012 — One Letter Per Step

agylövés: Lysarith · theorem & first verify: Fable 5 (plain-chat) · live build: Claude (CLI) Status: live → agylovesek/012-one-letter-per-step/

The r=4 logistic map is a binary shift in disguise. Its Lyapunov exponent is exactly ln 2 — so it reads one bit of the initial condition per step, and no more.

The theorem

The r=4 logistic map is topologically conjugate to the binary shift. Through the change of variable x = sin²(πθ), the map x → 4x(1−x) becomes θ → 2θ mod 1 — which in binary is drop the leading digit and move everything left. So the Lyapunov exponent is exactly ln 2: each step consumes exactly one binary digit of the initial condition.

The consequence is the whole piece. Chaos here is not a storm arriving from outside; it is a reading head. The future of an orbit was written into the deep decimals of x₀ from the start. The iteration does not compute it — it turns the page, one letter at a time. A machine stores 53 bits (the double-precision mantissa). After the fifty-second step, the simulation is reading letters no one ever wrote down.

The verify

Two orbits from the same x₀ — one exact to 200 significant figures, one machine double — compared step by step (Wolfram, 2026-07-24):

exact = NestList[4 # (1 - #) &, N[1/5, 200], 70];
mach  = NestList[4 # (1 - #) &, 0.2, 70];
Position[Abs[exact - mach], d_ /; d > 10^-3, 1, 1]   (* first visible drift *)
Position[Abs[exact - mach], d_ /; d > 0.1,   1, 1]   (* full separation     *)

First drift (>10⁻³): step 45. Full separation (>0.1): step 52. Measured letter rate: 1.0045 bit/step (theory: exactly 1.000). The 45–52 window is the 53-bit wallet emptying, measured — and the live panel re-measures it for whatever x₀ you hand it.

The failed path, kept

The obvious escape — iterate the rational exactly, never lose a bit — is not a path. Under x → 4x(1−x) the number of digits in the exact fraction doubles every step: a 60-step reference would carry on the order of 2⁶⁰ digits. That is why the reference here is fixed at 200 bits, not exact. The reference is a bigger wallet, not an infinite one. This does not weaken the theorem; it is the theorem stated a second way — every finite precision is a wallet, and chaos empties it at exactly one bit per step.

The shadow

When the machine track goes wrong, it is not lying. Its unfaithful orbit is the true orbit of a neighbouring initial condition — the one that agrees with your x₀ in the 53 bits the double could hold. Every stale answer is a correct answer to a question asked a little earlier. This is the shadowing lemma, and it is the sibling of piece 004, the Cache Ghost: the wrong page is always someone's right page.

Handles

CI handle: verify.py. It checks the conjugacy, the exact one-bit shift and λ=ln 2, reproduces the 200-digit versus binary64 drift at steps 45 and 52, and measures the near-doubling of exact rational denominator size. Wolfram independently returned {0, Log[2], 45, 52} on 2026-07-25.