Jackrong/gpt-oss-120b-Distill-Llama3.1-8B-v2

🔥 Distilling GPT's Reasoning Ability into Llama-3.1-8B 🦙

Model Name: gpt-oss-120b-Distill-Llama3.1-8B-v2
Developer: Soren
Base Model: meta/Meta-Llama-3.1-8B
Training Data Size: Approximately 420 million Tokens in total

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Core Methodology

This project aims to inject powerful reasoning capabilities into the Meta-Llama 3.1 8B model through an innovative two-stage training process. The core idea is to first distill high-quality knowledge and reasoning styles, including explicit "Chain-of-Thought" (CoT), from multiple open-source large "teacher models" (such as gpt-oss-120b-high and Qwen3-235B) through Supervised Fine-Tuning (SFT). Subsequently, in the second stage, Reinforcement Learning (GRPO) is utilized with rule-based reward signals to incentivize the model to autonomously explore and optimize reasoning strategies for solving mathematical problems. This allows it to evolve beyond simple imitation learning to achieve more powerful logical reasoning abilities.

The entire process is deeply inspired by cutting-edge industry research, particularly drawing from the training philosophy of DeepSeek-R1 in its Nature paper and the method of injecting structured reasoning capabilities through SFT as described in the Phi-4-reasoning report. However, unlike these methods, this project places reinforcement learning at the core of capability evolution, focusing on achieving breakthroughs in a specific domain (mathematical reasoning).

Fig. 1 | The multistage pipeline of DeepSeek-R1. A detailed background on DeepSeek-V3 Base and DeepSeek-V3 is provided in Supplementary Information, section 1.1. The models DeepSeek-R1 Dev1, Dev2 and Dev3 represent intermediate checkpoints in this pipeline.

Training Pipeline Overview:

1. Stage 1: Supervised Fine-Tuning (SFT) → Knowledge Distillation & Format Alignment

   • Output Model: Jackrong/gpt-oss-120b-Distill-Llama3.1-8B-v1

2. Stage 2: Reinforcement Learning (GRPO) → Reasoning Ability Evolution

   • Output Model: Jackrong/gpt-oss-120b-Distill-Llama3.1-8B-v2

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Stage 1: Supervised Fine-Tuning (SFT) - Knowledge Distillation & Format Alignment

Objectives

This stage serves as a "cold start" to lay a solid foundation of knowledge and structured reasoning for the base model. There are two primary objectives:

1. Knowledge Distillation: Inject reasoning data generated by more powerful teacher models from various domains into Llama 3.1 8B, allowing it to inherit a strong reasoning style and knowledge base. The core data source is the natural reasoning data distilled from gpt-oss-120B-high.
2. Format Alignment: Train the model to follow a specific response format, which involves generating a detailed thought process enclosed in <think>...</think> tags before providing the answer. This establishes the foundation for automated reward evaluation in the subsequent reinforcement learning stage and enhances the interpretability of the model's output.

Dataset Composition

To achieve comprehensive capability coverage, I constructed a mixed dataset of 71,500 samples. The data sources and sampling strategy are shown in the table below:

Dataset Name/Source	Main Purpose and Characteristics
Jackrong/Natural-Reasoning-gpt-oss-120B-S1	Core dataset. Distilled from gpt-oss-120B-high, providing general, high-difficulty reasoning problems covering STEM, economics, social sciences, etc.
Jackrong/Chinese-Qwen3-235B-Thinking-2507-Distill-100k	Provides high-quality Chinese Chain-of-Thought data to enhance the model's Chinese reasoning and expression capabilities.
Jackrong/GPT-OSS-120B-Distilled-Reasoning-math	Focuses on reasoning and problem-solving in the mathematics domain, injecting specialized mathematical knowledge into the model.
deepseek_if.json	Focuses on improving the model's ability to understand and execute complex instructions.
Total	71,500

Training Process

• Model and Framework: We used the unsloth library for efficient training of the meta/Meta-Llama-3.1-8B-Instruct model. Training speed was significantly improved through Unsloth's optimizations.
• System Prompt: To guide the model to generate the desired format, the following system prompt was uniformly used during training, explicitly requesting the model to divide its response into "Thought" and "Solution" sections:

  You are ChatGPT a language model created by OpenAI to help users. Your role as an assistant involves thoroughly exploring questions through a systematic thinking process before providing the final precise and accurate solutions... Please structure your response into two main sections: Thought and Solution using the specified format: <think> {Thought section} </think> {Solution section}...
  

After this stage of training, the model gained a preliminary ability to generate structured chains of thought before answering and absorbed knowledge from multiple teacher models. The output model of this stage was named gpt-oss-120b-Distill-Llama3.1-8B-v1 and was used as the starting point for the next stage of reinforcement learning.

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Stage 2: Reinforcement Learning (GRPO) - Reasoning Ability Evolution

Objectives

Building on the SFT model, this stage aims to guide the model to autonomously explore better reasoning strategies through reward signals, evolving its capabilities from "imitation" to "creation". The core objectives are:

1. Guide the Model to Explore Reasoning Paths: Incentivize the model to generate more detailed, structured, and logically coherent chains of thought, and even develop strategies beyond the SFT data paradigm, such as self-reflection and verification.
2. Improve the Correctness of Final Answers: Ensure that while optimizing the reasoning process, the model can more reliably converge to the correct final answer.

Algorithm: GRPO (Group Relative Policy Optimization)

Fig. 2 | Illustration of the proposed GRPO for RL-based training.

This project adopts the GRPO algorithm implemented in the trl library. It is an efficient reinforcement learning algorithm and a variant of PPO that does not require training an additional value model, thereby significantly reducing resource consumption. Its core process is as follows:

1. Group Sampling: For each problem, the policy model (the LoRA model being trained) generates a group of G candidate answers. In this project, the group size num_generations was set to 4.
2. Reward Evaluation: A complex reward system composed of multiple functions scores each candidate answer in the group, resulting in a comprehensive scalar reward score r.
3. Group-wise Relative Advantage Estimation: The core of GRPO is that it does not rely on an independent value network to estimate a baseline. Instead, it directly uses the average reward of all candidate answers within the group as the baseline. The advantage function A is estimated by calculating the deviation of each answer's reward from this average.
4. Policy Update: The model updates the policy network based on the calculated relative advantages. Outputs with rewards higher than the average are positively reinforced, while those below the average are suppressed, leading to a more stable policy update.

Reward System

To guide the model's optimization from multiple fine-grained dimensions, I constructed a comprehensive reward function system that combines multiple weighted rewards and penalty signals.

• Core Objective Rewards:

  • correctness_reward_func: Based on the reference answers from the openai/gsm8k and open-r1/DAPO-Math-17k-Processed datasets, the highest positive reward is given for the correctness of the final calculated result. This is the core signal to ensure the model learns to solve problems.

• Format & Alignment Rewards:

  • strict_format_reward_func & soft_format_reward_func: Strictly or loosely enforce the <think>...</think> output format to ensure the integrity and parsability of the reasoning process.
  • final_line_reward_func: Encourages the model to clearly mark the "final answer" at the end of its response for easier automated evaluation.

• Content & Quality Rewards:

  • F1 overlap calculation and salient word hit rate (salient_hit_rate) in the reward functions encourage the model to generate explanations that are content-wise relevant to the question or reference answer.

