Formulas and Calculations

A product described as “rounded” rounds half values toward positive infinity.

Time constants

Symbol	Meaning	Value
$D_y$	Days per year	365
$A_0$	Fresh-life age	14 years
$L_0$	Base lifespan	70 years
$S_0$	Base game speed	4 in-game days/real second

Age display

$$\operatorname{age}(d)=\left\lfloor \frac{d}{365}\right\rfloor$$

where $d$ is accumulated in-game days.

XP requirement

XP needed for the next level

$$\operatorname{maxXP}(b,\ell)= \operatorname{round}\left(b(\ell+1)(1.01)^\ell\right)$$

$b$ is the task’s base maximum XP and $\ell$ is its current non-negative integer level. Ordinary abilities use $b=100$. Resonance I–IV use the calibrated values $0.00030536056011010876$, $0.00003972362997298969$, $0.000004881366014841863$, and $0.00000045896650664729693$. Class bases are listed in the class catalog.

Ability level	XP needed
0	100
10	1,215
100	27,319
1,000	2,098,011,479

XP requirement calculator

Current level

XP required: —

When progress crosses a level boundary, excess XP carries forward. Up to 1,024 levels can be awarded in one fixed update, with the requirement recalculated after every new level; remaining excess continues on later updates. Extreme finite values saturate instead of overflowing.

XP gain and stacking order

Each task starts with 10 base XP per simulation unit. Every applicable modifier is multiplied, then the final result is rounded once:

$$\operatorname{xpGain}=\operatorname{round}\left(10\prod_i m_i\right)$$

The universal factors are:

1. task maximum-level bonus: $1+\text{maximum level}/10$;
2. one additive mastery multiplier containing every applicable mastery-item percentage;
3. Happiness;
4. Dark Influence;
5. Demon Training;
6. Void Influence;
7. Parallel Universe.

Classes then multiply Productivity, Dark Knowledge, and Personal Squire. Abilities instead multiply Concentration, Book, Study Desk, Library, Void Blade, Void Symbiosis, Universe Fragment, and Evil Incarnate.

One category-specific branch is applied afterward: Military class XP, Strength XP, Magic ability XP, Void Manipulation XP, Arcane class XP, Void class XP, Galactic class XP, Dark Magic XP, Almightiness XP, or Fundamentals XP. Dark Magic multiplies by current Evil; Almightiness multiplies by current Essence.

The expansion multiplier block is applied after the category-specific branch, in this order:

1. Essence VII multiplies all XP by 1.10. Essence I then multiplies ability XP by 1.05, or Essence II multiplies class XP by 1.05.
2. Doctrine II applies its selected class- or ability-XP branch. Crown of Null affects classes; Shadow Discipline and Blackstar Grimoire affect abilities. Root Key additionally affects Metaverse classes.
3. Doctrine IV’s all-XP benefit or drawback is applied, followed by Final Revision, Deathless Momentum, and Blinded By Darkness.
4. Magic abilities then receive Arcana Unbound and Enchanted Orb ranks. Darkness abilities receive the Darkness IX multiplier.
5. Great tasks finally receive $2^n$, where $n$ is the number of completed Great-XP Heroic milestones, up to 12.

Every applicable term multiplies; none of these percentage labels are combined additively unless its own formula says so.

Mastery bonuses do not multiply one another. If the applicable item percentages are $p_1,\ldots,p_n$, their single multiplier is:

$$M_{\text{mastery}}=1+\sum_{i=1}^{n}\frac{p_i}{100}$$

The current Copper, Silver, Gold, Platinum, Emerald, or Ruby value replaces the item’s previous tier value. See Mastery Items for the complete effect routing.

Mastery tier thresholds

Normalized work through class level $L$ is:

$$N(L)=\sum_{k=0}^{L-1}(k+1)(1.01)^k$$

For an item’s route level $P$, define $T=\max(N(15),2N(P))$. Copper, Silver, Gold, Platinum, Emerald, and Ruby target $T$, $2T$, $4T$, $8T$, $16T$, and $32T$. Each displayed requirement is the first integer $L$ whose $N(L)$ reaches its target. Only the separately recorded worked-class peak participates; current and inherited maximum levels do not.

