Volume 43 - KS→NS Piecewise Fourier-Shell Admissibility Law

Volume 43 certifies a finite local Fourier-shell admissibility object for the KS→NS comparison family under the inherited Sigma_41 instrument lineage. The study object is the piecewise shell-membership law over the declared width-by-halfspan family emitted by the production notebook, with certified constants q_crit = sqrt(20), q_sup = sqrt(37), q_branch_hi = 6, and q_branch_lo = 5. Within the certified region, the admissibility object has two branches: a primary branch with q_crit not in B and q_sup in B, and a support-free branch with q_crit not in B, q_sup not in B, q_branch_hi in B, and q_branch_lo not in B. The study uses fresh locked KS and NS rebuilds under the inherited geometry, explicit falsifier and atlas gates, support-shell recovery and uniqueness testing, and machine-readable synthesis and closeout artifacts under SHA-256 manifest discipline. The certified width family is evaluated against the declared halfspan slices emitted by the production notebook. Certified Results: Piecewise two-branch shell law certified on the declared width grid Zero violations for halfspan values {0.75, 1.00, 1.25, 1.50} First certified failure at halfspan = 1.75 Certified two-branch boundary interval: (1.50, 1.75) Unique recovered zero-violation support shell: q_sup = sqrt(37) Recovered support-shell candidate count: 13 Residual sector certified only as support-free high-shell region strictly above q_sup Certified Branch Structure: Branch 1 — q_crit not in B, q_sup in B Branch 2 — q_crit not in B, q_sup not in B, q_branch_hi in B, q_branch_lo not in B Canonical Constructs:Piecewise Fourier-shell admissibility law, support-shell falsifier, halfspan atlas, boundary interval certificate, third-sector geometry audit, Sigma_41 inherited instrument lineage. Continuity:Volume 43 continues the RSI shell-law program by moving from bounded frontier-family structure to a bounded cross-family Fourier-shell admissibility object. It certifies exact local branch structure, a unique support-shell isolation result, and a certified boundary interval, while leaving the residual sector beyond the two-branch region explicitly open. Reproducibility:Evidence for the certified claim set consists of a locked production notebook, fresh KS and NS rebuilds under the inherited geometry, explicit falsifier and atlas gates, machine-readable export records, synthesis and closeout artifacts, and SHA-256 manifesting of the final bundle. Scope Bound:This volume does not certify a closed-form third-branch law, an upper-q_lo boundary law for the residual sector, a general transport theorem, a universality claim beyond the declared finite shell object, or behavior outside the declared width family and certified halfspan interval. It certifies only the bounded shell-law object stated above under the inherited Sigma_41 instrument lineage.