling times. The resulting observation sets are disjoint in a stated nonzero-error regime, even though a competitor matches both endpoint stiffnesses. The guarantee persists over a nonzero target spring-tolerance box. An amplitude criterion and dimensional conversion give design requirements for the theoretical apparatus. The general fitting and set-separation principles are established methods; the claims concern this mechanical specialization. The study contains full derivations and deterministic certificates, with no experimental observations or material validation.