Ability effects

Linear ability

$$E_{\text{linear}}(\ell,c)=1+c\ell$$

$c$ is the ability coefficient from the ability catalog.

Expense-reduction ability

$$E_{\text{expense}}(\ell)= \max\left(0.1,\ 1-\frac{\log_7(\ell+1)}{10}\right)$$

The multiplier is clamped to a minimum of 0.1. Bargaining, Intimidation, Brainwashing, Abyss Manipulation, and Galactic Command each use this formula and multiply together.

Level	Expense multiplier
0	1.000000
10	0.876773
100	0.762830
1,000	0.644960

Logarithmic boost

$$E_{\text{boost}}(\ell,b)=1+\log_b(\ell+1)$$

Time Warping uses $b=13$; Life Essence uses $b=33$.

Cosmic Recollection is the special unshifted effect:

$$E_{\text{recall}}(\ell)=0.00065\ell$$

At level 1 it retains floor(0.00065 × maximum) during Transcend; the result can be zero.

Darkness ability

$$E_{\mathrm{dark}}(\ell,c)=1+c\log_{10}(\ell+1)$$

The six Darkness coefficients and channels are listed in Darkness and Collapse.

Great tasks and possessions

For Great rank $r$ and Great level $\ell$:

$$b_{\mathrm{Great}}=10^{10}(1.7)^r, \qquad \operatorname{maxXP}_{\mathrm{Great}}= \operatorname{round}\left(b_{\mathrm{Great}}(\ell+1)(1.01)^\ell\right).$$

A Great ability evaluates its effect at:

$$\ell_{\mathrm{effective}}=\max(P_{\mathrm{normal}},2000)+100\ell_{\mathrm{Great}}.$$

Great Cosmic Recollection alone caps its resulting effect at 2. For an original paid possession with ordinary expense $X_0$, ordinary effect $M_0$, and source-order rank $r$:

$$X_{\mathrm{Great}}=\max\left(10^6X_0,10^{20}10^{r/3}\right), \qquad M_{\mathrm{Great}}=1+4(M_0-1).$$

Class income

Class income

$$I=\operatorname{round}\left( I_0\left(1+\log_{10}(\ell+1)\right) M_{\text{wealth}}M_{\text{category}}M_{\text{mastery-income}} \right)$$

This multiplies base income $I_0$, the level term $1+\log_{10}(\ell+1)$, Demon’s Wealth, any applicable category income effect, and the additive mastery-income multiplier. Military adds Strength; Arcane Association adds All Seeing Eye. The final product is rounded once.

With base income 100 and no additional multipliers:

Level	Copper/day
0	100
10	204
100	300
1,000	400

Expenses, Happiness, and net display

Each active possession’s base daily expense is multiplied by the five expense-reduction ability effects, then rounded. Total expense is the property expense plus every active miscellaneous expense and, while enabled, the Shop Steward fee.

$$X_i=\operatorname{round}\left(X_{0,i} E_B E_I E_{Br} E_A E_G\right)$$

Happiness multiplies:

$$H= \frac{E_{\text{Meditation}}\, I_{\text{Butler}}\, I_{\text{Multiverse Fragment}}\, I_{\text{Property}}} {E_{\text{Mind Seize}}}$$

The Shop Steward fee is $0.10I$, and its recommended loadout budget is $0.90I$. Net magnitude is $\left|I-X\right|$; the sidebar prefixes it with + when income covers expenses and − when expenses are larger, with matching positive or negative styling.

Lifespan and game speed

$$L=70\cdot365\cdot E_{\text{Life Essence}} E_{\text{Astral Body}} E_{\text{Ceaseless Abyss}} E_{\text{Cosmic Longevity}} E_{\text{Higher Dimensions}}$$

Lifespan has no extra rounding or cap.

When time warping is enabled, actual warp is:

$$W_{\text{actual}} = E_{\text{Time Warping}}+ \left(E_{\text{Temporal Dimension}}E_{\text{Time Loop}}\right)$$

The sidebar display instead multiplies all three effects, so it can differ from the speed that actually advances the game.

