License: Creative Commons Attribution 4. 0 International (CC BY 4. 0) Keywords: constructive reverse mathematics, binary quadratic forms, trace-zero sublattice, totally real cubic fields, GL2 (Z) form class, Weil lattice, Gram matrix, number theory, constructive mathematics, BISH, exotic Weil class, quadratic resolvent, Artin conductor, computational number theory Related Identifiers: - Is supplement to: Paper 65, "Self-Intersection Patterns Beyond Cyclic Cubics" (CRM Series) - Is part of: Constructive Reverse Mathematics Series, Papers 1-66- References: Papers 50-53 (Atlas of Exotic Weil Classes), Papers 56-58 (Weil lattice Gram matrices) Description: This deposit contains the paper, computation scripts, dataset, and figures for Paper 66 of the Constructive Reverse Mathematics (CRM) series. SCIENTIFIC CONTENT For a totally real cubic number field F over Q, the trace-zero sublattice Lambda₀ = x in OF: Tr₅/ₐ (x) = 0 is a rank-2 positive-definite Z-lattice equipped with the trace pairing. Its Gram matrix G satisfies the determinant identity det (G) = 3 * disc (F), and its GL₂ (Z) -equivalence class is a well-defined arithmetic invariant of F. This invariant resolves the form-class question left open by Paper 65 for non-cyclic (S₃ Galois group) cubic fields. The paper establishes four results: Theorem A (Determinant Identity): For any totally real cubic F with a monogenic integral basis (index OF: Z[alpha] = 1), the integer Gram matrix of the trace-zero sublattice satisfies det (G) = 3 * disc (F). The proof proceeds via the Schur complement of the 3x3 trace matrix projected orthogonal to the Lefschetz component (the constant function 1), with the factor of 3 arising from the change-of-basis determinant between the rational Schur complement basis and the integral kernel basis. Theorem B (Cyclic Reduction): For cyclic cubics of conductor f, the trace-zero form reduces to 2f * (1, 1, 1), the hexagonal form x² + xy + y² scaled by 2f. This is forced by the Z/3Z Galois action on Lambda₀, which acts via the standard 2-dimensional representation and constrains the Gram matrix to be a scalar multiple of the unique invariant form. Theorem C (Non-Cyclic Uniqueness): Among the 51 non-cyclic totally real cubics with disc (F) = 3. 9, SymPy >= 1. 14, NumPy >= 1. 26, Matplotlib >= 3. 9 To reproduce: python3 p66computeᵥ2. pyTo compile paper: pdflatex paper66. tex (two passes) NOTES This paper was produced with AI assistance (Anthropic Claude) for computation and manuscript preparation. The author is not a domain expert in algebraic number theory; all claims are supported by exact computation or explicit proof. No Lean 4 formalization is included. The results are computational, verified by exact symbolic arithmetic over Z. A Lean 4 formalization of the determinant identity would require formalizing trace forms and Schur complements over Z-lattices, which is beyond the current scope of the Mathlib library.
Paul Chun-Kit Lee (Mon,) studied this question.