Airlock Sequence Rules: Eight Symbols and Three Directed Transitions

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Eight symbols and seven adjacent transitions

One round samples eight independent symbols, each uniformly chosen from A, B and C. The entire sequence is sampled once, kept hidden and revealed one symbol per second. Track the seven adjacent directed transitions between consecutive revealed symbols.

Terminal raw reward is one if AB, BC and CA have each appeared at least once among those seven transitions; otherwise it is zero. They need not appear consecutively or in that order. Reverse transitions AC, CB and BA do not count. Repeated identical symbols do not count. No closing transition from the final symbol back to the first is added. One captured stake covers the whole round.

Directed edges rather than a closing cycle

ABCABCAB contains AB, BC and CA and has terminal reward one. AAAAAAAA contains none and has terminal reward zero. ACBACBAC contains only reverse cyclic transitions and has terminal reward zero.

Once all three required transitions have appeared, every unseen completion qualifies. Before that, the quote reflects exactly how many unseen completions can still supply missing transitions. The terminal indicator one is a raw reward, not a payout of one coin or a guaranteed net profit.

Exact 6,561-sequence conditional quotes

There are 6,561 equally likely eight-symbol sequences. Exactly 1,668 contain all three required directed transitions, giving initial expected raw reward 1,668 divided by 6,561. At each revealed prefix, calculate the exact conditional mean terminal reward over all equally likely unseen completions.

The quote is 0.97 times that conditional mean divided by the initial mean. It starts at 0.97 and may rise or fall. Collection credits captured stake times the current quote as gross virtual return. The stake was already deducted: gross return is not net profit. Apply no second 0.97 factor.

This finite conditional-value process gives 97% expected gross return for permitted stopping policies using revealed information only. It is not a session guarantee or 97% winning chance. The maximum quote is 0.97 times 6,561 divided by 1,668, approximately 3.81545. Hidden symbols are not rerolled when computing a quote.

First crossing and full-quote settlement

Choose automatic collection target 1.2, 1.5, 2, 3 or 5 before starting. The first revealed quote reaching or exceeding the target collects at its actual full value, including an upward jump beyond the target. Payment is not capped at the selected target.

Target five is unreachable because the maximum quote is below four. A zero conditional mean ends the round immediately with zero credit. If no earlier collection or zero occurs, the eighth symbol settles automatically. Manual collection is available at revealed prefixes, including the initial 0.97 quote. Time between reveals adds no unseen information.