Lagrange

Lagrange is a Partner in Arcaea, designed after the Arcaea Song Contest 2020 Grand Prize winner Aegleseeker. 
== Character ==
=== Name ===
Lagrange shares her name with Italian mathematician and astronomer Joseph-Louis Lagrange. "Joseph-Louis Lagrange" is written on the left shoulder of her attire.
Her satellite companion Charon shares the name of the moon of the dwarf planet Pluto. The code S/1978 P 1 is written on Charon's body under its right eye. This was the temporary designation given to the moon following its discovery.
=== Appearance ===
Lagrange has lavender hair tied in a loose braid, tied with a dark purple bow. Over this braid there is a black string woven into it with small moons and stars on it. She has prominent strands of hair on both sides. She also wears indigo bows to attach her strands of hair. She wears a small decoration on her hair that has colors of white and purple. She has purple eyes, and wears a monocle with decorations of gold.
She wears a yellow dress and a white jacket connected by belts, with a decoration of white and purple, similar to her hairclip. Her jacket has colors of purple, even more belts and some decorations, such as ribbons. She also wears a white loose collar on her neck.
She has a companion called Charon. Charon has eyes with purple diamonds as its pupils. It has some line decorations under its eyes from pink to purple. It also wears some kind of robotic accessories with colors of white and gold. Charon wears an accessory similar to her hair decoration and the decoration on her belt.
She also wears lavender leggings and pointy shoes without laces. She also holds a book, which has an armillary sphere on a chain hanging from its spine.
=== Personality ===
Lagrange is a calm, curious and observant person, often observing and questioning to see how the world Arcaea around her works. She is intelligent, using what she has observed to pursue further knowledge. Lagrange is also honest, unafraid to criticize this "god" harshly.
==Story==
Lagrange is the main character of the Esoteric Order Side Story.
She also appears in Ilith's Short Story Dark Ambition.
==Gameplay==
===Stats by level===
=== Skill ===
Lagrange's skill is unlocked after experiencing Aegleseeker. Her skill replaces the Score display with a Pace display that shows the distance to the closest Grade threshold, based on a projected estimate of the final score and grade.
The Pace, or projected score, is calculated via the following:
* If all note judgements so far have been PURE:
\text{Pace} = 10'000'000 + \text{Shiny PUREs}\,.
* If there has been at least 1 FAR/LOST judgement:
\begin{align}
\text{Pace}&=8'600'000 + \frac{(1.4\cdot\text{PUREs} - 3.6\cdot\text{FARs} - 8.6\cdot\text{LOSTs}) \times 1'000'000}{\text{Total notes in chart}} + \text{Shiny PUREs}\\
&= \text{Actual Score} + 8'600'000 \times \left( 1 - \frac{\text{Notes so far}}{\text{Total notes}} \right) \,.
\end{align}
The Pace display will display this projected score in terms of the nearest projected grade.
The display works as follows:
* The projected grade will be one of C, B, A, AA, EX, EX+, or PM (see table below). Scores in the D range will default to C.
* The Pace (projected score) is displayed next to the grade in terms of its distance from the grade cutoff. This display uses 6 digits, with a plus or minus depending on whether the projected score is above or below the grade cutoff.
**e.g. "+001'568 PM" means that the projected score is 10'000'000 + 1'568 = 10'001'568.
**e.g. "−045'210 A" means that the projected score is 9'200'000 − 45'210 = 9'154'790.
** The projected grade will show PM only if all notes so far have been PURE.
*** The digits of Pace will display the amount of shiny PUREs obtained.
*** The Pace will display a tinted blue if the Pure Memory obtained is the maximum theoretical score of a chart, just like the normal Score.
*** Any LOST/FAR notes will result in a displayed grade of EX+ at best, i.e. a Pace of "−???'??? PM" is not possible.
** The displayed Pace will will not decrease below "−999'999 C" (corresponding to a projected score of 7'600'001 or lower). This may create confusion where, sometimes, continuously obtaining PUREs does not increase Pace when −999'999 C Pace is reached.
The projected score ranges for each possible projected grade as as follows:
Projected score	Projected score	Projected grade	Min cutoff to achieve projected grade
Min 	Max 	Projected grade	Min cutoff to achieve projected grade
10'000'000 	- 	PM 	10'000'000
9'900'000 	9'999'999 	EX+ 	9'900'000
9'650'000 	9'899'999 	EX 	9'800'000
9'350'000 	9'649'999 	AA 	9'500'000
9'050'000 	9'349'999 	A 	9'200'000
8'750'000 	9'049'999 	B 	8'900'000
0'000'000 	8'749'999 	C 	8'600'000
=== Notes ===
The Pace for non-PM scores can be expressed in terms of the current note accuracy:
\text{Pace}=(\text{Current Accuracy} \times 10'000'000) \times \frac{\text{Notes so far}}{\text{Total notes}} + 8'600'000 \times \left( 1 - \frac{\text{Notes so far}}{\text{Total notes}} \right) + \text{Shiny PUREs}\,,
where the current accuracy is defined as in other rhythm games:
\text{Current Accuracy}= \frac{\text{PUREs} + 0.5 \cdot \text{FARs}}{\text{Notes so far}} \,.
The Pace therefore serves as a weighted average of a fixed baseline score (8'600'000, i.e. the C grade cutoff) and the expected final score based on the current accuracy (ignoring shiny PUREs).
The accuracy-based estimated score is weighted more heavily as the chart progresses.
Unlike Score, the Pace can only increase with PURE note judgements. LOSTs and FARs will both decrease the Pace (if over "−999'999 C").
The amount by which a single LOST/FAR decreases the projected score depends on the current projected score, and the chart's total note count.
Due to the sizes of the score ranges for each grade, only projected scores below C will actually use the hundred-thousands digit of the displayed Pace.
As a consequence of the above, unless if the projected Grade is C, EX or EX+, the largest absolute value of the displayed numerical Pace is ±150'000.
Lagrange's skill functions differently from DORO*C's skill, which calculates the maximum possible score rather than a projected score.
==Trivia==
*Lagrange is the only partner in the game that is required to unlock their signature song, Aegleseeker.
**She is also the first partner to have their skill unlocked by playing a song.
*Lagrange, like Maya, and Kanae, was created based on an Arcaea Song Contest Grand Prize Winner.
*Lagrange is the second partner whose skill is based around 'Grade'; the first partner to do this is DORO*C.
**She is also the second partner to alter parts of the play screen UI, the first being Awakened Lethe.
*Her skill is the only locked skill in the game that depends on a player's local save file.
**As such, players who have experienced Aegleseeker on one device to unlock her skill is required to play Aegleseeker again on another device if they wish to unlock her skill on another device, granted the second device has not been synced with the save file that has her skill unlocked. This property is not exhibited in any other locked skill in the game.
==Artwork==
Aegleseeker.jpg|Aegleseeker jacket art
Far Away Light.jpg|Far Away Light jacket art
Seclusion.jpg|Seclusion jacket art
Divine Light of Myriad.jpg|Divine Light of Myriad jacket art
9-5.jpeg|Story Entry 9-5
45.png|Full body artwork
45 icon.png|Icon in the partner select screen
Pack Esoteric Order.png|Esoteric Order pack icon