J_Collatz Invariant Explorer

Im(τ) = v₂(context) / (a₀ + 1) · Gödel-encoded Collatz sequence analysis
Starting Number
Quick Examples
Sequence Statistics
length—
odd steps—
even steps—
Im(τ) start—
Im(τ) final—
net log ratio—
mean Im(τ)—
max Im(τ)—
The Two Rules (a₁ = 0 case)

E(a₀): Im → Im · (a₀²−1)/a₀²

even step; a₀ = #factors of 2

O(k): Im → Im · 1/(k+1)

odd step; k = v₂(3n+1)

—

E(2) = ×3/4   E(3) = ×8/9

O(1) = ×1/2   O(2) = ×1/3

All ratios are rational.

Key Findings
1. Arg invariance
arg(τ) = π/2 always when a₁=0. The sequence lives on the imaginary axis almost everywhere.

2. Rational grammar
Im(τ) transitions form a finite-alphabet string rewriting system over ℚ.

3. Terminal attractor
All sequences end at Im(τ) = 1/2 (at n=2). This is the fixed point.

4. Not a Lyapunov fn
E[ratio per step] ≈ 1.39 > 1. Im(τ) does not decrease monotonically — convergence comes from log(n), not Im(τ).
Im(τ) = v₂(context) / (a₀ + 1) over iteration steps
log Im(τ) — cumulative log-ratio walk
Transition ratio at each step (odd=pink, even=blue)
step n type a₀ a₁ v₂(ctx) Im(τ) rule ratio
Histogram: Im(τ) values in sequence
Im(τ)[i+1] vs Im(τ)[i] — phase portrait
Rule frequency (E vs O types)
Cumulative ratio product (net Im(τ) drift)