One curve rules them all: how long a roulette ladder survives

Your chance of finishing a session without losing the ladder depends on how many cycle profits your bankroll can absorb, not on which numbers you cover.

Chapter 5 of the spinner roulette research (chapters 0 to 9): One curve rules them all.

We put 53 ladders on one chart: Martingale on red, right-raise ladders on dozens, six-lines, corners and single numbers, and the widening and stacking ladders of the next chapter. Across: how many winning cycles a session plays, divided by how many cycle profits the bankroll can pay back. Up: the share of sessions that never lost the ladder.

Every ladder sits on or under one ceiling curve, and none beats it. Ladders with the same bankroll counted in profits tie, whatever the board. Where layout matters at all the effect is small, and covering more numbers or widening is slightly worse: the best layout beats red-style ladders by 3.3 points.

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The numbers

Share of sessions that never lost the ladder, for all 53 ladders (x = session length ÷ ladder cost)
Ladder	Group	x	Never capped	e^(−x)	k
e3a-L5	Martingale on red	7.85	0.0%	0.0%	1.15
e3a-L6	Martingale on red	3.86	0.7%	2.1%	1.18
e3a-L7	Martingale on red	1.92	10.2%	14.7%	1.21
e3a-L8	Martingale on red	0.95	31.6%	38.7%	1.24
e3a-L9	Martingale on red	0.48	54.8%	61.9%	1.27
e3a-L10	Martingale on red	0.24	74.0%	78.7%	1.31
e3b-L8-S250	Martingale on red	0.48	56.7%	61.9%	1.24
e3b-L8-S1000	Martingale on red	1.91	8.4%	14.8%	1.24
e3b-L8-S2000	Martingale on red	3.82	0.6%	2.2%	1.24
e3b-L10-S250	Martingale on red	0.12	86.7%	88.7%	1.31
e3b-L10-S1000	Martingale on red	0.48	53.3%	61.9%	1.31
e3b-L10-S2000	Martingale on red	0.95	29.0%	38.7%	1.31
e4-corner-g	Inside bets	2.35	6.0%	9.5%	1.10
e4-corner-x2	Inside bets	42.53	0.0%	0.0%	1.04
e4-dozen-g	Outside bets	1.90	10.6%	15.0%	1.16
e4-dozen-g-u5	Outside bets	0.37	63.8%	69.1%	1.22
e4-dozen-x2	Outside bets	10.38	0.0%	0.0%	1.07
e4-dozen-x2-u5	Outside bets	4.55	0.6%	1.1%	1.08
e4-sixline-g	Inside bets	2.17	7.8%	11.4%	1.12
e4-sixline-x2	Inside bets	31.79	0.0%	0.0%	1.04
e4-straights25-g	Inside bets	0.91	29.3%	40.3%	1.34
e4-straights35-g	Inside bets	0.37	24.8%	69.1%	3.79
e4-two-dozens-g	Outside bets	1.34	17.0%	26.2%	1.30
a2-sixline-streets-w27	Shapes (widen, stack, mix)	0.63	44.4%	53.3%	1.28
a3-splits-w22	Shapes (widen, stack, mix)	0.76	41.3%	46.8%	1.20
a4-straights-w17	Shapes (widen, stack, mix)	1.02	30.3%	36.1%	1.18
a5-corners-splits-w26	Shapes (widen, stack, mix)	1.50	15.2%	22.3%	1.23
a6-dozen-streets-w27	Shapes (widen, stack, mix)	1.40	18.0%	24.7%	1.25
a7-dozens-sixline-street-w33	Shapes (widen, stack, mix)	0.79	27.6%	45.4%	1.58
a8-streets-w24-x2	Shapes (widen, stack, mix)	1.89	9.3%	15.1%	1.18
a9-dozen-streets-default	Shapes (widen, stack, mix)	5.65	0.2%	0.4%	1.17
b2-corner-l42	Shapes (widen, stack, mix)	0.39	64.9%	67.7%	1.15
b4-sixline-l29	Shapes (widen, stack, mix)	0.41	61.5%	66.4%	1.17
b5-two-sixlines-l15	Shapes (widen, stack, mix)	0.37	63.9%	69.1%	1.22
b6-dozen-l15	Shapes (widen, stack, mix)	0.37	65.0%	69.1%	1.22
b7-three-sixlines-l9	Shapes (widen, stack, mix)	0.48	53.4%	61.9%	1.27
b8-red-l9	Shapes (widen, stack, mix)	0.48	55.4%	61.9%	1.27
b9-two-dozens-l6	Shapes (widen, stack, mix)	0.45	56.0%	63.8%	1.37
b10-street-l60	Shapes (widen, stack, mix)	0.22	79.5%	80.3%	1.16
b11-red-default-l7	Shapes (widen, stack, mix)	1.92	9.8%	14.7%	1.21
c1-streets-to24-stack	Shapes (widen, stack, mix)	0.63	46.7%	53.3%	1.25
c2-sixlines-to24-stack	Shapes (widen, stack, mix)	0.60	46.6%	54.9%	1.29
c3-corners-to24-stack	Shapes (widen, stack, mix)	0.60	46.9%	54.9%	1.27
c4-dozen-streets-to24-stack	Shapes (widen, stack, mix)	0.62	44.6%	53.8%	1.29
c5-dozen-to-two-dozens-stack	Shapes (widen, stack, mix)	0.22	73.8%	80.3%	1.39
c6-dozen-to-two-dozens-default	Shapes (widen, stack, mix)	6.15	0.0%	0.2%	1.19
c7-streets-to18-stack	Shapes (widen, stack, mix)	0.42	60.1%	65.7%	1.24
c10-sixlines-to12-stack	Shapes (widen, stack, mix)	0.40	62.8%	67.0%	1.21
a1-streets-w27 (3 seeds)	Shapes (widen, stack, mix)	0.47	55.1%	62.5%	1.28
b1-street-l53 (3 seeds)	Shapes (widen, stack, mix)	0.41	64.0%	66.4%	1.14
b3-two-streets-l29 (3 seeds)	Shapes (widen, stack, mix)	0.41	62.2%	66.4%	1.17
c8-sixlines-to18-stack (3 seeds)	Shapes (widen, stack, mix)	0.36	63.2%	69.8%	1.26
c9-streets-to12-stack (3 seeds)	Shapes (widen, stack, mix)	0.33	67.2%	71.9%	1.21

