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README.md

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 ---
 library_name: transformers
-tags: []
+tags:
+  - math
+  - reasoning
+  - reinforcement-learning
+  - qwen2
+  - mathematics
+  - chain-of-thought
+license: apache-2.0
+language:
+  - en
+  - zh
+base_model: Qwen/Qwen2.5-Math-1.5B-Instruct
+pipeline_tag: text-generation
 ---
 
-# Model Card for Model ID
+# Nexus-1.5B
 
-<!-- Provide a quick summary of what the model is/does. -->
+<p align="center">
+  <img src="https://img.shields.io/badge/Base%20Model-Qwen2.5--Math--1.5B--Instruct-orange" />
+  <img src="https://img.shields.io/badge/Parameters-1.54B-blue" />
+  <img src="https://img.shields.io/badge/Method-LPRO-green" />
+  <img src="https://img.shields.io/badge/MATH--500-80.2-red" />
+  <img src="https://img.shields.io/badge/GSM8K-85.2-red" />
+</p>
 
+**Nexus-1.5B** is a 1.54-billion-parameter mathematical reasoning model developed by [Neuriton](https://neuriton.ai), trained via **Length-Penalized Reward Optimization (LPRO)** — a novel reinforcement learning alignment method that improves both accuracy and response conciseness simultaneously.
 
+Built on top of `Qwen2.5-Math-1.5B-Instruct`, Nexus-1.5B achieves **80.2 on MATH-500** and **85.2 on GSM8K** (CoT), surpassing its base model by **+4.4 points** on MATH-500 while reducing average response length by **14%**.
 
-## Model Details
-
-### Model Description
-
-<!-- Provide a longer summary of what this model is. -->
-
-This is the model card of a 🤗 transformers model that has been pushed on the Hub. This model card has been automatically generated.
-
-- **Developed by:** [More Information Needed]
-- **Funded by [optional]:** [More Information Needed]
-- **Shared by [optional]:** [More Information Needed]
-- **Model type:** [More Information Needed]
-- **Language(s) (NLP):** [More Information Needed]
-- **License:** [More Information Needed]
-- **Finetuned from model [optional]:** [More Information Needed]
-
-### Model Sources [optional]
-
-<!-- Provide the basic links for the model. -->
-
-- **Repository:** [More Information Needed]
-- **Paper [optional]:** [More Information Needed]
-- **Demo [optional]:** [More Information Needed]
-
-## Uses
-
-<!-- Address questions around how the model is intended to be used, including the foreseeable users of the model and those affected by the model. -->
-
-### Direct Use
-
-<!-- This section is for the model use without fine-tuning or plugging into a larger ecosystem/app. -->
-
-[More Information Needed]
-
-### Downstream Use [optional]
-
-<!-- This section is for the model use when fine-tuned for a task, or when plugged into a larger ecosystem/app -->
-
-[More Information Needed]
-
-### Out-of-Scope Use
-
-<!-- This section addresses misuse, malicious use, and uses that the model will not work well for. -->
-
-[More Information Needed]
-
-## Bias, Risks, and Limitations
-
-<!-- This section is meant to convey both technical and sociotechnical limitations. -->
-
-[More Information Needed]
-
-### Recommendations
-
-<!-- This section is meant to convey recommendations with respect to the bias, risk, and technical limitations. -->
-
-Users (both direct and downstream) should be made aware of the risks, biases and limitations of the model. More information needed for further recommendations.
-
-## How to Get Started with the Model
-
-Use the code below to get started with the model.
-
-[More Information Needed]
-
-## Training Details
-
-### Training Data
-
-<!-- This should link to a Dataset Card, perhaps with a short stub of information on what the training data is all about as well as documentation related to data pre-processing or additional filtering. -->
-
-[More Information Needed]
-
-### Training Procedure
-
-<!-- This relates heavily to the Technical Specifications. Content here should link to that section when it is relevant to the training procedure. -->
+---
 
-#### Preprocessing [optional]
+## What is LPRO?
 
-[More Information Needed]
+Standard GRPO (Group Relative Policy Optimization) suffers from two key problems:
 
+1. **Length bias** — short responses receive disproportionately large gradient signals, implicitly penalizing long correct derivations.
+2. **Entropy collapse** — symmetric probability-ratio clipping causes the policy to converge to a narrow set of solution patterns, limiting further improvement.
 
