Week 5 fixed \(T^*=2\pi\) as a period constant standing outside the ladder. This week introduces the ladder itself: a sequence of seven named constants, each the dominant root \(\rho_n\) of an n-bonacci characteristic polynomial \(x^n - x^{{n-1}} - \cdots - x - 1 = 0\), climbing from Fibonacci toward the embodiment threshold \(\tau = 2\).
Why the limit is exactly 2: write the n-bonacci polynomial as \(x^n(1 - 1/x - 1/x^2 - \cdots - 1/x^n) = 0\). As \(n \to \infty\), the bracketed sum becomes the geometric series \(1/(x-1)\) for \(x>1\), so the equation becomes \(1 - 1/(x-1) = 0 \Rightarrow x = 2\). This is Theorem Ω.1 in the source chapter: \(\lim_{{n\to\infty}} \rho_n = 2\), and that limit is identified with the same \(\tau=2\) that appears in the canonical triple from Week 4 and in the noise-tolerance identity \(\tau \cdot \varepsilon_0 = 2/3\).
Notice the spacing as you climb: \(\eta - \varphi \approx 0.221\), \(\Delta - \eta \approx 0.089\), \(\Sigma - \Delta \approx 0.038\), \(\Omega - \Sigma \approx 0.018\). Each rung closes roughly half the remaining gap to \(\tau=2\) — a geometric convergence rate, not a linear one. Hexabonacci (\(\Omega \approx 1.984\)) is already within 1% of the limit after only six terms.
AXLE_v6.lean — theorem names and Lean identifiers above point at the actual source files.-- dm³ 101 · Week 06 · n-bonacci ladder, from AXLE_v6.lean + Book 3 -- g6 = τ^5 + 1 = 33 (crystal arithmetic, already proved, Week 4-adjacent) -- theorem tau_embodiment : tau^6 = 2^6 -- proved (AXLE_v6.lean) -- The ladder itself (ρ_n for n=2..6, and the n→∞ limit) is stated -- and numerically verified in the Book 3 chapter (chapters-pi-phi- -- mu-eta-delta-sigma-omega.html, Theorem Ω.1); a general Lean proof -- of lim ρ_n = 2 for arbitrary n is not yet in AXLE — an open -- formalization task, not a closed theorem, distinct from the -- already-proved specific facts about τ=2 itself. example : (2:ℝ)^6 = 64 := by norm_num