Run Merge Grammar: Rules, Modes, and Limits

Twelve independent fair bits produce an ordered list of maximal equal-bit run lengths. A single run has score zero. You count complete ordered full binary parenthesizations of this run list for which every internal node has strictly smaller total length on its left than on its right. Leaves keep their original order; every run is used exactly once. If there is no valid complete tree, the raw score is zero.

• Raw examples: [4,8] gives 1; [8,4] gives 0; and [1,2,3,6] gives 3.
• Score powers: Raw scores are raised to powers 1, 2, or 3.
• Unconditional means: For raw-score powers 1, 2, and 3, exact unconditional means are 0.626953125, 1.927734375, and 7.5.

You preselect a raw-score power (one, two, or three) and an automatic reveal cap (4, 8, or 12). These settings lock during the live round.

All 4096 equally likely complete words are enumerated. The policy is to bank at cap or earlier if the conditional value becomes zero.

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. The quote can rise or fall; zero conditional potential settles zero. Expected gross return under bounded stopping rules is 96%, before integer settlement with preserved fractional carry. This does not guarantee net profit.

Each semantic reveal follows 650ms. Manual bank settles the current conditional quote. The selected cap (4, 8, or 12) automatically banks the current quote; zero future potential settles zero earlier. No hidden future bit is exposed before its reveal. Controls include one whole captured stake with no row shares. Captured stake, power, and cap are locked during the live round. Sound remains available. Refill is idle-only; there is no Auto10 queue.

The initial balance is 1000 virtual coins. Manual idle-only restoration sets the balance to exactly 1000 when the balance is below 10. Fractional carry persists across settlement, refill, and reload. Settled balance persists locally, including zero. An unfinished round ends on reload without refund or resume. Stake, mode, and refill are locked while active.

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