Takeaways

No arrangement of numbers beats the ceiling: the shape of the bet is not the lever.
The lever is bankroll counted in cycle profits: a smaller base stake or a bigger committed bankroll.
Narrow stacks sit closest to the ceiling; wide coverage and widening sit lower.

The math

Let B be the cap loss divided by the mean cycle profit τ̄ (bankroll in cycle profits) and x = p·S/B, the expected cycles in S spins per unit of B. H3 says P(no cap) ≈ e^(−x). The sharper form is P(no cap) = e^(−k·x) with k = q·B/(1 − q) = 1 + 0.027 × (turnover per cycle) / (τ̄(1 − q)) ≥ 1, so e^(−x) is a ceiling.

Over 53 ladders the mean residual against e^(−x) is −6.0 points and none is above it; e^(−k·x) predicts to 0.8 points RMS. k runs from 1.14 (one street stacked) through 1.27 (Martingale red) and 1.37 (two dozens) to 1.58 (widening to 33 numbers) and 3.79 (35 straight-ups). The median horizon is S½ = ln 2 × B / (k·p), and the median is positive exactly when e^(−k·x) > ½, which held on all 65 recovering rows. Every positive median sits beside a negative mean.

x = p × S / B
P(no cap) = e^(−k·x) ≤ e^(−x),  k = q·B / (1 − q) ≥ 1
c9: x = 0.223 × 500 / 338.8 = 0.33, k = 1.21, e^(−0.40) = 0.672 (measured 0.672)

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