dm³ 103 · Week 6 — The Limit — n-bonacci → τ = 2 as n → ∞

dm³ 103 · Week 06 · Constant: τ

The Limit — n-bonacci → τ = 2 as n → ∞

Course: dm³ 103 · Constant: τ · Source: Book 3, Theorem Ω.1 (proved)

Theorem Ω.1 (proved). \(\lim_{{n\to\infty}}\rho_n=2\), where \(\rho_n\) is the dominant root of the n-bonacci polynomial \(x^n-x^{{n-1}}-\cdots-x-1=0\). Full derivation: factor as \(x^n\bigl(1-\tfrac1x-\tfrac1{{x^2}}-\cdots-\tfrac1{{x^n}}\bigr)=0\). For \(x>1\) fixed and \(n\to\infty\), the bracketed finite geometric sum converges to the infinite geometric series \(\sum_{{k=1}}^\infty x^{{-k}} = \tfrac{{1/x}}{{1-1/x}} = \tfrac{{1}}{{x-1}}\). So in the limit, the equation becomes \(1-\tfrac{{1}}{{x-1}}=0 \Rightarrow x-1=1 \Rightarrow x=2\).

Why this is a genuine limit, not a numerical trend

This isn’t “the numbers happen to be getting closer to 2” — it’s a proved statement about a sequence of exactly-defined algebraic numbers converging to a specific value, with the convergence mechanism (geometric series summation) fully exhibited. The same \(\tau=2\) is independently the embodiment threshold in the canonical triple \((T^*,\mu_{{\max}},\tau)=(2\pi,-2,2)\) from AXLE_v6.lean, and the same 2 that appears in tau_embodiment : tau^6 = 2^6. Three independent appearances of the same number is worth noticing, though — consistent with 102’s discipline — worth stating precisely rather than treating as automatically more significant than a single well-proved fact.

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 06 · Theorem Ω.1, full derivation

-- x^n - x^(n-1) - ... - x - 1 = 0
-- ⟺ x^n(1 - 1/x - 1/x² - ... - 1/x^n) = 0
-- As n→∞ (x>1 fixed): bracket → 1 - 1/(x-1)  (infinite geometric series)
-- ⟹ 1 - 1/(x-1) = 0 ⟹ x = 2

-- Same τ=2 appears in AXLE_v6.lean: tau_embodiment : tau^6 = 2^6
example : (2:ℝ)^6 = 64 := by norm_num

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