RSI Volume 44 - KS→NS Fourier-Shell Admissibility Law with Exact Three-Branch Certification and Branch-3 Triad Representative

Volume 44 certifies a finite three-branch Fourier-shell admissibility object for the KS→NS comparison family under the inherited Sigma_41 instrument lineage. The study object is the shell-membership law over the declared width-by-halfspan family emitted by the production notebook. Within the certified domain through halfspan 2.00, the admissibility object has three branches: a primary support-shell branch, a support-free substitute branch, and an exact higher-shell triad branch. The full declared family contains 1,695 transition cells across 66 declared (width, halfspan) pairs, and the certified three-branch law matches the measured admissibility oracle exactly, with zero mismatches, zero false positives, and zero false negatives. Branch 3 admits an exact triad representative based on the shell set {sqrt(45), sqrt(49), sqrt(50)}, that is, {sqrt(45), 7, sqrt(50)}. The former residual sector left open by Volume 43 is absorbed exactly by this third branch, and no fourth branch is required on the certified domain through halfspan 2.00. The study uses the inherited locked KS→NS shell-law setting under the Sigma_41 lineage, a rebuilt production-state admissibility oracle A(c), exact certification sweeps over the declared transition-cell family, and machine-readable artifact emission under the inherited RSI publication discipline. The certified object is a finite combinatorial shell-membership law on the declared scanning family, not a general transport theorem beyond that domain. Certified Results: Exact three-branch shell law certified on the declared width-by-halfspan family through halfspan 2.00 Production certification sweep covers 1,695 transition cells Exact agreement between the declared law and the measured admissibility oracle Zero mismatches Zero false positives Zero false negatives Full certified partition: EXCLUDE = 901, Branch 1 = 712, Branch 2 = 6, Branch 3 = 76 The inherited two-branch law leaves a 76-cell admissible residual on the certified domain Volume 44 certifies that all 76 residual cells are absorbed exactly by Branch 3 No fourth branch is required on the certified domain through halfspan 2.00 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 Branch 3 — sqrt(45) in B, sqrt(49) in B, sqrt(50) in B, sqrt(20) not in B, sqrt(37) not in B, 6 not in B, and 5 not in B Canonical Constructs: Three-branch Fourier-shell admissibility law, Branch-3 triad representative, residual-sector closure certificate, exact admissibility-oracle agreement on the declared family, no-fourth-branch certificate through halfspan 2.00, Sigma_41 inherited instrument lineage. Continuity: Volume 44 continues the RSI shell-law program by converting the structured residual sector identified in Volume 43 from residual geometry into exact branch structure. The inherited two-branch law is preserved where it was already exact, and the formerly uncovered admissible sector is absorbed exactly by an additional branch clause in the same shell-membership language. The result is therefore a closed three-branch shell law on the certified declared family through halfspan 2.00. Reproducibility: Evidence for the certified claim set consists of the locked production notebook, the rebuilt production-state KS→NS admissibility oracle A(c), the declared width-by-halfspan transition-cell sweep, exact certification totals emitted by the executable run, reader-facing export blocks, and the final artifact bundle under inherited RSI manifest discipline. Scope Bound: This volume does not certify uniqueness of the Branch-3 normal form, extension beyond halfspan 2.00, extension beyond the declared width range, or general transport beyond the declared KS→NS scanning family. It certifies only the bounded three-branch shell-law object stated above under the inherited Sigma_41 instrument lineage.