Sixth-order Bochner–Krall systems for regular complex functionals: a computer-assisted classification

Let u be a regular complex linear functional on C[x], let {Pₙ} be its monic orthogonal polynomial sequence, and let L = Σⱼ₌₀⁶ aⱼ(x)Dʲ, with deg aⱼ ≤ j, be a polynomial-coefficient differential operator for which LPₙ = λₙPₙ. We classify the systems whose least non-scalar spectral differential order is six. The proof first establishes, without positivity or a measure representation, the leading-coefficient theorem a₆(x) = Cσ(x)³ with C ≠ 0 and deg σ ≤ 2. The constant, linear, double-root quadratic, and split quadratic geometries are then treated separately. The first three reduce respectively to the Hermite, Laguerre, and generalized-Bessel Pearson equations. In the split quadratic case, exact local rank calculations, endpoint-defect elimination, and a singular-Jacobi analysis leave four Jacobi-type strata. Together with one Laguerre-type stratum, these give five classes, all described explicitly by their functionals, sixth-order operators, eigenvalues, and exact quasi-definiteness loci. Exact determinants exclude spectral operators of order four, and hence establish minimality. The infinite-dimensional steps are analytic; the finite branch eliminations are certified by reproducible symbolic calculations over rational-function fields. Individual positive members of the resulting families are classical generalized Jacobi or Laguerre examples; the purpose here is the exhaustive order-six statement in the setting of arbitrary regular complex functionals. Status. The manuscript presents an internally complete proposed classification under its stated hypotheses. The exact certificate suite and an independent finite-degree reconstruction of the orthogonal polynomial sequences both pass. The result has not been independently peer reviewed, and the package does not establish historical novelty. The manuscript is circulated as a draft for specialist review. Generative-AI disclosure and author responsibility. The proof search, symbolic derivations, checking scripts, audit, and initial manuscript drafting were carried out in an extended interaction with OpenAI's GPT-5.6 Sol Pro under prompts and supervision by Jonas Matuzas. The author is responsible for the theorem statement, materials, and any errors. External specialist review remains necessary. Previously published results remain credited to their original authors. 2020 Mathematics Subject Classification: Primary 33C45; Secondary 34A05, 42C05, 68V20. Files: the compiled manuscript Bochner_Krall_Order6_Classification_v1.pdf (32 pages) and the publication package Bochner_Krall_Order6_Publication_Package_v1.zip. The package contains the manuscript; Bochner_Krall_Order6_Audit_Report.pdf (6 pages), a second adversarial audit of the classification together with its residual-risk statement, which is an internal verification report and not an external referee report; source/ and audit_source/ with the self-contained LaTeX sources and the manual bibliography; supplement/ with the exact symbolic certificates and their saved reference outputs, organized by the round in which each finite lemma was first isolated; and SHA256SUMS.txt with integrity hashes for all other release files. The computations were rerun with Python 3.13.5 and SymPy 1.14.0. From supplement/, running ./run_audit.sh, python independent_ops_check.py and python rank_locus_refined.py reproduces the round-ten repair certificates, the independent moment and OPS checks, and the refined rank loci. The independent check reconstructs the monic orthogonal polynomials from exact moments and verifies the displayed differential equations through degree eight for rational test parameters in all five classes; it is a transcription check, not a proof of exhaustiveness. Every finite decision uses integer, rational, or rational-function arithmetic; floating-point computation is not used to decide proof branches. A few heavy eliminations, among them complex_mass_compatibility.py, take substantially longer than the fast audit; their exact sources and saved outputs are included.