(* Reconstructed complete Wolfram Language handle for 012.
   Historical inline drift source survives in origin.md. The conjugacy and
   Lyapunov checks were added on 2026-08-13 to make the recorded four-field
   verdict rerunnable as one exact source. *)
ClearAll[theta, xx, logistic, exact, machine, deltas, firstDrift,
  firstSeparation, conjugacyResidual, lyapunov, results];

logistic[z_] := 4 z (1 - z);

conjugacyResidual = FullSimplify[
  TrigExpand[logistic[Sin[Pi theta]^2] - Sin[2 Pi theta]^2],
  Assumptions -> Element[theta, Reals]
];

lyapunov = Integrate[
  Log[Abs[4 - 8 xx]]/(Pi Sqrt[xx (1 - xx)]),
  {xx, 0, 1},
  GenerateConditions -> False
];

exact = NestList[logistic, N[1/5, 200], 70];
machine = NestList[logistic, 0.2, 70];
deltas = Abs[exact - machine];

(* Position is one-based and NestList includes step zero. *)
firstDrift = First@First@Position[deltas, d_ /; d > 10^-3, 1, 1] - 1;
firstSeparation = First@First@Position[deltas, d_ /; d > 0.1, 1, 1] - 1;

results = <|
  "conjugacy residual" -> conjugacyResidual,
  "Lyapunov exponent" -> lyapunov,
  "first visible drift step" -> firstDrift,
  "first separation step" -> firstSeparation
|>;

Print[results];
Print["claim holds: ",
  TrueQ[
    conjugacyResidual === 0 &&
    lyapunov === Log[2] &&
    firstDrift === 45 &&
    firstSeparation === 52
  ]
];