-#### Training Hyperparameters
+**LPRO** fixes both with three targeted modifications:
 
-- **Training regime:** [More Information Needed] <!--fp32, fp16 mixed precision, bf16 mixed precision, bf16 non-mixed precision, fp16 non-mixed precision, fp8 mixed precision -->
+| Component | What it does |
+|---|---|
+| **Asymmetric clipping** | Decouples the lower and upper clip bounds (`ε_low=0.20`, `ε_high=0.28`) to preserve policy entropy |
+| **Token-level normalization** | Replaces per-response weight `1/G` with global weight `1/Σ|oᵢ|` to produce an unbiased gradient estimate |
+| **Length-penalized advantage** | Adds a group-standardized length penalty: `Aᵢ = (rᵢ - μᵣ)/(σᵣ + ε) - λ·(Lᵢ - μ_L)/(σ_L + ε)` |
 
-#### Speeds, Sizes, Times [optional]
+The final objective is:
 
-<!-- This section provides information about throughput, start/end time, checkpoint size if relevant, etc. -->
+$$\mathcal{J}_{\text{LPRO}}(\theta) = \mathbb{E}\left[\frac{1}{\sum_{i=1}^{G}|o_i|} \sum_{i=1}^{G}\sum_{t=1}^{|o_i|} \min\!\left(r_{i,t}(\theta)\,\hat{A}_{i,t},\ \text{clip}_{\text{asym}}(r_{i,t}(\theta))\,\hat{A}_{i,t}\right)\right]$$
 
-[More Information Needed]
+---
 
-## Evaluation
+## Model Details
 
-<!-- This section describes the evaluation protocols and provides the results. -->
+| Property | Value |
+|---|---|
+| **Base model** | `Qwen/Qwen2.5-Math-1.5B-Instruct` |
+| **Parameters** | 1.54B |
+| **Architecture** | Transformer Decoder (28 layers, GQA, RoPE, SwiGLU, RMSNorm) |
+| **Context length** | 8,192 tokens |
+| **Vocabulary size** | 128,256 |
+| **Training method** | LPRO (RL fine-tuning, no distillation) |
+| **Training data** | 100 difficulty-filtered problems from MATH-500 |
+| **Group size G** | 4 |
+| **Length penalty λ** | 0.10 |
+| **Learning rate** | 1e-6 |
+| **PPO epochs/iter** | 4 |
 
-### Testing Data, Factors & Metrics
+---
 
-#### Testing Data
+## Benchmark Results
 
-<!-- This should link to a Dataset Card if possible. -->
+### Chain-of-Thought (CoT)
 
-[More Information Needed]
+| Model | GSM8K | MATH-500 | MMLU-STEM | CMATH | GaoKao Cloze | GaoKao QA |
+|---|---|---|---|---|---|---|
+| Qwen2-Math-1.5B-Instruct | 84.2 | 69.4 | 54.9 | 79.6 | 59.7 | 50.7 |
+| Qwen2.5-Math-1.5B-Instruct | 84.8 | 75.8 | 57.5 | 83.0 | 65.5 | 54.1 |
+| **Nexus-1.5B** | **85.2** | **80.2** | **60.3** | **83.5** | **67.2** | **56.9** |
 
-#### Factors
+### Tool-Integrated Reasoning (TIR)
 
-<!-- These are the things the evaluation is disaggregating by, e.g., subpopulations or domains. -->
+| Model | MATH-500 | Minerva Math | GaoKao 2023 EN | Olympiad Bench | College Math |
+|---|---|---|---|---|---|
+| Qwen2.5-Math-1.5B-Instruct | 80.0 | 34.0 | 68.0 | 49.0 | 54.0 |
+| **Nexus-1.5B** | **84.0** | **40.0** | **74.0** | **56.0** | **57.0** |
 
-[More Information Needed]
+### Ablation: Effect of Length Penalty (λ)
 
-#### Metrics
+| λ | MATH-500 Acc. | Avg. Response Length |
+|---|---|---|
+| 0.0 (GRPO baseline) | 77.4 | 312 tokens |
+| **0.1 (Nexus-1.5B)** | **80.2** | **268 tokens** |
+| 0.3 (over-penalized) | 78.0 | 201 tokens |
 
-<!-- These are the evaluation metrics being used, ideally with a description of why. -->
+> **Key insight:** At λ=0.1, accuracy and conciseness improve simultaneously. The length penalty acts as a de-noising regularizer — discouraging redundant steps rather than suppressing genuinely long derivations.
 
-[More Information Needed]
+---
 
-### Results
+## How to Use
 
-[More Information Needed]
+```python
+from transformers import AutoModelForCausalLM, AutoTokenizer
 
-#### Summary
+model_name = "Dat1710/nexus-1.5b"
 
+tokenizer = AutoTokenizer.from_pretrained(model_name)
+model = AutoModelForCausalLM.from_pretrained(
+    model_name,
+    torch_dtype="auto",
+    device_map="auto"
+)
 
+# Chain-of-Thought prompt
+system_prompt = "Please reason step by step, and put your final answer within \\boxed{}."
 