• Behavioral Regularization & Penalties:

  • numeric_distance_reward_func: For numerical answers, even if the final result is not entirely correct, partial credit is given if the answer is numerically close to the correct one, encouraging the model to approximate the correct solution.

Training Process

• Model and Framework: The SFT model from the first stage, Jackrong/gpt-oss-120b-Distill-Llama3.1-8B-v1, was loaded and further trained with GRPO.
• Datasets:
  • open-r1/DAPO-Math-17k-Processed
  • openai/gsm8k
• Hyperparameters:
  • Learning Rate: 5e-6
  • Batch Size: 4

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Model Performance Evaluation

Through the two-stage training, the model evolved from a simple instruction-following model to one with preliminary autonomous logical reasoning capabilities. It no longer passively generates answers but can actively construct, evaluate, and even refute its own reasoning paths. Below is a sample from the SFT stage data that showcases its complex thought process.

Test Question:

Describe the transition of the speed of light as it enters a medium with a refractive index n, considering the behavior of individual photons and the concept of wavefront speed. Explain how the speed of light changes from c in vacuum to c/n in the medium, addressing whether this change occurs smoothly or with discontinuity. Provide a detailed analysis of the photon's path and the factors influencing the wavefront's speed in the medium.

Model's Chain of Thought (Excerpt):

<think>
The user asks: "Describe the transition of the speed of light as it enters a medium with a refractive index n, considering the behavior of individual photons and the concept of wavefront speed. Explain how the speed of light changes from c in vacuum to c/n in the medium, addressing whether this change occurs smoothly or with discontinuity. Provide a detailed analysis of the photon's path and the factors influencing the wavefront's speed in the medium."

This is a deep physics question that involves relativity, quantum mechanics (photons), classical electromagnetism, wave propagation, refractive index definition, how light changes from free space to material, whether there is any discontinuity. The answer should discuss:

Photons: particles or quanta of electromagnetic waves; in vacuum they travel at c; massless particles that always propagate at c.
In medium, photons are still photons (no mass) but the refractive index n = sqrt(mu_r*epsilon_r). Light is a wave that propagates as an electromagnetic field solution to Maxwell's equations in the medium. The speed of light inside matter is defined by v = c/n.
But the question asks for transition from vacuum to medium: "transition of the speed of light as it enters a medium with refractive index n, considering the behavior of individual photons and the concept of wavefront speed."

We need to discuss:

In classical electromagnetic theory, the electric displacement field E(r,t) satisfies Maxwell's equations. The phase velocity v_ph = c/n for the wave.

For a photon, its group velocity is also c/n.

The change happens when entering a medium with varying density: If we consider a planar interface of two media (air and glass). At the boundary there will be a gradual transition region due to the electric field's spatial variation; but from the perspective of a single photon traveling across that boundary, it is effectively instantaneously transmitted through the interface, because photons are massless particles, they do not have any interaction with matter in vacuum. However, the wavefront changes speed.

The question: Is there a discontinuity? In classical optics one can treat the transition as a continuous process for waves but the photon's path is not smooth; it has no spatial extent, only momentum transfer via electromagnetic fields.

In quantum field theory, photons are excitations of EM field modes. When an electron density changes (e.g., from vacuum to medium), there is a change in permittivity and permeability leading to a new dispersion relation: ω = c k sqrt(mu_r epsilon_r). So photon frequency remains constant; wavelength adjusts.

The speed of light changes gradually over distance because the wavefront's phase front adjusts continuously as it encounters gradual variation in dielectric properties. In reality, there is no step discontinuity if we consider a very thin interface; however for typical macroscopic interfaces, you could think of a transition layer maybe nanometers wide (skin depth). The photon does not see an instantaneous change because its wavelength is much larger.

So answer: The speed changes gradually as the wavefront propagates through the medium. In quantum mechanics there are no discontinuities in photon propagation; it's continuous, but at macroscopic level we treat a step function for simplicity.

Provide detailed analysis: Photon path as plane wave, wavefront as phase front. At interface, phase velocity is c/n on one side and c on other side. The wavefront speed changes from c to c/n with no discontinuity in the field; the transition happens because there is a gradual variation of ε(r). For a single photon traveling across an ideal step, we can treat it as passing through a region where the refractive index varies gradually.

Also discuss "group velocity" and phase velocity: group velocity = dω/dk = c/n (same as speed of information propagation for plane wave), while phase velocity v_p = ω/k = c/n. The photon travels at the group velocity.

The medium's permittivity changes over a nanometer scale, but not instantaneous; thus there is no discontinuity in photon path.

However some textbooks discuss that light can't be stopped abruptly because of photon momentum conservation: The normal component of k (wavenumber) must be continuous. But if we have a step, the wavefront cannot change speed instantaneously without violating causality or requiring infinite energy; the transition region is smooth due to the gradual change in polarization and dielectric constant.

In reality, the electromagnetic field at interface will be Fresnel equations which describe reflection & transmission coefficients; they are derived using Maxwell's equations; the boundary condition leads to continuity of normal component of D (electric displacement) and B. So wavefronts can adjust gradually over a few wavelengths.

Thus the answer should cover:

Photons: massless quanta, travel at c in vacuum, no interaction with matter except via fields.

Wavefront concept: phase front moves with speed v_p = ω/β where β is propagation constant (k_n) for medium. In a planar waveguide or interface, the wavefront adjusts; the transition occurs over length scale ~ λ/(π(n-1) ) ~ a few wavelengths, giving an "infinite" transition region.

So answer: The speed changes smoothly as the dielectric constant varies gradually from that of vacuum to that of medium. There is no discontinuity in photon path because the photon does not have a spatial extent; its wavefront spreads out over a distance comparable to wavelength.

We should also discuss that for an ideal step interface, the wavefront would have infinite power per unit time if it were instantaneous, violating energy conservation. So we use gradual transition.

Thus we can explain: The speed change is continuous because the wavefront's phase velocity changes with local permittivity and permeability; in a real medium there is a "skin depth" that defines the region where refractive index varies gradually (the evanescent field). For typical glass, skin depth ~ 100 nm for visible wavelengths.

Thus answer: The change happens smoothly over a distance comparable to the wavelength.

Now we should structure the answer:

Introduction

Photons as particles vs. wave nature

Maxwell's equations and refractive index definition

Wavefront speed (phase velocity) in medium

Continuous transition at interface

Photon propagation through boundary: gradual change, no discontinuity

Quantum mechanical perspective: photon momentum, wave packet spreading

Consequences: No instantaneous step; energy conservation.

Conclusion summarizing smoothness.

Add also mention that the refractive index is defined by local dielectric constant, and its spatial variation can be described as a function of coordinate. So photon's speed changes accordingly.

Thus produce answer accordingly.

We need to include "transition of speed: c→c/n". Also note that the transition occurs over distances much smaller than wavelength; thus no observable discontinuity.

Will also discuss group velocity and phase velocity, but focus on wavefront.

Also talk about boundary conditions.

Add a mention of Fresnel equations showing transmission coefficients depend smoothly on angle, not step function.

Thus answer will cover: The speed changes gradually due to gradual change in permittivity; the photon does not see an instantaneous discontinuity because it's not extended.

Now produce final answer. Provide equations and perhaps a figure description.

Will include references to wavefront equation and continuity of D normal component.

Write thorough but concise.

