Scoring V3

This document specifies the ScoreV3 calculation rules used by Milthm.

ScoreV3 uses the ordinary judgment sequence $\Xi = \{\xi_1, \xi_2, \cdots, \xi_N\}$ defined in Judgment Definitions. Lightning judgments are isolated from this sequence, but Lightning Miss overflow can still affect the combo score state used by the score multiplier.

Input Symbols

These symbols are defined in Judgment Definitions and are used as inputs in this document.

Symbol	Meaning
$\mathcal{J}$	Ordinary judgment set.
$\Xi$	Ordinary judgment sequence.
$\Phi$	Lightning judgment sequence.
$\xi_i$	The $i$-th ordinary judgment.
$N$	Ordinary judgment count.
$O_f$	Lightning overflow from a Lightning Miss.

Score Symbols

This document defines the following score-specific symbols.

Symbol	Name	Meaning
$S_{\mathrm{judge}}$	Stepwise judgment score function	Maps an ordinary judgment grade to its base judgment score value.
$AS_n$	Average judgment score	Average judgment score up to the $n$-th ordinary judgment.
$B$	Base combo score mapping	Maps a judgment grade and ordinary judgment count to combo score bounds and increment.
$\Theta$	Combo score sequence	Running combo score state recorded after ordinary judgments.
$b_{\max}$	Maximum combo score	Upper bound reachable by the combo score.
$\mathring{\kappa}_n$	Base compensation multiplier	Unadjusted combo compensation multiplier up to the $n$-th ordinary judgment.
$\kappa_n$	Combo compensation multiplier	Adjusted combo compensation multiplier up to the $n$-th ordinary judgment.
$CS_n$	Combo bonus	Combo bonus up to the $n$-th ordinary judgment.
$AP_n$	AP bonus	AP bonus up to the $n$-th ordinary judgment.
$S_n$	Score	Score up to the $n$-th ordinary judgment.
$TS$	Total score	Final ScoreV3 score.

Judgment Score

The stepwise judgment score function $S_{\mathrm{judge}}: \mathcal{J} \to \mathbb{R}$ maps each ordinary judgment grade to a score value.

Grade	Judgment score
$s$	$1{,}000{,}000$
$p$	$990{,}000$
$g$	$600{,}000$
$n$	$300{,}000$
$b$	$150{,}000$
$m$	$0$

The average judgment score up to the $n$-th ordinary judgment is:

$$AS_n = \frac{\sum_{i=1}^{n} S_{\mathrm{judge}}(\xi_i)}{n}$$

Combo Compensation Multiplier

The base combo score mapping $B: \mathcal{J} \times \mathbb{N}^{+} \to \mathbb{R}^{3}$ maps a judgment grade $\xi$ and ordinary judgment count $N$ to:

• $b_{\xi,N}$: combo base score upper bound
• $d_{\xi,N}$: combo base score lower bound
• $a_{\xi,N}$: judgment increment

ScoreV3 uses the following mapping.

Judgment $\xi$	Upper bound $b_{\xi,N}$	Lower bound $d_{\xi,N}$	Increment $a_{\xi,N}$
$s$	$\min(\max(\lfloor 0.24N \rfloor, 1), 192)$	$0$	$2$
$p$	$\min(\max(\lfloor 0.24N \rfloor, 1), 192)$	$0$	$1$
$g$	$\min(\max(\lfloor 0.16N \rfloor, 1), 128)$	$0$	$0$
$n$	$\min(\max(\lfloor 0.12N \rfloor, 1), 96)$	$0$	$0$
$b$	$\min(\max(\lfloor 0.10N \rfloor, 1), 80)$	$0$	$0$
$m$	$\min(\max(\lfloor 0.08N \rfloor, 1), 64)$	$0$	$0$

ScoreV3 maintains a running combo score state $c$. The combo score sequence $\Theta$ records the initial state and the value of $c$ after ordinary judgments only:

$$\Theta = \{\theta_0, \theta_1, \cdots, \theta_N\}$$

At the beginning of a chart:

$$c = \theta_0 = b_{s,N}$$

For the $i$-th ordinary judgment, let $c_i^-$ be the value of $c$ after applying any preceding Lightning overflow events:

$$\begin{aligned} c_i^- &= c, \\ c &\leftarrow \min\left(\max\left(c + a_{\xi_i,N}, d_{\xi_i,N}\right), b_{\xi_i,N}\right), \\ \theta_i &= c. \end{aligned}$$

For a Lightning Miss with overflow $O_f > 0$, the overflow is applied to the same running combo score state:

$$c \leftarrow \max(c - O_f, 0)$$

This Lightning overflow update does not add an element to $\Theta$ and does not advance the ordinary judgment index $i$. It also updates the combo score compensation state at the current ordinary judgment index, so an overflow can reduce the final score even when it occurs after the last ordinary judgment.

