The Koide C₃ Closure — Tautology, Saddle Point, and the Real Problem: The spacing is automatic. The potential is a saddle. The amplitude is the question.

The Koide C₃ Closure — Tautology, Saddle Point, and the Real Problem: The spacing is automatic. The potential is a saddle. The amplitude is the question. This paper is part of the HOWL research archive—a collection of physics papers exploring integer fraction derivations across multiple domains using exact arithmetic and automated comparison. Abstract The Koide formula K = (mₑ + m_μ + m_τ) / (√mₑ + √m_μ + √m_τ) ² = 2/3 has held for charged leptons to six significant figures since 1982. The natural attempt to derive it is through C₃ symmetry: the three phases in the Koide parametrization √mₖ = M (1 + a cos (θ₀ + 2πk/3) ) are equally spaced at 120°, which looks like the ground state of a frustrated C₃ potential. This paper proves the C₃ path is dead, with two independent arguments either of which is sufficient. First, the 120° spacing is a tautology — three parameters (M, a, θ₀) fitting three data points (mₑ, m_μ, m_τ) is an exactly determined system, and ANY three positive masses have this form. The spacing is a property of the parametrization, not of the physics. Second, K = 2/3 is a saddle point of the Koide ratio evaluated on the C₃ phase landscape — perturbing the phases in one direction increases K, in another direction decreases it. The C₃ potential does not select K = 2/3 as a preferred value. The path is doubly dead: its success (120° spacing) is tautological, and its failure (saddle point, no amplitude selection) is on the only thing that matters — why a² = 2 for charged leptons. The real problem is the amplitude. Every known reformulation of K = 2/3 (as a = √2, CV (√m) = 1, Var (√m) = Mean (√m) ², midpoint of allowed range) is algebraically equivalent. None is a derivation. The open question is: derive a² = 2 from physics, or explain why charged leptons satisfy it while quarks do not (a²down = 2. 39, a²ᵤp = 3. 09). Falsification Criteria All papers in this archive are subject to falsification through direct comparison to published experimental measurements. Each derived value is tested against independent data with explicit PASS/FAIL criteria. Any derived value that fails its comparison is documented and published alongside the successes. Research Context This archive documents an ongoing research program in integer fraction physics. The methodology is: derive values from gauge group integers using exact fraction arithmetic, compare to published measurements, and document all results including failures. The archive spans multiple physics domains connected through the soliton boundary framework described in the constituent papers. Package Contents manuscript. md: The complete derivation and supporting analysis. README. md: Navigation, dependencies, and citation (Registry: HOWL-PHYS-23-2026). Dependencies: HOWL-PHYS-1-2026, HOWL-PHYS-10-2026, HOWL-PHYS-11-2026, HOWL-PHYS-12-2026, HOWL-PHYS-13-2026, HOWL-PHYS-14-2026, HOWL-PHYS-15-2026, HOWL-PHYS-17-2026, HOWL-PHYS-18-2026, HOWL-PHYS-19-2026, HOWL-PHYS-2-2026, HOWL-PHYS-20-2026, HOWL-PHYS-21-2026, HOWL-PHYS-22-2026, HOWL-PHYS-6-2026, HOWL-PHYS-7-2026, HOWL-PHYS-8-2026, HOWL-PHYS-9-2026 Motto: Derive. Compare. Publish. Status: Complete