Run Merge Grammar: Rules, Math, and Demo

The game generates twelve independent fair bits, producing an ordered list of maximal equal-bit run lengths. A single run yields a score of zero. The raw score is the count of complete ordered full binary parenthesizations of this run list where every internal node has a strictly smaller total length on its left than on its right. Leaves keep their original order, and every run is used exactly once. No valid complete tree results in zero.

For raw-score powers 1, 2, and 3, the exact unconditional means are 0.626953125, 1.927734375, and 7.5, respectively.

Players select a raw-score power of one, two, or three, and an automatic reveal cap of 4, 8, or 12. The captured stake, power, and cap are locked during the live round. Each semantic reveal follows a 650ms interval. Manual banking is available immediately while the game is live. There is no Auto10 queue. The selected cap automatically banks the current quote; if zero future potential remains, it settles zero earlier. No hidden future bit is exposed before its reveal.

The live gross bank quote is 0.96 times the captured whole stake times the exact conditional mean of the terminal powered score divided by its own unconditional mean. All 4096 equally likely complete words are enumerated. The quote can rise or fall; zero conditional potential settles zero. The expected gross return under bounded stopping rules is 96%, before integer settlement with preserved fractional carry. This does not guarantee net profit.

This demo uses a virtual wallet with an initial balance of 1000. Fractional carry persists across settlement, refill, and reload. Settled balance persists locally, including zero. If an unfinished round ends on reload, it ends without refund or resume. Manual restoration is available when idle and the balance is below 10, setting the balance to exactly 1000. Stake, mode, and refill are locked while active and through any auto queue; sound remains available.

The source uses browser localStorage for the settled demo balance and fractional carry. This does not establish anonymity or exclude other processing. The template declares a Google Fonts stylesheet. Actual browser request capture is pending; this is not a claim that other requests or processing are absent. There is no demo account or payment flow.

This guide targets an 18+ audience as a policy, not a claim about local law or an implemented age gate. Responsible-use pages can describe optional pauses, a personal time/virtual-coin limit, and finding appropriate local qualified help. This does not promise account-level blocking, a timer, or self-exclusion tools that the interface does not have.