L222: Unique Ruler scoring

Draw three independent uniform integer gaps g1,g2,g3 from1 through6. Form marks0,g1,g1+g2,g1+g2+g3. Compute all six distances between distinct pairs of marks. If all six distance values differ, raw score is the total span g1+g2+g3; if any values repeat, score0. Distances from different pairs are still required to differ. This is one whole captured stake, without row shares or partial-prefix payouts.

Gaps1,2,4 produce marks0,1,3,7 with six distances1,3,7,2,6,4, all distinct, and score7. Gaps1,2,3 have distinct neighboring gaps but repeated distance3, so score0. Gaps1,1,1 also score0. Players do not choose the gaps or adjust an already sampled ruler.

Of216 ordered triples,120 score0 and96 score positively. Scores6,7,8,9,10,11,12,13,14,15 have counts2,6,8,18,10,18,10,12,6,6. Linear, squared and cubed modes do not change positive frequency. A positive gross credit may be below its captured stake and need not be a net gain. Earlier rulers do not predict new independent gaps.

For p=1,2,3, gross multiplier=0.96×rawScore^p/mean(rawScore^p). Exact unconditional means are4.7592592592592595,53.27777777777778,620.4814814814815. Each mode has96% expected gross credit of captured stake, not winning frequency or a guaranteed final wallet. Zero remains zero. Squared and cubed are powers, not score multiplied by2 or3.

Capture stake and mode and debit once before480ms settlement. Settings and refill lock during settlement and an active Auto queue. Auto schedules at most10 rounds when balance permits. Stop cancels future rounds; an already begun round finishes and settles. There is no early cashout, resampling or refund of an active round.

A new browser wallet starts at 1000 virtual coins. Settled integer balance and fractional payout carry are saved in localStorage. A gross credit is added to any saved fraction; the integer part is credited and the remaining fraction is carried forward. These coins have no cash value, deposits, withdrawals or prizes. Storage belongs to this browser and origin, not to an account or an external payment balance.

Only an idle balance below ten offers Free refill. Clicking it sets the balance to 1000; it does not add 1000 to the existing balance. The fractional carry is preserved. Refill is unavailable during settlement or an active automatic queue. Changing the stake cannot force a refill while the balance is ten or more. A previously saved zero balance remains zero after reload until the user clicks the available refill.

Reload preserves saved balance and fractional carry when browser storage remains available. A draw debits its stake before the short settlement animation: leaving or reloading during that unfinished animation can discard its pending credit. Reload after settlement preserves the completed result. Clearing cache alone may not delete localStorage; removing site storage can erase the wallet. Private browsing and browser settings can change storage availability and retention.

Enumeration of216 gap triples verifies Unique Ruler and its normalization only. It does not establish an operator, licence, application, payment ability, privacy, legal eligibility or another game’s mathematics. Domain age, search ranking, licence logos, HTTPS and unidentified support replies do not establish business identity.