Effective game speed is $4W_{\text{actual}}$ while alive and unpaused, otherwise zero. Essence V multiplies effective game speed by 1.10; this milestone bonus is not part of the sidebar’s displayed-warp number.

Rebirth rewards

Touch Eye’s next-life per-task multiplier is:

$$M_{\text{task}}=1+\frac{\max(\ell,\ell_{\max})}{10}$$

There is no rebirth-count-based global XP multiplier. Evil gain multiplies Evil Control, Blood Meditation, Absolute Wish, Void Embodiment, Yin Yang, the additive mastery Evil-gain multiplier, and Essence VI’s 1.15 multiplier.

Essence gain multiplies Yin Yang, Essence Collector, the additive mastery Essence-gain multiplier, Essence IV’s 1.10, Essence IX’s 1.15, and 1 + 0.1 × Essence Bargain rank. Transcend gains another ×1.5 after current Essence reaches 30,000 and completes Essence III.

Echoes and Evil perks

With remembered Resonance peak $p_i$ and target $T_i\in\{1000,1250,1500,1750\}$, matching Echo efficiency is:

$$E_i(p_i)= \begin{cases} 0.25+0.75p_i/T_i,&p_i\le T_i,\\ 1+0.05\log_2(p_i/T_i),&p_i>T_i. \end{cases}$$

Echo XP uses the target ability’s complete current XP gain multiplied by $E_i(p_i)$, including Great XP when the target is promoted. See Echoes and Resonance for routing rules.

With Evil $V$, Essence $S$, and Essence VIII multiplier $M_8\in\{1,1.25\}$:

$$\operatorname{EPP/day}= \frac{\log_{10}(1+V)\left(1+0.25\log_{10}(1+S)\right)}{365}M_8.$$

For a base age $A$, perk rank $r$, and reduction $d$ years per rank, the effective requirement is $\max(0,A-rd)$. Current maximum ranks produce Touch Eye 15, Embrace Evil 100, Void 100, and Galactic Council 1,000.

Collapse and Dark Orbs

For current Essence $S$, $x=S/(5\times10^{10})$, and the product of all Dark Matter multipliers $M_{\mathrm{DM}}$:

$$\operatorname{Collapse}(S)= \begin{cases} 0,&x<1,\\ \left\lfloor x^{1.12}(1+0.15\log_{10}x)M_{\mathrm{DM}}\right\rfloor,&x\ge1. \end{cases}$$

Extreme positive results saturate at the largest finite value. The Dark Orb generator’s next-rank cost and rate are:

$$C_{\mathrm{generator}}(r)=\left\lceil3^r\right\rceil,$$ $$\operatorname{Orbs/day}=0.01(2^r-1)(1+0.25\sqrt{D})M_{11},$$

where $D$ is lifetime Dark Matter and $M_{11}$ is 1.5 after Darkness XI, otherwise 1. For an Orb upgrade with base cost $B$, the next rank costs $B4^r$.

Metaverse and Hypercubes

For gross Dark Matter earned this run $R$, $y=R/1000$, and Metaverse gain multiplier $M_{\mathrm{MP}}$:

$$\operatorname{Metaverse}(R)= \begin{cases} 0,&y<1,\\ \left\lfloor y^{1.1}(1+0.1\log_{10}y)M_{\mathrm{MP}}\right\rfloor,&y\ge1. \end{cases}$$

With Hypercube Altar rank $a$, active item multiplier $M_i$, and perk multiplier $M_p$:

$$\operatorname{Hypercubes/day}=0.03M_iM_p(1.25)^a, \qquad \operatorname{cap}=10{,}000(1+\mathrm{lifetime\ MP})^2.$$

An altar with base cost $B$ costs $B3^r$ for rank $r+1$. Reality Boost duration is $60+10d$ active seconds, and its cooldown is $\max(120,600(0.95)^c)$ active seconds.

Empire career score

$$C=p\ell_{\text{peak}}^2$$

$p$ is the class career-prestige value, equal to that class’s base max XP. $\ell_{\text{peak}}$ is the highest level reached while that class was worked.