Core Mechanics and Scoring
The game begins by generating twelve independent fair bits. These bits are processed to produce an ordered list of maximal equal-bit run lengths. For example, a sequence of bits might produce a run list such as [4, 8] or [1, 2, 3, 6].
Scoring is determined by counting the number of complete ordered full binary parenthesizations of the run list. A valid tree must satisfy the condition that every internal node has a strictly smaller total length on its left branch than on its right branch. Leaves retain their original order, and every run is used exactly once. A single run in the list yields a score of zero. If there is no valid complete tree, the raw score is zero.
Raw Score Examples
| Run List | Raw Score |
|---|---|
| [4, 8] | 1 |
| [8, 4] | 0 |
| [1, 2, 3, 6] | 3 |
Modes and Scoring Powers
Players select a raw-score power mode before the round begins. The available modes are 1, 2, and 3. Each mode corresponds to a different mathematical transformation of the raw score.
Exact Unconditional Means by Mode
| Raw-Score Power | Exact Unconditional Mean |
|---|---|
| 1 | 0.626953125 |
| 2 | 1.927734375 |
| 3 | 7.5 |
These values represent the exact unconditional means for the respective raw-score powers across all 4096 equally likely complete words.
Controls, Caps, and Stopping Rules
The game offers specific controls for managing the live round. A single whole captured stake is used; there are no row shares.
Auto-Cap Limits
Players preselect a raw-score power and an automatic reveal cap, which can be set to 4, 8, or 12. These selections are locked during the live round. The cap determines the maximum number of semantic reveals before the system automatically banks the current quote.
Manual Banking and Reveals
Each semantic reveal follows a fixed delay of 650ms. Players may manually bank the current quote at any time while the round is live. The captured stake, power, and cap remain locked during the round. Sound controls remain available independently. There is no Auto10 queue; refill is available in an idle-only state.
Zero Potential Settlement
If the conditional potential for a future score becomes zero, the round settles at zero earlier than the cap is reached. The quote can rise or fall during the reveal process.
Payout Calculation and Gross Return
The live gross bank quote is calculated based on the captured whole stake and the statistical properties of the revealed runs. The formula is:
- Live Gross Quote = 0.96 × Captured Whole Stake × (Exact Conditional Mean of Terminal Powered Score / Exact Unconditional Mean)
The quote fluctuates as more bits are revealed. If the conditional potential settles to zero, the payout is zero. All 4096 equally likely complete words are enumerated in the underlying mathematics.
Expected Gross Return
The expected gross return under the bounded stopping rules is exactly 96%. This is calculated before integer settlement with preserved fractional carry. This percentage does not guarantee net profit in a single session.
Mathematical Evidence and Simulation
The following table presents simulation measurements from 1,000,000 rounds for each mode and auto-cap combination. These are supporting sample observations and not the mathematical payout law. The exact gross RTP remains 0.96 for all configurations.
| Mode index (0/1/2 = powers 1/2/3) | Auto Cap | Exact Gross RTP (ratio) | Measured RTP (%) | Policy |
|---|---|---|---|---|
| 0 | 4 | 0.96 | 95.96803588829849 | Bank at cap or earlier zero conditional value |
| 0 | 8 | 0.96 | 96.00070818689349 | Bank at cap or earlier zero conditional value |
| 0 | 12 | 0.96 | 96.17591745828824 | Bank at cap or earlier zero conditional value |
| 1 | 4 | 0.96 | 95.9419175686869 | Bank at cap or earlier zero conditional value |
| 1 | 8 | 0.96 | 96.0273772156897 | Bank at cap or earlier zero conditional value |
| 1 | 12 | 0.96 | 96.22776374427421 | Bank at cap or earlier zero conditional value |
| 2 | 4 | 0.96 | 96.07712330011776 | Bank at cap or earlier zero conditional value |
| 2 | 8 | 0.96 | 95.66615839944734 | Bank at cap or earlier zero conditional value |
| 2 | 12 | 0.96 | 96.14968320009302 | Bank at cap or earlier zero conditional value |
Wallet, Persistence, and Privacy
Wallet Mechanics
- Initial Balance: The demo wallet starts with a balance of 1000.
- Restoration: A manual idle-only restoration sets the balance to exactly 1000. This is eligible when the balance is below 10 and the user is idle. It is not an automatic replenishment or a deposit.
- Fractional Carry: Fractional carry persists across settlement, refill, and reload.
- Settled Balance: The settled balance persists locally, including if it reaches zero.
- Unfinished Round: If the page reloads during an unfinished round, it ends without refund or resume.
Storage and Privacy
This demo 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, but actual browser request capture is pending. There is no account login flow or payment flow in this demo. Virtual coins have no cash value.
Frequently asked questions
What happens if the conditional potential becomes zero?
If the conditional potential for a future score becomes zero, the round settles at zero earlier than the auto-cap is reached. The quote can rise or fall during the reveal process, but a zero conditional potential locks the outcome at zero.
How is the fractional carry handled?
Fractional carry persists across settlement, refill, and reload. This means that any fractional value remaining after a settlement is retained in the wallet for future use, rather than being rounded down immediately.
Can I resume an unfinished round after reloading?
No. If the page reloads during an unfinished round, it ends without refund or resume. The balance state persists, but the specific round state does not.
The public HTML also declares a Statinside script (https://statinside.com/counter.js) and a Statinside noscript pixel. These declarations are not a complete network capture or evidence of retention policies.