-## Model Examination [optional]
+messages = [
+    {"role": "system", "content": system_prompt},
+    {"role": "user", "content": "Find all functions f: ℝ⁺ → ℝ⁺ such that for each x ∈ ℝ⁺, there is exactly one y ∈ ℝ⁺ satisfying xf(y) + yf(x) ≤ 2."}
+]
 
-<!-- Relevant interpretability work for the model goes here -->
+text = tokenizer.apply_chat_template(
+    messages,
+    tokenize=False,
+    add_generation_prompt=True
+)
 
-[More Information Needed]
+model_inputs = tokenizer([text], return_tensors="pt").to(model.device)
 
-## Environmental Impact
+generated_ids = model.generate(
+    **model_inputs,
+    max_new_tokens=2048,
+    temperature=0.7,
+    do_sample=True,
+)
 
-<!-- Total emissions (in grams of CO2eq) and additional considerations, such as electricity usage, go here. Edit the suggested text below accordingly -->
+generated_ids = [
+    output_ids[len(input_ids):]
+    for input_ids, output_ids in zip(model_inputs.input_ids, generated_ids)
+]
 
-Carbon emissions can be estimated using the [Machine Learning Impact calculator](https://mlco2.github.io/impact#compute) presented in [Lacoste et al. (2019)](https://arxiv.org/abs/1910.09700).
+response = tokenizer.batch_decode(generated_ids, skip_special_tokens=True)[0]
+print(response)
+```
 
-- **Hardware Type:** [More Information Needed]
-- **Hours used:** [More Information Needed]
-- **Cloud Provider:** [More Information Needed]
-- **Compute Region:** [More Information Needed]
-- **Carbon Emitted:** [More Information Needed]
+### Tool-Integrated Reasoning (TIR)
 
-## Technical Specifications [optional]
+```python
+system_prompt = (
+    "Please integrate natural language reasoning with programs to solve the problem above, "
+    "and put your final answer within \\boxed{}."
+)
+```
 
-### Model Architecture and Objective
+---
 
-[More Information Needed]
+## Evaluation Prompt Format
+
+**CoT (8-shot for GSM8K, 4-shot for MATH-500):**
+```
+<|im_start|>system
+Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
+<|im_start|>user
+{problem}<|im_end|>
+<|im_start|>assistant
+```
+
+**TIR (zero-shot):**
+```
+<|im_start|>system
+Please integrate natural language reasoning with programs to solve the problem above,
+and put your final answer within \boxed{}.<|im_end|>
+<|im_start|>user
+{problem}<|im_end|>
+<|im_start|>assistant
+```
 
-### Compute Infrastructure
+---
 
-[More Information Needed]
+## Training Details
 
-#### Hardware
+### Data Curation
 
-[More Information Needed]
+Training problems are sourced from **MATH-500** and filtered by difficulty using a learnable-zone criterion: a problem is retained if, among 8 sampled solutions from the base model, **between 2 and 5 are correct**. This yields 100 training problems that provide meaningful gradient signal — neither trivially easy nor intractably hard.
 
-#### Software
+### Training Procedure
 
-[More Information Needed]
+1. **Group sampling:** For each prompt, sample G=4 responses from the current policy.
+2. **Reward computation:** Rule-based binary reward (correctness via symbolic answer matching) + small format bonus (α=0.1) for well-formed `\boxed{}` output.
+3. **Advantage computation:** Compute length-penalized group z-score advantages.
+4. **Policy update:** Maximize LPRO objective for 4 epochs per iteration.
+5. **Iterate:** Set old policy ← new policy and repeat.
 
-## Citation [optional]
+### Reward Function
 
-<!-- If there is a paper or blog post introducing the model, the APA and Bibtex information for that should go in this section. -->
+$$r_i = \mathbf{1}[\hat{a}(o_i) = a^*] + 0.1 \cdot \mathbf{1}[\text{format}(o_i)]$$
 
-**BibTeX:**
+where $\hat{a}(o_i)$ is the extracted answer from the last `\boxed{}` expression, verified via symbolic equivalence.
 
-[More Information Needed]
+---
 
-**APA:**
+## Limitations
 
-[More Information Needed]
+- **Scale:** Nexus-1.5B operates at 1.54B parameters. Hard olympiad problems (e.g., AIME) remain challenging for models at this scale.
+- **Language:** Primarily optimized for English and Chinese mathematical text. Performance on other languages is not evaluated.
+- **Domain:** Designed for mathematical reasoning. General language understanding or instruction-following tasks are outside the model's training distribution.
+- **TIR dependency:** Tool-integrated reasoning requires a sandboxed Python interpreter at inference time.
 
-## Glossary [optional]
+---
 
-<!-- If relevant, include terms and calculations in this section that can help readers understand the model or model card. -->
+## Citation
 
-[More Information Needed]
+If you use Nexus-1.5B in your research, please cite:
 
-## More Information [optional]
+```bibtex
+@techreport{neuriton2026nexus,
+  title     = {Nexus-1.5B: Length-Penalized Reward Optimization for Robust Mathematical Reasoning},
+  author    = {Neuriton Team},
+  institution = {Neuriton},
+  year      = {2026},
+  month     = {Summer},
+  note      = {Technical Report}
+}
+```
 
-[More Information Needed]
+---
 
-## Model Card Authors [optional]
+## Acknowledgements
 
-[More Information Needed]
+We thank the Qwen Team at Alibaba Group for open-sourcing the Qwen2.5-Math model family, and the authors of DAPO for the asymmetric clipping insight that is central to LPRO.
 
-## Model Card Contact
+---
 
-[More Information Needed]
+*Developed by [Neuriton](https://neuriton.ai) · Summer 2026*