dm³ 103 · Week 5 — Ω — The Hexabonacci Constant

dm³ 103 · Week 05 · Constant: Ω (n=6)

Ω — The Hexabonacci Constant

Course: dm³ 103 · Constant: Ω (n=6) · Source: Book 3, Chapter Ω

The Hexabonacci sequence sums the previous six terms; dominant root \(\Omega\approx1.9837\) — also with no closed radical form, per Week 3’s Abel–Ruffini argument (degree 6, same obstruction as degree 5). \(\Omega\) is the fixed point of the recurrence ladder in a specific sense: the limit of all n-bonacci dominant roots as \(n\to\infty\) is exactly \(2\), and at \(n=6\), \(\Omega\) already sits within 1% of that limit.

The full ladder, assembled

\(n{{=}}2\): \(\varphi\approx1.618\) · \(n{{=}}3\): \(\eta\approx1.839\) · \(n{{=}}4\): \(\Delta\approx1.928\) · \(n{{=}}5\): \(\Sigma\approx1.966\) · \(n{{=}}6\): \(\Omega\approx1.984\) · \(n\to\infty\): \(\rho_\infty=2=\tau\). Six of six named rungs are now covered across this three-course sequence. The spacing between consecutive rungs shrinks roughly geometrically — each rung closes about half the remaining gap to \(\tau\).

This week’s content is grounded directly in the AXLE/Book 3/5 sources cited above — no material in this page depends on the external, unverified source removed from dm³ 102.

-- dm³ 103 · Week 05 · Ω, Hexabonacci — the ladder complete

-- Ω ≈ 1.9837 (n=6), no closed radical form (degree 6, Abel-Ruffini)
-- Full ladder: φ(1.618) η(1.839) Δ(1.928) Σ(1.966) Ω(1.984) → τ(2)
example : (2:ℝ) - 1.984 < 0.02 := by norm_num  -- within 1% of τ

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