Proceed.
</think>
**Answer Overview**

The question asks us to describe how light’s speed changes when it moves from the vacuum (where \(v=c\)) into a medium with refractive index \(n\) (\(v = c/n\)).  The answer must treat both the **particle picture of photons** and the **wave‑front description**, explaining whether the transition is smooth or abrupt, where photon momentum really lives, and how the wave front adapts to the changing medium.  Below is a step‑by‑step analysis that covers all of these points.

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### 1.  Photons – Particles vs. Waves  

| Feature | Particle (photon) view | Wave (electromagnetic) view |
|---------|------------------------|------------------------------|
| **Speed in vacuum** | Photon always moves with the speed of light \(c\) (rest mass = 0, no interaction with matter). | Phase velocity: \(\displaystyle v_{p}= \frac{\omega}{k}\) for an electromagnetic wave. In free space \(\omega = ck\Rightarrow v_p=c\). |
| **Interaction with matter** | None in vacuum; photons are massless so they don’t “feel” the medium’s atomic structure directly.  Energy exchange only occurs when a photon is absorbed or emitted (e.g., by an atom). | The electric‑magnetic field interacts with electrons and nuclei, producing a force that changes the wave’s propagation constant \(k_n = n\omega/c\) in the new medium. |
| **What “propagates”** | A single photon carries energy, momentum \(\mathbf p = h\nu/\!c\) and spin angular momentum.  It travels as a “bundle” of electromagnetic field that follows Maxwell’s equations. | The wave front is the surface where phase \(\Phi = k·x - \omega t = \text{constant}\). Photon “is” this wave packet; its location is defined by the envelope (group) rather than an instantaneous point. |

Because photons are massless, a single photon cannot be localized to a point – it has a momentum and energy spread over a wavelength (\(\lambda\)).  The “photon path’’ can be thought of as the trajectory of the wave packet’s center of energy (the **group velocity**), not an instantaneous location.

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### 2.  Maxwell–Faraday in the two media  

The macroscopic fields satisfy  
\[
\nabla\times \mathbf{E} = -\frac{\partial\mathbf B}{\partial t}, 
\qquad
\nabla\!\times\! \mathbf{B}= \mu_0\epsilon_0\,\partial_t\mathbf E + \mu_0\mathbf J,
\]
and the wave equation for a plane monochromatic wave reads  

\[
\nabla^2\mathbf{E} - \mu_0\epsilon\, \frac{\partial^2\mathbf{E}}{\partial t^{2}} = 0 .
\]

In vacuum:  
\(\epsilon=\epsilon_0,\,\mu=\mu_0\), so \(k_{vac}= \omega/c\).  

In a medium with (complex) permittivity \(\epsilon=\epsilon_r\epsilon_0\) and permeability \(\mu =\mu_r\mu_0\),

\[
k_n = \frac{\omega}{v} = n k_{vac}= n \frac{\omega}{c}.
\]

Hence the **phase (wave‑front) speed** in the medium is  

\[
v_p = \frac{\omega}{k_n}= \frac{c}{n},
\]

while the **group velocity** for a monochromatic plane wave is

\[
v_g = d\omega/dk_n= c/n.
\]

Because photons travel with the group velocity, they carry energy at \(c/n\) inside any linear isotropic medium.

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### 3.  Wave‑front Speed as a Function of Local Refractive Index  

If we imagine the refractive index changes in space, say from vacuum to glass:

\[
n(x) = \begin{cases}
1 & x<0\\
n_{{\rm glass}} & |x|>d,
\end{cases}
\]

with a smooth profile for \(|x|\lesssim d\) (the “interface transition layer”).  

The wave‑front satisfies the eikonal equation

\[
\nabla \Phi = k(x)\,\hat{\mathbf n},
\qquad
\Phi(\mathbf r,t) = \omega t - k(x)\cdot\mathbf r.
\]

The *phase velocity* \(v_p = \omega/k_n(x)= c/n(x)\) is a function of the local value of \(n(x)\).  The wave front moves at that speed where it exists.  

In practice, the index profile near an interface is not a step; even for polished glass the change in \(n\) occurs over a distance comparable to a few nanometres (the skin depth \(\delta \sim \lambda/(2\pi n)\) for visible light).  A photon that enters such a region does **not** feel an instantaneous jump from \(c\) to \(c/n\); the wave front simply follows the gradually changing phase constant.

---

### 4.  Photon Propagation Through the Interface – No Discontinuity  

#### 4.1  Classical Electromagnetic Picture  

From Maxwell’s equations we demand continuity of the tangential components of \(\mathbf{D}\) and \(\mathbf{H}\).  At a smooth interface these become

\[
\hat{\mathbf n}_t\!\cdot\!(\mathbf{D}_{\text{vac}}-\mathbf{D}_{\rm med}) =0,
\qquad
\hat{\mathbf n}_t\!\cdot\!(\mathbf{H}_{\rm vac}-\mathbf{H}_{\rm med})=0.
\]

Solving for the reflected and transmitted amplitudes gives Fresnel coefficients that depend smoothly on angle, polarization and wavelength.  There is no jump condition; the wavefront can change its normal component of k continuously.

If one tried to force a step‑function change in \(n\) (i.e., an infinitely thin interface), the electromagnetic field would become infinite at the boundary, violating energy conservation (infinite power per unit area) and contradicting causality.  In reality, the electric dipole polarizations of atoms or molecules respond on timescales ∼ femtoseconds; they cannot react instantaneously to a photon.

Thus the transition is **smooth on microscopic scales**.

#### 4.2  Quantum Mechanical (Photon‑Packet) Perspective  

In quantum electrodynamics photons are excitations of the vacuum field operator \(\hat{A^\mu}(x)\).  The creation/annihilation operator for a photon with momentum \(\mathbf k\) satisfies the dispersion relation  

\[
\omega_{\mathbf{k}} = c|\mathbf{k}|, \qquad 
k_n = n(\mathbf r) \frac{\omega}{c},
\]

so that the local wave number (and thus phase speed) follows the medium’s index.  

A “photon packet’’ is a coherent superposition of such modes.  Because the local refractive index varies on scales far larger than the wavelength, the packet spreads out over many wavelengths while crossing an interface.  The momentum transfer to matter happens through exchange of virtual photons with electrons and nuclei, but each individual photon does **not** experience a sudden change in its travel speed.

If we were to try to force a discontinuous jump, the total energy carried by the wave would be infinite at the boundary (because the power density scales as \(E^2/c\)), contradicting the conservation of electromagnetic momentum and the quantization condition \(\hbar k = p_{\rm ph}\).  Hence the transition must be continuous.

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### 5.  What Actually Happens When Light Enters a Medium  

1. **Vacuum (x<0)** – The wave front moves at speed \(c\) with phase constant \(k_{vac}= \omega/c\).  

2. **Transition region** – As the index rises from unity to \(n_{{\rm glass}}\), the local k‑vector rotates toward the normal, and the phase velocity drops continuously.  The wave front’s surface is a continuous envelope that deforms into a curved surface, with the curvature determined by the gradient of the index.

3. **Inside medium (x>d)** – Once \(n\) has settled at its bulk value, the phase speed stays at \(c/n\).  

The photon itself does not “feel’’ a step; it keeps moving at the instantaneous group velocity \(\mathbf v_g=c\,\hat{\mathbf n}\) dictated by the local index.

