dm³ 103 · Week 08 · Milestone

Milestone VI — What “τ = 2 Is Reached” Actually Means

Separating the proved ladder limit from the interpretive language around it
dm³ 103 · Week 08 · Milestone
Milestone VI — What “τ = 2 Is Reached” Actually Means
Course: dm³ 103  ·  Milestone  ·  Source: Weeks 6–7 synthesis

Milestone task: reproduce Theorem Ω.1’s derivation from Week 6 by hand (the geometric-series limit argument), then write a short, precise statement of what “the system reaches \(\tau=2\)” means and doesn’t mean. It means: the n-bonacci ladder’s dominant roots converge to 2 as \(n\to\infty\) (proved), and \(\tau=2\) is a load-bearing structure constant with real downstream consequences in AXLE_v6.lean (proved, given the structure). It does not mean — without further argument this course hasn’t made — that contact curvature, the ladder, and any other named construction all provably converge as one unified phenomenon; that stronger claim was removed in Week 7 precisely because its previous form depended on unverified external axioms.

Why this milestone matters for the rest of 103
Weeks 9–16 build \(\rho\) (spectral radius) and the course’s closing synthesis on top of this exact distinction. Getting sloppy about “reached” here would compound into overclaiming later. State it precisely now.
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 08 · Milestone VI — precise statement of "τ=2 reached"
--
-- Reproduce Theorem Ω.1's geometric-series limit argument by hand.
-- Then state precisely:
--   PROVED: lim ρ_n = 2 (ladder); tau_embodiment and consequences
--           (given τ=2 as structure data)
--   NOT PROVED (removed, Week 7): any unified 3-way convergence
--   theorem bundling contact curvature with the ladder limit
example : True := trivial