Cardinals

Cardinals
An assignable value in Ordinal Markup, and when assigned can provide boosts by Aleph Power.
Prestige Layer	Collapse

Cardinals, referred to in-game by their symbol ℵ (pronounced aleph), are a mid-game unit obtained by performing a Collapse. They are powerful producers of a variety of benefits. They are used to purchase Aleph Upgrades.

Cardinal gain formula

The formula that determines the amount of Cardinals gained upon Collapsing is:

Let $x=\lfloor \max(f-24,1)^{0.5+0.8\cdot (5-\mathrm {log} _{10}{\widehat {t}})^{1.5}}\rfloor$, where $f$ is the number of Factor Boosts you have performed and ${\widehat {t}}$ is the time elapsed since your last Collapse in seconds, capped between 1000 and 100,000 (in other words, if $t$ seconds have elapsed, then the capped time is ${\widehat {t}}=\min(\max(t,1000),100000)$).

Cardinal Gain Exponent Table

Collapse Time in seconds	Cardinal Gain Exponent	Cardinal Gain (at 10,000,000,024 Factor Boosts)	Factor Boosts Require (for 1e10 Cardinal)
1,000.00	2.762742	4.24e27	4,189
1,089.29	2.7	1.00e27	5,079
1,250.50	2.6	1.00e26	7,042
1,438.75	2.5	1.00e25	10,024
1,659.24	2.4	1.00e24	14,702
1,918.35	2.3	1.00e23	22,230
2,223.96	2.2	1.00e22	35,136
2,585.82	2.1	1.00e21	57,821
3,016.13	2	1.00e20	100,024
3,530.23	1.9	1.00e19	183,323
4,147.69	1.8	1.00e18	359,406
4,893.61	1.7	1.00e17	762,723
5,800.67	1.6	1.00e16	1,778,304
6,911.69	1.5	1.00e15	4,641,613
8,285.39	1.4	1.00e14	13,894,979
10,000.00	1.3	1.00e13	49,238,851
12,166.72	1.2	1.00e12	215,443,494
14,945.77	1.1	1.00e11	1,232,846,764
18,577.85	1	1.00e10	10,000,000,024
23,444.42	0.9	1.00e9	129,154,966,526
30,198.06	0.8	1.00e8	3,162,277,660,193
40,100.50	0.7	1.00e7	193,069,772,888,350
56,234.13	0.6	1.00e6	46,415,888,336,127,813
100,000.00	0.5	1.00e5	100,000,000,000,000,000,024

If $x$ is greater than or equal to the currently pending Cardinals, your new pending Cardinals will become $x$, otherwise your pending Cardinals will remain unchanged. This means that any pending Cardinals gained at a lower Collapse time will not be lost when your Collapse time increases.

Note that, for very large values of $f$ and for values of $t$ that are less than 1000 seconds, the equation above is capable of being simplified down to approximately $x=f^{0.5+0.8\cdot 2^{1.5}}\approx f^{2.7627}$.

Also note that, if you have AU9 ("Keep 10% of Factor Boosts upon Collapse") the value of $x$ in the next Collapse will use 10% of your current value of $f$; hence, to estimate the number of Cardinals gained upon Collapse twice in a row, you should divide your current value of $f$ by 10.

Types

There are five total types of Cardinals. Those are the unassigned, ℵ₀, ℵ₁, ℵ₂ and ℵ_ω Cardinals.

Unassigned

Unassigned Cardinals are Cardinals you gain directly from Collapse. They do nothing on their own and must be assigned to either ℵ₀, ℵ₁, or ℵ₂ Cardinals.

ℵ₀, ₁ and ₂

The three ℵs each produce their respective ℵ Power at a rate of $n^{\max(2,{\sqrt {\log _{10}(n)}})}$ per second. Each ℵ Power provides a multiplier of their bonus at the rate of $\log _{10}(p+10)$, where $p$ is the amount of ℵ power you have. ℵ₀ boosts Tier 1 Automation, ℵ₁ boosts Tier 2 Automation, and ℵ₂ boosts OP gain. Some ℵ Multipliers give additional bonuses after the right Aleph Upgrade is purchased.

ℵ Power, unlike ℵs themselves, does not get lost, reset or spent at any point, meaning that spending all ℵ to get ℵ_ωs does not reset or use up any ℵ power.

The multiplier on each type of aleph is:

Changes	Content	Type	Detail	Note
	log10(ℵi + 10)	i	Base aleph power multiplier
multiply	(ℵω+1)^(1/32)	1	Aleph Upgrade 7 multiplier	If you have AU7.
multiply	1.5	all		If you have SFU52
multiply	1+0.4*Unspent Functions	2		If you have SFU71
power	max(1,2-e^(0.5-0.5*OC5 completion))	all		OC5 completion = your OC5 completion
power	0	all		If you are in Omega Challenge 5
divide	10,000	all

ℵ_ω

You can get an ℵ_ω by spending 1 of each of the 3 other ℵ. ℵ_ω does nothing on its own until you get the 7th Aleph Upgrade, but you can spend them on Aleph Upgrades.

Omega Challenge Effects

• Omega Challenge 5 effectively fixed your ℵs' power to x0.0001, equal to $\frac{1}{10000}$, meaning that:
  • Tier 1 Automation is 10,000x slower.
  • Tier 2 Automation is 10,000x slower.
  • Globally, you gain 10,000x less OP.
  • You gain 10,000x less Incrementy. (with AU5)
  • Dynamic Factor cap is divided by 10,000. (with AU6)
  • Challenge 8 goals are 10,000x higher. (with SFU63)

This also leaves your Cardinals disabled.

Trivia

• The Most Cardinal Collapsed at once variable was introduced in v0.32, which calculates AU8's effectiveness. However, if you do the 1^st Collapse in v0.32 or v0.33 (not v0.30 or v0.31, as this variable not even exist), this variable doesn't update; it will stay at 0 instead.
• The Aleph symbol is usually substituted by N in text chats.
• There used to be a Cardinal mechanic known as Cardinal Decay, where Cardinals gain is based on the current Cardinal gain instead of best Cardinal gain in a Collapse. However, it was so disliked by the community and even the developers that they had to remove the mechanic in the Singularity Update since the player always had to play actively and then Collapse in a short period. I just played to e7 Cardinal in single collapse, but the collapse run only took me around 40 minutes ~ WongTingHo05
• When It out, the time border is using 10 and 100,000 seconds, this was changed when v0.31 is released.
• Prior to Update v0.34, the Cardinal gain at base is max(x-24,2) instead of max(x-24,1), where x is your Factor Boosts, this is mean you can gain more Cardinals then expected if you have exactly 25 Factor Boosts within 18,577.9 seconds in the Collapse. (From always 1 to 2^(0.5+0.8*(5-log10(t))^1.5), where t is Collapse time in seconds)

Glitches

Note: These happened in previous versions.

Post-Collapse screenshot at v0.32, but the Most Cardinals Collapsed at once is unchanged (0).