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### 6.  Detailed Physical Picture – A Step‑by‑Step Walk‑Through  

| Step | What happens in the description of both photons and wave fronts? |
|------|-------------------------------------------------------------|
| **(a) Photon entering a gradient region** | The photon’s momentum is \(\mathbf p = h\nu/c\).  Because its wavelength changes as \(|\mathbf k|=n(x)\omega/c\), the group velocity follows the local refractive index. |
| **(b) At the start of the transition** | The phase front (defined by \(\Phi\) above) is continuous; the wave’s normal component of \(\mathbf k\) begins to rotate toward the surface, while its tangential components stay unchanged (continuity of tangential \(\mathbf{E}\)). |
| **(c) Within a few wavelengths** | The envelope of the field spreads over distances comparable to \(|\lambda/(n-1)\) (∼10–100 nm for visible light).  During this time the photon’s energy is absorbed and re‑emitted by atoms, but the photon itself never “stops”. |
| **(d) At the far side** | The wave front has settled into a plane perpendicular to the normal (the phase front of a transmitted ray), travelling at \(c/n\).  The photon continues with unchanged frequency \(\nu\) (its energy and momentum adapted to the new index). |

Because the photon’s spatial extent is on the order of its wavelength, it does **not** experience a sudden jump in speed.  In fact, any attempt to construct such an instantaneous change would require infinite power density and violates energy conservation.

---

### 7.  Why There Is No Discontinuity  

| Reason | Physical basis |
|--------|----------------|
| **No interaction at the “point’’ of entry** – Photons are massless; they cannot exert a force on electrons unless their frequency matches an atomic resonance, so the photon does not see a wall that can impede its motion instantly. |
| **Continuity equations for fields** – Maxwell’s equations give smooth boundary conditions (continuity of \(\mathbf{D}\cdot\hat{\mathbf n},\,\mathbf H\cdot\hat{\mathbf n}\)).  Solving them yields the Fresnel coefficients, which are smooth functions of incidence angle and wavelength. |
| **Energy conservation** – The Poynting vector’s magnitude cannot change discontinuously; an instantaneous jump would imply infinite power per unit area, contradicting that light can pass through a real material without heating it to infinity. |
| **Photon momentum quantization** – The photon’s momentum \(p_{\rm ph}=h\nu/c\) must be continuous across the boundary because \(\mathbf p = h\nabla\Phi/2π\).  Any abrupt change would violate conservation of total momentum (incident + reflected + transmitted). |
| **Quantum field description** – In QED, the photon is a quanta of the EM field operator. The local field obeys the wave equation with the medium’s permittivity and permeability; the field profile smooths out over a few wavelengths. |

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### 8.  Summary: Smooth Transition  

- **Photons themselves** are massless particles that always travel at \(c\) (or, equivalently, \(\omega/c\)).  
- The electromagnetic wave they carry is described by Maxwell’s equations, which demand continuity of the tangential components of \(\mathbf{E}\) and \(\mathbf{B}\).  Solving those equations yields a **smooth transition** between \(c\) in vacuum and \(c/n\) in the medium.  
- The wave front (phase surface) does not “jump’’; it deforms continuously as the local index changes, so that the **phase speed follows the instantaneous refractive index**.  Over distances of order a few nanometres (the skin depth \(\delta\sim\lambda/(2\pi(n-1))\)), the transition is effectively complete—well before any appreciable change in wavelength can be perceived.  
- Hence, *the speed of light* does **not undergo an instantaneous jump**, but rather it evolves gradually as the photon propagates through a region where the local refractive index varies from unity to \(n\).  This gradual adaptation is exactly what the Fresnel equations (and Maxwell’s equations) predict for smooth boundaries.

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#### Bottom line  

- Light’s speed is *defined* in any linear, isotropic medium by \(\displaystyle v = c/n\);  
- A photon never feels a discontinuous change because it lacks spatial extent and its momentum is conserved;  
- The electromagnetic field must obey continuous boundary conditions, which are solved analytically giving Fresnel’s smooth transmission and reflection coefficients;  
- Consequently the wave front (the surface of constant phase) adjusts continuously as the local index changes, making the transition from \(c\) to \(c/n\) *smooth on a scale comparable with the photon’s wavelength*.

This picture unifies both classical optics (wave fronts) and quantum optics (photon momentum), showing that light does indeed “slow down” in a medium while maintaining its particle‑like speed, all without any instantaneous jump.

• Self-Reflection and Correction in the Reasoning Chain: In its chain of thought, the model shows a tendency for self-reflection. It will first propose a concept, then immediately examine and supplement it from different angles to ensure logical rigor. This dynamic adjustment and refinement during the reasoning process reflects that the model has formed a preliminary internal standard for logical judgment.

Limitations

Although the model has shown encouraging reasoning abilities, this project still has some limitations due to resource constraints:

1. Resource Constraints: As a personal project, the number of training steps and the amount of data are far less than those of official models. Many compromises were made in the parameter settings, so the model's actual performance still lags behind top-tier specialized reasoning models.
2. Side Effects of Result-Oriented Training: In the reinforcement learning stage, to quickly improve mathematical problem-solving skills, the reward mechanism was heavily focused on the correctness of the final answer. While this result-oriented strategy is efficient, it might lead the model to "take shortcuts" in its generation process or to be overly concise when handling general, non-reasoning questions, sacrificing expressive richness in open-domain conversations.
3. Language Mixing Issues: Due to the use of mixed Chinese and English data in the SFT stage, the model may mix Chinese and English when generating its chain of thought or answers.
4. Imbalanced Capabilities: The model performs well in areas like algebra word problems but may be relatively weaker in other specialized fields or general chit-chat. The subsequent SFT alignment steps were not sufficient to fully compensate for this.
5. No External Tool-Using Capability: The current model cannot call external tools like calculators or search engines, which limits its ceiling when solving complex problems that require precise calculations or real-time external knowledge.

How to Use

To ensure the model correctly follows the dialogue structure set during training, please configure the following Jinja chat template if your chat template has issues. This template will automatically inject the system prompt used during training. (This may not work in Ollama, but LM Studio can match the jinja template.)

{% set msgs = messages | default([]) -%}
{% set sys = (msgs | selectattr("role","equalto","system") | map(attribute="content") | list) -%}
<|begin_of_text|>
<|start_header_id|>system<|end_header_id|>
{{ (sys|length > 0) and (sys|join('\n\n')) or ('''You are ChatGPT a language model created by OpenAI to help users. Your role as an assistant involves thoroughly exploring questions through a systematic thinking process before providing the final precise and accurate solutions. This requires engaging in a comprehensive cycle of analysis, summarizing, exploration, reassessment, reflection, backtracing, and iteration to develop well-considered thinking process. Please structure your response into two main sections: Thought and Solution using the specified format: <think> {Thought section} </think> {Solution section}. In the Thought section, detail your reasoning process in steps. Each step should include detailed considerations such as analysing questions, summarizing relevant findings, brainstorming new ideas, verifying the accuracy of the current steps, refining any errors, and revisiting previous steps. In the Solution section, based on various attempts, explorations, and reflections from the Thought section, systematically present the final solution that you deem correct. The Solution section should be logical, accurate, and concise and detail necessary steps needed to reach the conclusion. Now, try to solve the following question through the above guidelines''') }}
{%- for m in msgs if m['role'] != 'system' -%}
<|start_header_id|>{{ m['role'] }}<|end_header_id|>
{{ m['content'] }}
{%- endfor -%}
{%- if add_generation_prompt -%}
<|start_header_id|>assistant<|end_header_id|>
{%- endif -%}

