PAR Sheet: Crash Minimal¶

Submission status: pre-cert. The analytical derivation below tracks games/crash-minimal/src/outcome.ts and settle.ts. The 10M-round regression run, the tail-distribution confidence interval, and independent-reviewer sign-off remain open before GLI-11 submission.

The analytical specification of the Crash Minimal game: rules, probability model, payout model, RTP derivation, volatility, and change-management rules. A cert lab (GLI, BMM, iTech Labs) would test against the numbers below using a multi-million-round simulation and compare observed behaviour against the theoretical distribution declared here.

A change to any of the parameters in this document, or to packages/rng-core/src/ (shared primitives) or games/crash-minimal/src/outcome.ts (game-specific mapping), triggers re-certification in a real deployment.

Machine-readable companions: - par-sheet.json, probabilities, payouts, RTP and volatility as structured JSON (diff-friendly for certification drift detection). - rng-test-vectors.md + ../fixtures/rng-test-vectors.json: fixed (serverSeed, clientSeed, nonce, config) → (crashMultiplier, u) tuples. TBD, generate with scripts/generate-test-vectors.ts crash-minimal before submission.

Game description¶

Crash Minimal is a single-multiplier crash game: every round produces one crashMultiplier ∈ [1.00, maxMultiplier] with a heavy-tailed distribution. The player commits to a cashoutMultiplier up front; wins if crashMultiplier ≥ cashoutMultiplier. See game-rules.md for the complete rule set.

Distribution¶

Given uniform u_raw ∈ [0, 1) from the first 52 bits of the HMAC:

P(bust at 1.00)               = houseEdge
P(crashMultiplier ≥ K | no bust) = 1 / K     for K ∈ (1.00, maxMultiplier]
P(crashMultiplier ≥ K)        = (1 − houseEdge) / K

The 1/K survival function is the classic Bustabit-family crash curve; the pre-roll bust branch is the standard way to inject house edge without distorting the tail.

RTP¶

For any cashout multiplier K ∈ (1.00, maxMultiplier]:

RTP(K) = K · P(crashMultiplier ≥ K)
       = K · (1 − houseEdge) · (1/K)
       = 1 − houseEdge

The RTP is identical for every cashout choice: this is the defining property of the 1/K crash curve. Players cannot improve their expected return by picking a different cashout; they can only change the variance.

`houseEdge`	Theoretical RTP	House edge
0.00	100.00 %	0.00 %
0.01 (default)	99.00 %	1.00 %
0.03	97.00 %	3.00 %

The maxMultiplier clamp subtracts a negligible amount from the theoretical RTP because the truncated tail mass is (1 − houseEdge) / maxMultiplier; at the default maxMultiplier = 1000 this is < 0.001: below cert-lab measurement noise.

Volatility¶

`cashoutMultiplier`	Hit frequency	StdDev of return per unit stake	Classification
1.50	0.660	≈ 0.91	low
2.00	0.495	≈ 1.00	medium
5.00	0.198	≈ 1.99	high
10.00	0.099	≈ 2.98	very high

Values above use houseEdge = 0.01 and ignore the maxMultiplier clamp. Precise values for submission are computed by the regression run against the production code path.

Max exposure per bet¶

maxPayoutMicro = maxBetMicro · maxMultiplier − commissionMicro

At the default maxMultiplier = 1000 and a maxBetMicro set by the operator, the max exposure per bet is 1000× the stake. Operators cap this via maxBetMicro and, optionally, per-player per-round concurrency limits (not implemented in this repo).

How cert labs test this¶

A GLI-19 / ISO/IEC 17025 cert lab would run the following tests against the specification above. The repo ships the regression test at tests/games/crash-minimal/rtp-regression.spec.ts.

RTP regression¶

Simulate 10,000,000 rounds at the declared houseEdge using the production RNG path (determineCrashOutcome in games/crash-minimal/src/outcome.ts).
For each round, score a fixed basket of cashout multipliers (e.g. 1.50×, 2.00×, 5.00×, 10.00×) and aggregate observed RTP per bucket.
Assert each bucket is within ±0.5 pp of 1 − houseEdge.

Tail-distribution goodness-of-fit¶

Bucket outcomes by multiplier range and compute a chi-square against the theoretical P(crashMultiplier ∈ [a, b)) = (1 − houseEdge) · (1/a − 1/b).
Assert p-value > 0.01.

Bust-rate audit¶

Count rounds with crashMultiplier == 1.00.
Assert observed bust rate matches houseEdge within ±3σ.

Determinism¶

For fixed (serverSeed, clientSeed, nonce, config), assert the outcome is bit-identical across runs, processes, and machines.

Change management¶

The following changes require re-certification. The repo encodes this policy in CI: any PR touching these files fails without a CERT-ATTEST-CRASH_MINIMAL: line in the PR body.

File	Reason
`packages/rng-core/src/`	Shared commit-reveal primitives, also invalidates every other game's cert
`games/crash-minimal/src/outcome.ts`	Crash-specific derivation: 52-bit uniform, house-bust branch, 100/(1-u) mapping
`games/crash-minimal/src/settle.ts`	Win condition and payout arithmetic
`games/crash-minimal/src/config.ts`	`houseEdge` / `maxMultiplier` schema and defaults
This file	Declared probabilities / RTP are the cert-lab input