Final assessment, two parts, mirroring the honesty discipline built up since Week 1. Part 1 — operator-chain essay (600–900 words): explain \(G = U \circ F \circ K \circ C\), state which pieces are proved (Theorems A–D) versus open (Theorem T1, entropy), and explain precisely how \(\pi\) and \(\varphi\) relate to that chain — as a period constant and a subcritical ladder rung respectively, not as additional operators, per the framing notes in Weeks 5 and 9. Grading rewards precision about what is proved versus argued over enthusiasm about the framework’s scope.
Part 2 — Lean proof sketch: submit a hand-written (not necessarily machine-checked) proof sketch of Fibonacci ratio convergence \(F_{{n+1}}/F_n \to \varphi\), mirroring the structure of the existing TribonacciRatioConvergence.lean file (Week 12) with \(n=2\) in place of \(n=3\). This closes the diagnostic gap identified in Week 12 at the level this course can reasonably ask for — a real proof structure, even if not yet formally verified in the AXLE codebase.
| Constant | Value | Status |
|---|---|---|
| \(T^*\) | \(2\pi\) | Structure data; Reeb-flow argument prose-argued |
| \(\varphi\) (n=2) | 1.618… | Subcritical, proved (classical); no AXLE Lean file yet |
| \(\mu_{{\max}}\) | −2 | Proved (double-root derivation) |
| \(\eta\) (n=3) | 1.839… | Critical rung; proved, 0 sorry (3 Lean files) |
| \(\varepsilon_0\) | 1/3 | Proved (symmetric analytical bound) |
| \(r^*\) | 0.77594059 | Value certified (8dp); general asymmetry theorem still an axiom — AXLE #13 |
| \(\tau\) | 2 | Proved; ladder limit as \(n\to\infty\) |
Portfolio submission: essay + proof sketch + this table, annotated with your own check marks. dm³ 102 picks up at the criticality bridge in full — the Collatz correspondence, and the still-open Gronwall asymmetry problem.
FibonacciRatioConvergence.lean exercise are the same task at two levels of rigor. If you want the milestone to be more than an assignment, finish the Lean file for real, get sorry_count: 0, and open a pull request against AXLE alongside your portfolio submission. The inner_basin_is_asymmetric axiom (AXLE #13 — the numeric value \(r^*=0.776\) is certified, but the general theorem isn’t yet proved from first principles) and the Reeb-flow-in-Lean gap from Week 7–8 are two more open items in the same repo, for whenever you’re ready to go past Fibonacci — this course’s open gaps are real project gaps, and closing one is a legitimate way to finish it.
AXLE_v6.lean — theorem names and Lean identifiers above point at the actual source files.-- dm³ 101 · Week 16 · Milestone II — full constants audit -- -- Submit alongside essay + proof sketch: -- T* = 2π (structure data; Reeb argument prose-argued) -- φ (n=2) ≈ 1.618 (subcritical; classical proof; no Lean file) -- μ_max = -2 (proved, double-root derivation) -- η (n=3) ≈ 1.839 (critical; proved, 0 sorry, 3 files) -- ε₀ = 1/3 (proved, symmetric bound) -- r* = 0.77594059 (certified value, 8dp; asymmetry -- theorem itself still an axiom — AXLE Issue #13) -- τ = 2 (proved; ladder limit n→∞) -- -- Course complete. dm³ 102 begins at the criticality bridge. example : True := trivial