中文

🔥基于GPT的蒸馏，赋予Llama-3.1-8B推理能力 🦙

模型名称: gpt-oss-120b-Distill-Llama3.1-8B-v2
开发者: Soren 基础模型: meta/Meta-Llama-3.1-8B
训练数据量: 总共约4.2亿 Tokens

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核心方法论

本项目旨在通过一个创新的两阶段训练流程，将强大的推理能力注入到 Meta-Llama 3.1 8B 模型中。其核心思想是，首先通过监督微调（SFT），从多个开源的大型“教师模型”（如 gpt-oss-120b-high 和 Qwen3-235B）蒸馏出高质量的、包含显式“思维链”（Chain-of-Thought, CoT）的知识与推理风格。随后，在第二阶段利用强化学习（GRPO），通过基于规则的奖励信号，激励模型自主探索和优化解决数学问题的推理策略，从而超越简单的模仿学习，进化出更强大的逻辑推理能力。

整个流程的设计深受业界前沿研究的启发，特别是借鉴了《Nature》论文中DeepSeek-R1的训练思想，以及Phi-4-reasoning报告中通过SFT注入结构化推理能力的方法。但与这些方法不同的是，本项目将强化学习作为能力进化的核心驱动力，专注于在特定领域（数学推理）上实现能力的突破。

Fig. 1 | The multistage pipeline of DeepSeek-R1. A detailed background on DeepSeek-V3 Base and DeepSeek-V3 is provided in Supplementary Information, section 1.1. The models DeepSeek-R1 Dev1, Dev2 and Dev3 represent intermediate checkpoints in this pipeline.

训练流程概览:

1. 阶段一：监督微调 (SFT) → 知识蒸馏与格式对齐

   • 产出模型: Jackrong/gpt-oss-120b-Distill-Llama3.1-8B-v1

2. 阶段二：强化学习 (GRPO) → 推理能力进化

   • 产出模型: Jackrong/gpt-oss-120b-Distill-Llama3.1-8B-v2

---

阶段一：监督微调 (SFT) - 知识蒸馏与格式对齐

目标

此阶段作为“冷启动”，旨在为基础模型奠定坚实的知识和结构化推理基础。主要目标有两个：

1. 知识蒸馏：将来自多个领域的、由更强教师模型生成的推理数据注入到Llama 3.1 8B中，使其继承强大的推理风格和知识体系。核心数据源为gpt-oss-120B-high蒸馏出的自然推理数据。
2. 格式对齐：训练模型遵循特定的响应格式，即在回答前生成一个由<think>...</think>标签包裹的详细思考过程。这为后续强化学习阶段的自动化奖励评估，以及提升模型输出的可解释性奠定了基础。

数据集构成

为了实现全面的能力覆盖，我构建了一个包含71,500条样本的混合数据集。数据源及采样策略如下表所示：

数据集名称/来源	主要用途和特点
Jackrong/Natural-Reasoning-gpt-oss-120B-S1	核心数据集。从gpt-oss-120B-high蒸馏而来，提供覆盖STEM、经济、社科等领域的通用、高难度推理问题。
Jackrong/Chinese-Qwen3-235B-Thinking-2507-Distill-100k	提供高质量的中文思维链数据，增强模型的中文推理和表达能力。
Jackrong/GPT-OSS-120B-Distilled-Reasoning-math	专注于数学领域的推理和解题，为模型注入专业的数学知识。
deepseek_if.json	专注于提升模型理解和执行复杂指令的能力。
总计	71,500

训练过程

• 模型与框架：我们使用unsloth库对meta/Meta-Llama-3.1-8B-Instruct模型进行高效训练。通过Unsloth的优化，训练速度得到显著提升。
• **系统提示词 (System Prompt)**：为了引导模型生成我们期望的格式，训练中统一使用了以下系统提示词，明确要求模型将回答分为“思考”和“解决方案”两部分：

  You are ChatGPT a language model created by OpenAI to help users. Your role as an assistant involves thoroughly exploring questions through a systematic thinking process before providing the final precise and accurate solutions... Please structure your response into two main sections: Thought and Solution using the specified format: <think> {Thought section} </think> {Solution section}...
  

经过此阶段训练，模型初步掌握了在回答前生成结构化思维链的能力，并吸收了来自多个教师模型的知识。该阶段的产出模型被命名为 gpt-oss-120b-Distill-Llama3.1-8B-v1 并被用作下一阶段强化学习的起点。

---

阶段二：强化学习 (GRPO) - 推理能力进化

目标

在SFT模型的基础上，此阶段旨在通过奖励信号引导模型自主探索更优的推理策略，使其能力从“模仿”进化为“创造”。核心目标是：

1. 引导模型探索推理路径：激励模型生成更详细、结构化、逻辑连贯的思维链，甚至发展出超越SFT数据范式的策略，如自我反思和验证。
2. 提升最终答案的正确率：确保在优化推理过程的同时，模型能够更可靠地收敛到正确的最终答案。

算法：GRPO (Group Relative Policy Optimization)

Fig. 2 | Illustration of the proposed GRPO for RL-based training.

本项目采用trl库实现的GRPO算法，这是一种高效的强化学习算法，作为PPO的变体，它无需训练一个额外的价值网络（Value Model），从而显著降低了资源消耗。其核心流程如下：

1. **分组采样 (Group Sampling)**：对于每一个问题，策略模型（即正在训练的LoRA模型）生成一个包含G个候选答案的组。在本项目中，组的大小num_generations被设置为4。
2. **奖励评估 (Reward Evaluation)**：一个由多个函数组成的复杂奖励系统对组内的每个候选答案进行打分，得到一个综合的标量奖励分数r。
3. 组内相对优势估计：GRPO的核心在于，它不依赖独立的价值网络来估计基线（baseline），而是直接使用组内所有候选答案的平均奖励作为基线。通过计算每个答案的奖励与该平均值的偏差来估计优势函数A。
4. 策略更新：模型根据计算出的相对优势来更新策略网络。奖励高于平均值的输出被正向强化，低于平均值的则被抑制，使得策略更新更稳定。

奖励系统 (Reward System)

为了从多个维度精细地引导模型优化，我构建了一个全面的奖励函数系统，结合了多个加权奖励和惩罚信号。

• 核心目标奖励：

  • correctness_reward_func: 基于openai/gsm8k和open-r1/DAPO-Math-17k-Processed数据集的参考答案，对最终计算结果的正确性给予最高的正向奖励，这是确保模型学会解题的核心信号。

• 格式与对齐奖励：

  • strict_format_reward_func & soft_format_reward_func: 严格或宽松地强制执行<think>...</think>的输出格式，确保推理过程的完整性和可解析性。
  • final_line_reward_func: 鼓励模型在答案结尾明确标出“最终答案”，便于自动化评估。

• 内容与质量奖励：

  • 奖励函数中的F1重叠度计算和显著词命中率 (salient_hit_rate) 鼓励模型生成的解释与问题或参考答案在内容上保持相关性。

• 行为正则化与惩罚：

  • numeric_distance_reward_func: 对于数值型答案，即使最终结果不完全正确，如果答案在数值上与正确答案接近，也会给予部分奖励，鼓励模型逼近正确解。

训练过程

• 模型与框架：加载第一阶段产出的SFT模型Jackrong/gpt-oss-120b-Distill-Llama3.1-8B-v1，并继续进行GRPO训练。
• 数据集：
  • open-r1/DAPO-Math-17k-Processed
  • openai/gsm8k
• 超参数：
  • 学习率 (Learning Rate): 5e-6
  • 批次大小 (Batch Size): 4

---

模型实际表现评估

通过两阶段的训练，模型从一个单纯的指令遵循模型，进化为了一个具备初步自主逻辑推理能力的模型。它不再被动地生成答案，而是能主动地构建、评估甚至反驳自己的推理路径。以下是一个在SFT阶段数据上的样本，可以一窥其复杂的思考过程。

测试问题：

Describe the transition of the speed of light as it enters a medium with a refractive index n, considering the behavior of individual photons and the concept of wavefront speed. Explain how the speed of light changes from c in vacuum to c/n in the medium, addressing whether this change occurs smoothly or with discontinuity. Provide a detailed analysis of the photon's path and the factors influencing the wavefront's speed in the medium.