The maximum combo score is:

$$b_{\max} = b_{s,N}$$

The base compensation multiplier up to the $n$-th ordinary judgment is:

$$\mathring{\kappa}_n = \frac{\sum_{i=1}^{n} \theta_i}{n \cdot b_{\max}}$$

Multiplier Adjustment

The maximum combo score recovery rate is:

$$a_{\max} = a_{s,N}$$

The predicted combo score sequence describes the maximum combo score attainable when processing an ordinary judgment:

$$\theta_i^* = \min(c_i^- + a_{\max}, b_{\max})$$

The combo score recovery sequence is:

$$P = \{\rho_0, \rho_1, \cdots, \rho_N\}$$

where $\rho_i$ is the minimum number of ordinary judgments required after the $i$-th ordinary judgment to restore the combo score to its maximum value, excluding the $i$-th ordinary judgment itself:

$$\rho_i = \max\left( \left\lceil \frac{b_{\max} - \theta_i}{a_{\max}} - 1 \right\rceil, 0 \right)$$

The combo score compensation sequence is:

$$\hat{\Theta} = \{\hat{\theta}_0, \hat{\theta}_1, \hat{\theta}_2, \cdots, \hat{\theta}_N\}$$

with $\hat{\theta}_0 = 0$. For $i \ge 1$:

$$\hat{\theta}_i = \begin{cases} \hat{\theta}_{i-1}, & i + \rho_i \le N \ \text{or}\ \theta_i = \theta_i^*, \\ \dfrac{ \left(2(b_{\max} - \theta_i) - a_{\max}(N + 1 - i + \rho_i)\right)(i + \rho_i - N) }{2}, & \text{otherwise.} \end{cases}$$

When a Lightning overflow event occurs between ordinary judgments, the same compensation rule is evaluated at the current ordinary judgment index using the current value of $c$ after the overflow. This keeps the end-of-chart score loss consistent with ordinary combo score loss.

The adjusted combo compensation multiplier up to the $n$-th ordinary judgment is:

$$\kappa_n = 0.4 + 0.6 \cdot \frac{ \max\left(\left(\sum_{i=1}^{n} \theta_i\right) - \hat{\theta}_n, 0\right) }{ n \cdot b_{\max} }$$

Combo Bonus

The combo judgment mapping $C: \mathcal{J} \to \mathbb{B}$ maps ordinary judgment grades to Boolean values:

$$C(\xi) = \begin{cases} 1, & \xi \in \{s, p, g, n\}, \\ 0, & \text{otherwise.} \end{cases}$$

The current combo sequence is:

$$\Gamma = \{\gamma_0, \gamma_1, \cdots, \gamma_N\}$$

with:

$$\begin{aligned} \gamma_0 &= 0, \\ \gamma_i &= \begin{cases} \gamma_{i-1} + 1, & C(\xi_i) = 1, \\ 0, & C(\xi_i) = 0. \end{cases} \end{aligned}$$

The running maximum combo sequence is:

$$\Gamma^* = \{\gamma_0^*, \gamma_1^*, \cdots, \gamma_N^*\}$$

with:

$$\begin{aligned} \gamma_0^* &= 0, \\ \gamma_i^* &= \max(\gamma_i, \gamma_{i-1}^*). \end{aligned}$$

The combo bonus up to the $n$-th ordinary judgment is:

$$CS_n = 5000 \cdot \frac{\gamma_n^*}{N}$$

AP Bonus

The AP bonus up to the $n$-th ordinary judgment is:

$$AP_n = \begin{cases} 5000 \cdot \dfrac{n}{N}, & \forall i \in \{1, 2, \cdots, n\},\ \xi_i \in \{s, p\}, \\ 0, & \text{otherwise.} \end{cases}$$

Lightning judgments, Lightning HP, and Lightning overflow do not change the AP bonus condition because they are not ordinary judgments.

Result Conditions

Lightning judgments do not change ordinary judgment counters and do not change ordinary combo counters. Therefore, a Lightning Miss can reduce the ScoreV3 total only through Lightning overflow applied to the combo score state; it does not affect All Perfect or All Combo conditions.

Total Score

The score up to the $n$-th ordinary judgment is:

$$S_n = \kappa_n \cdot AS_n + CS_n + AP_n$$

The total ScoreV3 score is:

$$TS = S_N$$

The theoretical maximum score is $1{,}010{,}000$, achieved if and only if every ordinary judgment is Exact and no score is lost because of Lightning notes:

$$\forall \xi \in \Xi,\ \xi = s \qquad\text{and}\qquad \kappa_N = 1$$

Therefore, Lightning judgments do not all have to be Lightning Exact; the theoretical maximum is still attainable as long as Lightning Misses cause no score loss.