Milestone task, two parts. Part 1: starting only from \(V(q)=q^3-cq\), derive \(c^*=3\) as the unique value producing a double root at \(q=1\) — the full argument from Week 7, reproduced without looking it up: find \(V'(q)\), solve for the critical point, impose \(q^*=1\), solve for \(c\), verify \(V''(1)\neq0\), factor \(V(q)+2\). Part 2: write one paragraph connecting this to the Collatz map precisely — not just “Collatz has \(c=3\) too” as a slogan, but stating exactly what “coefficient\(=3\)” means for the odd branch \(T(n)=3n+1\) and why that identification is meaningful rather than superficial (hint: it’s the same linear coefficient in both the continuous potential and the discrete map, not merely two unrelated appearances of the digit 3).
-- dm³ 102 · Week 08 · Milestone III — Theorem C.1 by hand -- -- Part 1: reproduce the c*=3 derivation without reference material. -- Part 2: one paragraph — what "Collatz has c=3" precisely means. -- -- Submit: your derivation + paragraph. example : True := trivial