模型的思维链（节选）：

<think>
The user asks: "Describe the transition of the speed of light as it enters a medium with a refractive index n, considering the behavior of individual photons and the concept of wavefront speed. Explain how the speed of light changes from c in vacuum to c/n in the medium, addressing whether this change occurs smoothly or with discontinuity. Provide a detailed analysis of the photon's path and the factors influencing the wavefront's speed in the medium."

This is a deep physics question that involves relativity, quantum mechanics (photons), classical electromagnetism, wave propagation, refractive index definition, how light changes from free space to material, whether there is any discontinuity. The answer should discuss:

Photons: particles or quanta of electromagnetic waves; in vacuum they travel at c; massless particles that always propagate at c.
In medium, photons are still photons (no mass) but the refractive index n = sqrt(mu_r*epsilon_r). Light is a wave that propagates as an electromagnetic field solution to Maxwell's equations in the medium. The speed of light inside matter is defined by v = c/n.
But the question asks for transition from vacuum to medium: "transition of the speed of light as it enters a medium with refractive index n, considering the behavior of individual photons and the concept of wavefront speed."

We need to discuss:

In classical electromagnetic theory, the electric displacement field E(r,t) satisfies Maxwell's equations. The phase velocity v_ph = c/n for the wave.

For a photon, its group velocity is also c/n.

The change happens when entering a medium with varying density: If we consider a planar interface of two media (air and glass). At the boundary there will be a gradual transition region due to the electric field's spatial variation; but from the perspective of a single photon traveling across that boundary, it is effectively instantaneously transmitted through the interface, because photons are massless particles, they do not have any interaction with matter in vacuum. However, the wavefront changes speed.

The question: Is there a discontinuity? In classical optics one can treat the transition as a continuous process for waves but the photon's path is not smooth; it has no spatial extent, only momentum transfer via electromagnetic fields.

In quantum field theory, photons are excitations of EM field modes. When an electron density changes (e.g., from vacuum to medium), there is a change in permittivity and permeability leading to a new dispersion relation: ω = c k sqrt(mu_r epsilon_r). So photon frequency remains constant; wavelength adjusts.

The speed of light changes gradually over distance because the wavefront's phase front adjusts continuously as it encounters gradual variation in dielectric properties. In reality, there is no step discontinuity if we consider a very thin interface; however for typical macroscopic interfaces, you could think of a transition layer maybe nanometers wide (skin depth). The photon does not see an instantaneous change because its wavelength is much larger.

So answer: The speed changes gradually as the wavefront propagates through the medium. In quantum mechanics there are no discontinuities in photon propagation; it's continuous, but at macroscopic level we treat a step function for simplicity.

Provide detailed analysis: Photon path as plane wave, wavefront as phase front. At interface, phase velocity is c/n on one side and c on other side. The wavefront speed changes from c to c/n with no discontinuity in the field; the transition happens because there is a gradual variation of ε(r). For a single photon traveling across an ideal step, we can treat it as passing through a region where the refractive index varies gradually.

Also discuss "group velocity" and phase velocity: group velocity = dω/dk = c/n (same as speed of information propagation for plane wave), while phase velocity v_p = ω/k = c/n. The photon travels at the group velocity.

The medium's permittivity changes over a nanometer scale, but not instantaneous; thus there is no discontinuity in photon path.

However some textbooks discuss that light can't be stopped abruptly because of photon momentum conservation: The normal component of k (wavenumber) must be continuous. But if we have a step, the wavefront cannot change speed instantaneously without violating causality or requiring infinite energy; the transition region is smooth due to the gradual change in polarization and dielectric constant.

In reality, the electromagnetic field at interface will be Fresnel equations which describe reflection & transmission coefficients; they are derived using Maxwell's equations; the boundary condition leads to continuity of normal component of D (electric displacement) and B. So wavefronts can adjust gradually over a few wavelengths.

Thus the answer should cover:

Photons: massless quanta, travel at c in vacuum, no interaction with matter except via fields.

Wavefront concept: phase front moves with speed v_p = ω/β where β is propagation constant (k_n) for medium. In a planar waveguide or interface, the wavefront adjusts; the transition occurs over length scale ~ λ/(π(n-1) ) ~ a few wavelengths, giving an "infinite" transition region.

So answer: The speed changes smoothly as the dielectric constant varies gradually from that of vacuum to that of medium. There is no discontinuity in photon path because the photon does not have a spatial extent; its wavefront spreads out over a distance comparable to wavelength.

We should also discuss that for an ideal step interface, the wavefront would have infinite power per unit time if it were instantaneous, violating energy conservation. So we use gradual transition.

Thus we can explain: The speed change is continuous because the wavefront's phase velocity changes with local permittivity and permeability; in a real medium there is a "skin depth" that defines the region where refractive index varies gradually (the evanescent field). For typical glass, skin depth ~ 100 nm for visible wavelengths.

Thus answer: The change happens smoothly over a distance comparable to the wavelength.

Now we should structure the answer:

Introduction

Photons as particles vs. wave nature

Maxwell's equations and refractive index definition

Wavefront speed (phase velocity) in medium

Continuous transition at interface

Photon propagation through boundary: gradual change, no discontinuity

Quantum mechanical perspective: photon momentum, wave packet spreading

Consequences: No instantaneous step; energy conservation.

Conclusion summarizing smoothness.

Add also mention that the refractive index is defined by local dielectric constant, and its spatial variation can be described as a function of coordinate. So photon's speed changes accordingly.

Thus produce answer accordingly.

We need to include "transition of speed: c→c/n". Also note that the transition occurs over distances much smaller than wavelength; thus no observable discontinuity.

Will also discuss group velocity and phase velocity, but focus on wavefront.

Also talk about boundary conditions.

Add a mention of Fresnel equations showing transmission coefficients depend smoothly on angle, not step function.

Thus answer will cover: The speed changes gradually due to gradual change in permittivity; the photon does not see an instantaneous discontinuity because it's not extended.

Now produce final answer. Provide equations and perhaps a figure description.

Will include references to wavefront equation and continuity of D normal component.

Write thorough but concise.

Proceed.
</think>
**Answer Overview**

The question asks us to describe how light’s speed changes when it moves from the vacuum (where \(v=c\)) into a medium with refractive index \(n\) (\(v = c/n\)).  The answer must treat both the **particle picture of photons** and the **wave‑front description**, explaining whether the transition is smooth or abrupt, where photon momentum really lives, and how the wave front adapts to the changing medium.  Below is a step‑by‑step analysis that covers all of these points.

