v2.2.0 — Major mathematical revision. Φ definition unified as (J∘σ⋆∘J)∘ρ⁻ʳ throughout, consistent with Corollary 4.3 (dihedral pairing now uses Φ(σ⋆∘ρʳ) = (J∘σ⋆∘J)∘ρ⁻ʳ, consistent with the proof of Corollary 4.3), Definition 4.1 (algebraic introduction of J and equivalence with tail reversal), Proposition 5.2 (reformulated as conditional reduction, removing the incorrect global factor-of-2 claim), Observation 5.3.2 (reformulated as open problem with motivating heuristic), Observation 5.2 (constant-factor reduction within cubic complexity, not asymptotic), Section 5.3 Toeplitz (restricted to subfamilies satisfying offset-multiset symmetry). Notation unified: σ⋆ for canonical representative, σ̂ for companion, Φ for dihedral map throughout. Added extended verification scripts for n=6 and n=7. We introduce the ARE Method, a framework that organizes the terms of the Leibniz expansion of the determinant of an n×n matrix into orbits under the action of the cyclic group Cₙ on the symmetric group Sₙ. Each orbit contains exactly n monomials related by cyclic shifts of column indices, and is paired with a companion orbit via a dihedral involution Φ. This structure generalizes the classical Sarrus rule for 3×3 matrices to arbitrary dimension. We establish the algebraic properties of the orbital decomposition, characterize the conditions under which dihedral cancellation occurs, and provide computational verification for n = 3, 4, 5, 6, 7. The method yields the determinant and the classical adjugate simultaneously, and admits an exact arithmetic implementation without floating-point error. Limitations with respect to asymptotic computational complexity are explicitly discussed. AI Disclosure: Parts of the Python verification code were drafted with the assistance of generative AI (Claude) and then reviewed, edited, and validated by the author. No generative AI was used to generate scientific content or to create/modify figures.
Ramón Moya (Fri,) studied this question.