How Halving Journey Scores One Finite Sequence

Six independent fair bits select ONE uniformly distributed starting integer from one through sixty-four. They do not select six independent integers or use six separate random number generators. Starting from that integer, divide an even current value by two; replace an odd current value by three times that value plus one. Repeat until the first arrival at one and stop there.

Every division by two and every three-times-plus-one operation counts as one step. An odd operation followed by division counts as TWO operations, not a combined step. Starting at one takes zero steps: do not continue the one-four-two-one cycle. Raw score is max(0, steps minus five). A trajectory that takes five or fewer operations scores zero.

Choose linear, squared or cubed score before drawing. Gross multiplier equals 0.96 times raw score raised to the chosen power divided by that power’s exact unconditional mean. The means are 21.734375, 1478.890625 and 141493.578125 respectively. Seven of the sixty-four starting values score zero; fifty-seven score positively. A positive score can still return less than stake. Each mode has 96% exact unrounded expected gross credit of captured stake, not winning frequency, a promised session result or an external product’s odds.

One whole captured stake prices the complete draw and is debited once. One round is active at a time. Stake and power controls lock during settlement and an active Auto queue. Normal animation lasts 480 milliseconds; reduced motion changes duration rather than probability. Auto schedules at most ten rounds. Stop cancels future rounds while the current round finishes and settles. There is no early collection, cashout or stake divided among rows.