dm³ 103 · Week 01 · Review

Review — The Ladder So Far: π, φ, μ, η, Δ

Five constants, two proved theorems, one honest open problem, entering the home stretch
dm³ 103 · Week 01 · Review
Review — The Ladder So Far: π, φ, μ, η, Δ
Course: dm³ 103  ·  Review  ·  Source: dm³ 101–102

Precise recap, since this is the final course and precision compounds. \(T^*=2\pi\): structure data, geometric derivation prose-argued. \(\varphi\approx1.618\): proved subcritical, no dedicated AXLE Lean file. \(\mu_{{\max}}=-2\): proved by two independent derivations (fold double-root; toy-ODE linearisation). \(\eta\approx1.839\): proved via Cardano’s closed form, three Lean files, 0 sorry. \(\Delta\approx1.928\): quartic root, depth-supercritical yet potential-subcritical. Theorem C.1 (\(c^*=3\)): derived fully by hand in 102. \(r^*=0.77594059\): certified numerically to 8dp; the general asymmetry theorem is still an axiom.

What 103 actually covers
\(\Sigma\) (Pentanacci) and \(\Omega\) (Hexabonacci) close out the individually-named n-bonacci ladder. \(\rho\) (spectral radius) opens a genuinely different, real research thread on the Collatz/Syracuse transfer operator, including an honestly-labelled open problem (“Bridge 0”). The course closes with \(\tau=2\) itself and AXLE v6.1’s own audited status: 8 verified constants, 9 honestly-named open sorrys, 0 axioms beyond Mathlib4.
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 01 · Carried forward from 101-102
--   μ_max=-2 (proved, 2 derivations)   ε₀=1/3 (proved)
--   η≈1.839 (Cardano, 3 files 0 sorry)  Δ≈1.928 (quartic root)
--   Theorem C.1: c*=3 (derived by hand)  r*=0.77594059 (certified 8dp)
example : True := trivial
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