Wavenumber 6: The Orthogenetic Stability Generator of Nested Infinities

Wavenumber 6 derives the orthogenetic stability generator of nested infinities from first principles in contact geometry. The derivation proceeds in three steps: (1) the orthogenetic recurrence w(k+3) = w(k+2) + w(k+1) + w(k) is the minimal recurrence consistent with three generative axes on a contact 3-manifold; (2) the geometric ansatz w(k) = η⁻ᵏ yields the Tribonacci polynomial P(η) = η³ − η² − η − 1, with dominant real root η ≈ 1.839286755; (3) the weight η⁻ᵏ geometrizes both the Hilbert-space inner product ⟨j|k⟩ = η⁻ᵏδⱼₖ and the geometric Born rule PG(k) = |cₖ|²η⁻ᵏ.

The dimensionless stability threshold g = 33 appears across three empirically distinct domains: Wigner crystallization of the electron gas (r*s ≈ 30–40), coherent conformational waves in neuronal microtubules (minimum stable segment 33 × 8 nm = 264 nm), and Fibonacci lock-in in plant phyllotaxis (8–13 primordia). Each domain carries an explicit falsifiability condition.

Wavenumber m = 6 is derived as the fundamental azimuthal mode selected by the orthogenetic generator on compact azimuthal topology: the only mode that closes consistently under the 3-step recurrence with forward/backward symmetry satisfies m = 2 × 3 = 6. The Saturn north polar hexagon is recorded as an interpretive consistency, not a derivation.

On the contact distribution ker α, non-Abelian anyons (Ising and Fibonacci) supply the microscopic realization. Three independent frameworks converged on the same Tribonacci spine within three weeks (February–April 2026): this paper (contact 3-manifold and Reeb flow), Palmer (2026, PNAS, rational quantum mechanics, invariant set theory), and Navrátil (2026, geometric quantum mechanics, SL(3,Z) Tribonacci algebra). The convergence is structural.

All operator algebra is formally stated in Lean 4 via the AXLE engine (github.com/TOTOGT/AXLE). Companion matrix and characteristic polynomial are verified without sorry. Monster Reflection Lemma, Monster Regeneration Theorem, club filter properties, and Poincaré-to-Collatz correspondence are formally stated with honest admits (open proof obligations, not false theorems).

This paper is part of the Principia Orthogona series (Volume IV continued) and is a companion to The Number 33 (Zenodo 10.5281/zenodo.19431918) and the DNLS paper (Zenodo 10.5281/zenodo.20026942).