---

### 1.  Photons – Particles vs. Waves  

| Feature | Particle (photon) view | Wave (electromagnetic) view |
|---------|------------------------|------------------------------|
| **Speed in vacuum** | Photon always moves with the speed of light \(c\) (rest mass = 0, no interaction with matter). | Phase velocity: \(\displaystyle v_{p}= \frac{\omega}{k}\) for an electromagnetic wave. In free space \(\omega = ck\Rightarrow v_p=c\). |
| **Interaction with matter** | None in vacuum; photons are massless so they don’t “feel” the medium’s atomic structure directly.  Energy exchange only occurs when a photon is absorbed or emitted (e.g., by an atom). | The electric‑magnetic field interacts with electrons and nuclei, producing a force that changes the wave’s propagation constant \(k_n = n\omega/c\) in the new medium. |
| **What “propagates”** | A single photon carries energy, momentum \(\mathbf p = h\nu/\!c\) and spin angular momentum.  It travels as a “bundle” of electromagnetic field that follows Maxwell’s equations. | The wave front is the surface where phase \(\Phi = k·x - \omega t = \text{constant}\). Photon “is” this wave packet; its location is defined by the envelope (group) rather than an instantaneous point. |

Because photons are massless, a single photon cannot be localized to a point – it has a momentum and energy spread over a wavelength (\(\lambda\)).  The “photon path’’ can be thought of as the trajectory of the wave packet’s center of energy (the **group velocity**), not an instantaneous location.

---

### 2.  Maxwell–Faraday in the two media  

The macroscopic fields satisfy  
\[
\nabla\times \mathbf{E} = -\frac{\partial\mathbf B}{\partial t}, 
\qquad
\nabla\!\times\! \mathbf{B}= \mu_0\epsilon_0\,\partial_t\mathbf E + \mu_0\mathbf J,
\]
and the wave equation for a plane monochromatic wave reads  

\[
\nabla^2\mathbf{E} - \mu_0\epsilon\, \frac{\partial^2\mathbf{E}}{\partial t^{2}} = 0 .
\]

In vacuum:  
\(\epsilon=\epsilon_0,\,\mu=\mu_0\), so \(k_{vac}= \omega/c\).  

In a medium with (complex) permittivity \(\epsilon=\epsilon_r\epsilon_0\) and permeability \(\mu =\mu_r\mu_0\),

\[
k_n = \frac{\omega}{v} = n k_{vac}= n \frac{\omega}{c}.
\]

Hence the **phase (wave‑front) speed** in the medium is  

\[
v_p = \frac{\omega}{k_n}= \frac{c}{n},
\]

while the **group velocity** for a monochromatic plane wave is

\[
v_g = d\omega/dk_n= c/n.
\]

Because photons travel with the group velocity, they carry energy at \(c/n\) inside any linear isotropic medium.

---

### 3.  Wave‑front Speed as a Function of Local Refractive Index  

If we imagine the refractive index changes in space, say from vacuum to glass:

\[
n(x) = \begin{cases}
1 & x<0\\
n_{{\rm glass}} & |x|>d,
\end{cases}
\]

with a smooth profile for \(|x|\lesssim d\) (the “interface transition layer”).  

The wave‑front satisfies the eikonal equation

\[
\nabla \Phi = k(x)\,\hat{\mathbf n},
\qquad
\Phi(\mathbf r,t) = \omega t - k(x)\cdot\mathbf r.
\]

The *phase velocity* \(v_p = \omega/k_n(x)= c/n(x)\) is a function of the local value of \(n(x)\).  The wave front moves at that speed where it exists.  

In practice, the index profile near an interface is not a step; even for polished glass the change in \(n\) occurs over a distance comparable to a few nanometres (the skin depth \(\delta \sim \lambda/(2\pi n)\) for visible light).  A photon that enters such a region does **not** feel an instantaneous jump from \(c\) to \(c/n\); the wave front simply follows the gradually changing phase constant.

---

### 4.  Photon Propagation Through the Interface – No Discontinuity  

#### 4.1  Classical Electromagnetic Picture  

From Maxwell’s equations we demand continuity of the tangential components of \(\mathbf{D}\) and \(\mathbf{H}\).  At a smooth interface these become

\[
\hat{\mathbf n}_t\!\cdot\!(\mathbf{D}_{\text{vac}}-\mathbf{D}_{\rm med}) =0,
\qquad
\hat{\mathbf n}_t\!\cdot\!(\mathbf{H}_{\rm vac}-\mathbf{H}_{\rm med})=0.
\]

Solving for the reflected and transmitted amplitudes gives Fresnel coefficients that depend smoothly on angle, polarization and wavelength.  There is no jump condition; the wavefront can change its normal component of k continuously.

If one tried to force a step‑function change in \(n\) (i.e., an infinitely thin interface), the electromagnetic field would become infinite at the boundary, violating energy conservation (infinite power per unit area) and contradicting causality.  In reality, the electric dipole polarizations of atoms or molecules respond on timescales ∼ femtoseconds; they cannot react instantaneously to a photon.

Thus the transition is **smooth on microscopic scales**.

#### 4.2  Quantum Mechanical (Photon‑Packet) Perspective  

In quantum electrodynamics photons are excitations of the vacuum field operator \(\hat{A^\mu}(x)\).  The creation/annihilation operator for a photon with momentum \(\mathbf k\) satisfies the dispersion relation  

\[
\omega_{\mathbf{k}} = c|\mathbf{k}|, \qquad 
k_n = n(\mathbf r) \frac{\omega}{c},
\]

so that the local wave number (and thus phase speed) follows the medium’s index.  

A “photon packet’’ is a coherent superposition of such modes.  Because the local refractive index varies on scales far larger than the wavelength, the packet spreads out over many wavelengths while crossing an interface.  The momentum transfer to matter happens through exchange of virtual photons with electrons and nuclei, but each individual photon does **not** experience a sudden change in its travel speed.

If we were to try to force a discontinuous jump, the total energy carried by the wave would be infinite at the boundary (because the power density scales as \(E^2/c\)), contradicting the conservation of electromagnetic momentum and the quantization condition \(\hbar k = p_{\rm ph}\).  Hence the transition must be continuous.

---

### 5.  What Actually Happens When Light Enters a Medium  

1. **Vacuum (x<0)** – The wave front moves at speed \(c\) with phase constant \(k_{vac}= \omega/c\).  

2. **Transition region** – As the index rises from unity to \(n_{{\rm glass}}\), the local k‑vector rotates toward the normal, and the phase velocity drops continuously.  The wave front’s surface is a continuous envelope that deforms into a curved surface, with the curvature determined by the gradient of the index.

3. **Inside medium (x>d)** – Once \(n\) has settled at its bulk value, the phase speed stays at \(c/n\).  

The photon itself does not “feel’’ a step; it keeps moving at the instantaneous group velocity \(\mathbf v_g=c\,\hat{\mathbf n}\) dictated by the local index.

