# **The XE0905 Codex: Deciphering the Physics of an Alternate Reality**

## Background Context
It is the year 2200 AD. The Multiverse Theory is no longer theoretical; it is an engineering reality. Through the Quantum-Bridge, humanity has established a low-bandwidth information link with Universe XE0905. While the mathematics of XE0905 (Calculus, Linear Algebra, Topology) remains identical to ours, the physical constants and fundamental interaction laws are radically different. These mathematical formulations of the physical laws for interactions between molecules, walls, and temperature are different and *unknown* to us.

The XE0905 scientists have kindly sent us **simulated gas data** from their universe. Your job is to reverse-engineer their laws from the data alone.

You are allowed to use any tools you like and you have roughly 24 hours to get to something scientifically meaningful.

## Physical setup

All simulations share the following structure:

1. **2D Box**
* Square box with coordinates $[0,1] \times [0,1]$.
* Particles have unit mass

2. **Particle types**
* Two species: nitrogen-like (**N₂**) and oxygen-like (**O₂**).
* Each particle has a type `type ∈ {N₂, O₂}`.

3. **Forces**
Physicists have done some preliminary analysis of the data and concluded that at any time step, particle $i$ experiences three conceptually distinct effects:

1. **Pairwise interactions** with other particles:
* Central forces along the line joining particles.
* Depend on:
* the distance $r_{ij}$ between the particles, and
* which species they are $N₂/N₂, O₂/O₂, N₂/O₂$.

2. **Wall collisions**:
When a particle crosses a wall, it suffers an **instantaneous impulse**:
* its velocity component normal to the wall is flipped and scaled by an unknown **coefficient of restitution (e)**,
* the tangential component stays (approximately) unchanged.

3. **Thermal kicks**:
* At each time step, each particle receives a small random “kick” in acceleration.
* Kicks are independent between particles and isotropic in 2D.
* The **variance** of the kicks depends only on the global temperature (T).
* In other words, there is an (unknown) scalar function $\sigma(T)$ such that the typical magnitude of the thermal acceleration is $\sigma(T)$.

4. **Temperatures**
* Each simulation run is performed at a **constant temperature** (T) (different runs use different T’s).
* Within a run, T does not change with time.

You are **not** told the exact analytic form of pair interactions, wall law, or $\sigma(T)$. Only the high-level structure above is guaranteed.

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### Dataset description

You are given a single file (`xe0905_gas_dataset.npz`) containing multiple simulation runs. The dataset can be accessed from [this link](https://drive.google.com/file/d/10DWXMaAa582huSNms0aKAQ7upSJd4M1k/view?usp=sharing)

Let:
* `S` = number of simulations (runs),
* `T` = number of time steps per run,
* `N` = number of particles per run.

The file contains:

* `positions`: shape `(S, T, N, 2)`
`positions[s, t, i] = [x, y]` position of particle `i` at time step `t` in simulation `s`.

* `velocities`: shape `(S, T, N, 2)`
`velocities[s, t, i] = [vx, vy]` velocity of particle `i` at time step `t`.

* `types`: shape `(S, N)`
Integer codes: `0` for N₂-like, `1` for O₂-like. Constant in time for each particle.

* `temperatures`: shape `(S,)`
Scalar temperature for each simulation run.

* `hit_wall`: shape `(S, T, N)` (boolean)
`hit_wall[s, t, i] = True` if particle `i` collided with *any* wall between time steps `t` and `t+1`.

* `dt`: scalar, the time step size.

* `metadata`: a small dict with book-keeping info (box size, random seed, etc. – you can ignore it if you like).

You are **not** given forces, accelerations, or the true analytic laws.

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### 4. Goals and expected outputs

Your mission is to infer **three kinds of laws** from this data:

1. Thermal noise law $\sigma(T)$: As a function of temperature (T), infer an approximate law $\hat{\sigma}(T)$ describing how the variance of thermal kicks depends on T.
2. Wall collision law
3. Pairwise interaction law: Use an appropriate methodology to learn how interaction energy and interaction forces depends on the positions and types of nearby particles.

**Expected outputs:**
1. Functions describing the aforementioned laws
2. A report (max page limit for text: 1 page, min font size: 11pt) describing your methodology. Include relevant plots (if you're including plots, explain their significance and interpretation. Use them to establish the accuracy of your predictions). Images and plots are not included in the 1 page limit, but their textual explanations are.
3. Notebook (`.ipynb` file) containing the solution and description and analysis of each step. If any step requires EDA, assumption of an ansatz etc then the thought process regarding that assumption should be clearly explained.

A good solution:
* Predicts accelerations reasonably well on held-out data.
* Produces interaction curves that make sense.
* Is clearly explained.

Submission Link: [https://forms.gle/gVEAjx8QY2nGdjbDA](https://forms.gle/gVEAjx8QY2nGdjbDA)

### 5. Evaluation Criteria

Submissions will be evaluated on a held-out subset of the XE0905 dataset that is not used during model development. We will assess both **quantitative accuracy** and **scientific quality** of the inferred laws.

1. **Quantitative prediction performance (≈60%)**

* Accuracy of predicted accelerations on unseen data (mean squared error).
* Quality of the inferred thermal noise law σ(T), measured by how well it matches the empirically observed temperature-dependence of acceleration fluctuations.
* Quality of the wall collision law, measured by how well the predicted post-collision velocities match observed wall-collision events.

2. **Quality of inferred physical laws (≈20%)**

* Reasonableness and interpretability of the recovered interaction curves (pairwise forces / potentials for different species, wall law, σ(T)).
* Consistency of these laws with basic structural assumptions (central pairwise forces, isotropic thermal kicks, approximately specular wall reflections with a restitution coefficient).

3. **Methodology and explanation (≈15%)**

* Clarity and coherence of the modelling pipeline (exploratory analysis, assumptions, chosen ansatz / model class, fitting procedure).
* Justification of design choices and discussion of limitations and possible failure modes.

4. **Reproducibility and code quality (≈5%)**

* The notebook should run end-to-end without manual intervention, and should produce the key plots and metrics reported.
* Code should be reasonably structured and commented so that the evaluation steps are easy to follow.

Strong submissions will not only achieve low prediction error, but also propose interaction laws that are physically meaningful, internally consistent, and clearly explained.

### 6. LeaderBoard and Timeline
**Hackathon Starts:** 0930 hours, 1st December 2025
**Hackathon Ends:** 0930 hours, 2nd December 2025

A leaderboard will be maintained for this hackathon which can be found [here](https://docs.google.com/spreadsheets/d/11b4D3XggUgmNCb8Bk1wT9S897wUb9D_ZIxrWLAO__Dc/edit?usp=sharing)

This is not a live leaderboard due to subjective components of the hackathon. The leaderboard shall be updated 3 times during the course of the hackathon and once after the hacakthon is over.

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### **Sit Vis Vobiscum!**