Binary Representation
x = 127
2-adic fuel: 7 bits
×3
+1
÷2
7
Fuel (trailing 1s)
0
Danger steps (v₂=1)
127
Current value
7
Initial fuel
SAFE
DANGER: 100%
MAX FUEL
Trajectory Log
#
Value
Fuel
v₂
Hensel's Lemma for Collatz: Starting from odd x,
the next d half-steps T(x) = (3x+1)/2 all produce odd results
(i.e., v₂ = 1 at each step)
if and only if 2d+1 | (x + 1).
In binary, x must end in at least d + 1 consecutive 1-bits. Each dangerous step consumes one bit of fuel. Among odd numbers, only 1/2d can sustain d dangerous steps.
In binary, x must end in at least d + 1 consecutive 1-bits. Each dangerous step consumes one bit of fuel. Among odd numbers, only 1/2d can sustain d dangerous steps.