Pak Super: Trace rules and Closed Bridge Energy

Each captured drop begins at position0 and takes twelve independent fair left-or-right steps. After row i the position is x_i. A trace whose final position x_12 is not0 scores0. A returning trace scores the sum x_1²+...+x_12². The entire trace is generated at launch and settles after its animation. No path decisions, cashout or endpoint paytable are offered.

The alternating trace RLRLRLRLRLRL returns to zero and scores6: its odd-row positions are1 and its even-row positions0. The trace RRRRRRLLLLLL also returns and scores146. Twelve consecutive right steps finish at12, so score0 despite their large intermediate positions. Only the final-return gate admits the energy score.

Twelve fair binary choices give4096 equally likely words. Exactly924 return to zero and have a positive score; the other3172 score0. A positive gross credit need not exceed the captured stake. Linear, square and cube preserve the same returning words and positive frequency, while changing their normalized credit amounts. Previous drops do not change this probability law.

The alternating trace RLRLRLRLRLRL returns to zero and scores6: its odd-row positions are1 and its even-row positions0. The trace RRRRRRLLLLLL also returns and scores146. Twelve consecutive right steps finish at12, so score0 despite their large intermediate positions. Only the final-return gate admits the energy score.

Twelve fair binary choices give4096 equally likely words. Exactly924 return to zero and have a positive score; the other3172 score0. A positive gross credit need not exceed the captured stake. Linear, square and cube preserve the same returning words and positive frequency, while changing their normalized credit amounts. Previous drops do not change this probability law.

Before a launch choose exponent p=1,2 or3. Gross multiplier=0.96 × score^p / mean(score^p). The exact means are 5.865234375, 246.33984375, 15199.3359375. Each mode therefore has expected gross credit96% of its captured stake. This is not a96% winning chance or a guaranteed final-wallet balance. Zero stays zero, and saved fractional carry accompanies integer settlement.

One whole stake prices one trace, with no row shares. At most six paid drops run concurrently. Each retains its own stake and mode at launch; changing these controls affects later launches only. A seventh launch is disabled until space is available. Auto schedules up to ten launches if the balance can pay. Stop cancels only future launches: already priced drops still settle. Refill remains locked while any drop or queue is active.

Enumerating the4096 direction words and checking the normalization establishes only the stated Closed Bridge Energy model. It does not certify external randomness, a commercial catalogue, a licence or another service’s safety. This guide provides no outside audit, official endorsement or prediction of real-money outcomes.