Milestone task, two parts. Part 1 — by hand: starting from the contact form \(\alpha = dz - p\,dq\) on \(M = S^1 \times \mathbb{{R}}\), verify that the Reeb vector field \(R = \partial_z\) satisfies \(\alpha(R) = 1\) and \(\iota_R\, d\alpha = 0\), and show that because \(z\) is compactified as \(z \in \mathbb{{R}}/2\pi\mathbb{{Z}}\), the flow \(\phi_t(z) = z+t\) is periodic with minimal period \(T^*=2\pi\). This is exactly Theorem π.1 from Week 5 — the milestone is reproducing the three-line argument yourself, not looking it up.
Part 2 — the honesty check: write one paragraph distinguishing three separate claims about \(2\pi\) in this framework: (a) \(T^*=2\pi\) as a definitional structure field in AXLE_v6.lean — trivially true by assignment; (b) \(T^*=2\pi\) as a geometric consequence of Reeb-flow compactification — argued in prose (Book 3), reproduced by you in Part 1, not yet formalized in Lean; (c) downstream numeric facts like tau_embodiment : tau^6 = 2^6 — fully proved, 0 sorry, in Lean. Confusing these three is the single most common way a framework like this gets oversold. Keeping them separate is the actual skill this milestone is testing.
| Claim | Status |
|---|---|
| \(T^*=2\pi\) as structure data | Definitional (trivial) |
| \(T^*=2\pi\) from Reeb-flow compactification | Argued in prose — not yet in Lean |
| \(\tau^6=2^6\), spectral period facts | Proved, 0 sorry |
Submission: your hand derivation plus the one-paragraph status note, timestamped. Phase II is now closed — Week 9 opens Phase III with the same discipline applied to \(\varphi\).
AXLE_v6.lean — theorem names and Lean identifiers above point at the actual source files.-- dm³ 101 · Week 08 · Milestone I — π closed out -- -- Reproduce by hand (not in Lean): Theorem π.1's three-line -- argument (α(R)=1, ι_R dα=0, z-compactification ⇒ T*=2π). -- -- Then classify each claim: -- T_star := 2*Real.pi -- definitional -- Reeb-flow argument -- prose-argued, not Lean-formalized -- tau_embodiment : tau^6=2^6 -- proved, 0 sorry -- -- Submit: hand derivation + one-paragraph status note. example : True := trivial