dm³ 102 · Week 13 — G⁵ — Fifth Iteration and Convergence Rate

dm³ 102 · Week 13 · Synthesis

G⁵ — Fifth Iteration and Convergence Rate

Course: dm³ 102 · Synthesis · Source: AXLE_v6.lean + Weeks 2–4

101 covered \(G^2\); this course has now defined \(\mu\), \(\eta\), \(\Delta\) on top of \(\pi\), \(\varphi\) — five of the nine named G1–G9 constants. At a fixed point \(x^*\), \(n\) applications of \(G\) contract deviations at rate \(n\mu_{{\max}}\) in the exponent (Week 2): \(\xi(t) \sim \xi_0 e^{{n\mu_{{\max}} t}}\). For \(G^5\) with \(\mu_{{\max}}=-2\): \(\xi(t) \sim \xi_0 e^{{-10t}}\) — a perturbation that took \(t\approx1.15\) to lose 90% of its size under a single \(G\) now loses that same 90% in \(t\approx0.23\), five times faster, exactly as the linear contraction-rate scaling predicts.

Distance to τ, stated carefully

“Closer to \(\tau=2\)” needs to mean something specific here, and it’s not the ladder rungs themselves (those are fixed constants \(\varphi,\eta,\Delta,\ldots\), not moving under iteration) — it’s the orbit under repeated \(G\) that gets closer to the fixed point \(\Gamma^*\) exponentially fast, at the rate set by \(\mu_{{\max}}=-2\), independent of which ladder rung you’re studying at a given phase of the course. Don’t conflate the two convergences: the n-bonacci ladder’s convergence (\(\rho_n\to\tau\) as \(n\to\infty\), a sequence of distinct constants) and \(G\)’s orbit convergence (a single trajectory approaching \(\Gamma^*\) under repeated application) are structurally similar (both exponential, both tied to the same \(\mu_{{\max}}=-2\)/\(\tau=2\) machinery) but are different mathematical objects.

This week’s content is grounded in AXLE_v6.lean and Weeks 2–4 of this course cited above — no material in this page depends on the external, unverified source removed from earlier drafts of this course.

-- dm³ 102 · Week 13 · G⁵ convergence rate

-- Single G: ξ(t) ~ ξ₀·e^{μ_max·t}, μ_max=-2
-- G⁵:       ξ(t) ~ ξ₀·e^{5·μ_max·t} = ξ₀·e^{-10t}   -- 5x faster in the exponent

-- Two DIFFERENT convergences, both real, not the same object:
--   ladder:  ρ_n → τ=2 as n→∞  (sequence of distinct constants φ,η,Δ,...)
--   orbit:   G^k(x) → Γ* as k→∞  (single trajectory, rate μ_max=-2)
example : (5:ℝ) * (-2) = -10 := by norm_num

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