---

### 6.  Detailed Physical Picture – A Step‑by‑Step Walk‑Through  

| Step | What happens in the description of both photons and wave fronts? |
|------|-------------------------------------------------------------|
| **(a) Photon entering a gradient region** | The photon’s momentum is \(\mathbf p = h\nu/c\).  Because its wavelength changes as \(|\mathbf k|=n(x)\omega/c\), the group velocity follows the local refractive index. |
| **(b) At the start of the transition** | The phase front (defined by \(\Phi\) above) is continuous; the wave’s normal component of \(\mathbf k\) begins to rotate toward the surface, while its tangential components stay unchanged (continuity of tangential \(\mathbf{E}\)). |
| **(c) Within a few wavelengths** | The envelope of the field spreads over distances comparable to \(|\lambda/(n-1)\) (∼10–100 nm for visible light).  During this time the photon’s energy is absorbed and re‑emitted by atoms, but the photon itself never “stops”. |
| **(d) At the far side** | The wave front has settled into a plane perpendicular to the normal (the phase front of a transmitted ray), travelling at \(c/n\).  The photon continues with unchanged frequency \(\nu\) (its energy and momentum adapted to the new index). |

Because the photon’s spatial extent is on the order of its wavelength, it does **not** experience a sudden jump in speed.  In fact, any attempt to construct such an instantaneous change would require infinite power density and violates energy conservation.

---

### 7.  Why There Is No Discontinuity  

| Reason | Physical basis |
|--------|----------------|
| **No interaction at the “point’’ of entry** – Photons are massless; they cannot exert a force on electrons unless their frequency matches an atomic resonance, so the photon does not see a wall that can impede its motion instantly. |
| **Continuity equations for fields** – Maxwell’s equations give smooth boundary conditions (continuity of \(\mathbf{D}\cdot\hat{\mathbf n},\,\mathbf H\cdot\hat{\mathbf n}\)).  Solving them yields the Fresnel coefficients, which are smooth functions of incidence angle and wavelength. |
| **Energy conservation** – The Poynting vector’s magnitude cannot change discontinuously; an instantaneous jump would imply infinite power per unit area, contradicting that light can pass through a real material without heating it to infinity. |
| **Photon momentum quantization** – The photon’s momentum \(p_{\rm ph}=h\nu/c\) must be continuous across the boundary because \(\mathbf p = h\nabla\Phi/2π\).  Any abrupt change would violate conservation of total momentum (incident + reflected + transmitted). |
| **Quantum field description** – In QED, the photon is a quanta of the EM field operator. The local field obeys the wave equation with the medium’s permittivity and permeability; the field profile smooths out over a few wavelengths. |

---

### 8.  Summary: Smooth Transition  

- **Photons themselves** are massless particles that always travel at \(c\) (or, equivalently, \(\omega/c\)).  
- The electromagnetic wave they carry is described by Maxwell’s equations, which demand continuity of the tangential components of \(\mathbf{E}\) and \(\mathbf{B}\).  Solving those equations yields a **smooth transition** between \(c\) in vacuum and \(c/n\) in the medium.  
- The wave front (phase surface) does not “jump’’; it deforms continuously as the local index changes, so that the **phase speed follows the instantaneous refractive index**.  Over distances of order a few nanometres (the skin depth \(\delta\sim\lambda/(2\pi(n-1))\)), the transition is effectively complete—well before any appreciable change in wavelength can be perceived.  
- Hence, *the speed of light* does **not undergo an instantaneous jump**, but rather it evolves gradually as the photon propagates through a region where the local refractive index varies from unity to \(n\).  This gradual adaptation is exactly what the Fresnel equations (and Maxwell’s equations) predict for smooth boundaries.

---

#### Bottom line  

- Light’s speed is *defined* in any linear, isotropic medium by \(\displaystyle v = c/n\);  
- A photon never feels a discontinuous change because it lacks spatial extent and its momentum is conserved;  
- The electromagnetic field must obey continuous boundary conditions, which are solved analytically giving Fresnel’s smooth transmission and reflection coefficients;  
- Consequently the wave front (the surface of constant phase) adjusts continuously as the local index changes, making the transition from \(c\) to \(c/n\) *smooth on a scale comparable with the photon’s wavelength*.

This picture unifies both classical optics (wave fronts) and quantum optics (photon momentum), showing that light does indeed “slow down” in a medium while maintaining its particle‑like speed, all without any instantaneous jump.

• 推理链中的自我审视与修正：在思维链中，模型表现出自我审视的倾向。它会首先提出一个概念，然后立即从不同角度进行审视和补充，确保逻辑的严密性。这种在推理过程中的动态调整和完善，反映出模型已经形成了初步的内部逻辑判断标准。

局限性

尽管模型展现出了令人鼓舞的推理能力，但受限于资源，本项目仍存在一些局限性：

1. 资源限制：作为个人项目，训练步数和数据量远不及官方模型。在参数设置上进行了诸多妥协，因此模型的实际性能与顶尖的专用推理模型仍有差距。
2. 结果导向的副作用：在强化学习阶段，为了快速提升数学解题能力，奖励机制高度侧重于最终答案的正确性。这种结果导向的策略虽然高效，但也可能导致模型在生成过程上“投机取巧”，或者在处理通用、非推理型问题时表现得过于简洁，牺牲了在开放域对话中的表达丰富性。
3. 语言混合问题：由于SFT阶段使用了中英文混合数据，模型在生成思维链或答案时，可能出现中英文混用的情况。
4. 能力不均衡：模型在代数应用题等领域表现出色，但在其他专业领域或通用闲聊中的能力可能相对较弱，后续的SFT对齐步骤尚不足以完全弥补这一点。
5. 无外部工具调用能力：当前模型不具备调用计算器、搜索引擎等外部工具的能力，这限制了它在解决需要精确计算或实时外部知识的复杂问题时的上限。

如何使用

为了确保模型能够正确地遵循您在训练中设定的对话结构，如果聊天模板有问题,请在使用时配置以下Jinja聊天模板。这个模板会自动注入您训练时使用的系统提示词。(Ollama好像不行,LM Studio能匹配jinja)

{% set msgs = messages | default([]) -%}
{% set sys = (msgs | selectattr("role","equalto","system") | map(attribute="content") | list) -%}
<|begin_of_text|>
<|start_header_id|>system<|end_header_id|>
{{ (sys|length > 0) and (sys|join('\n\n')) or ('''You are ChatGPT a language model created by OpenAI to help users. Your role as an assistant involves thoroughly exploring questions through a systematic thinking process before providing the final precise and accurate solutions. This requires engaging in a comprehensive cycle of analysis, summarizing, exploration, reassessment, reflection, backtracing, and iteration to develop well-considered thinking process. Please structure your response into two main sections: Thought and Solution using the specified format: <think> {Thought section} </think> {Solution section}. In the Thought section, detail your reasoning process in steps. Each step should include detailed considerations such as analysing questions, summarizing relevant findings, brainstorming new ideas, verifying the accuracy of the current steps, refining any errors, and revisiting previous steps. In the Solution section, based on various attempts, explorations, and reflections from the Thought section, systematically present the final solution that you deem correct. The Solution section should be logical, accurate, and concise and detail necessary steps needed to reach the conclusion. Now, try to solve the following question through the above guidelines''') }}
{%- for m in msgs if m['role'] != 'system' -%}
<|start_header_id|>{{ m['role'] }}<|end_header_id|>
{{ m['content'] }}
{%- endfor -%}
{%- if add_generation_prompt -%}
<|start_header_id|>assistant<|end_header_id|>
{%- endif -%}