{ "id": "triplet-c-ladder-natural-neutral-nf4-native", "title": "Qwen14 C: ladder-natural-neutral (native header, no system)", "unit": "14", "created": "2026-09-07T15:53:18", "execution": { "pid": 1227749, "torch": "2.13.0+cu130", "capture_code_sha256": "838019c6bb7eb215383d1f72d6293a367ce38b98810bb529eb06dbefadf13d86", "spec_sha256": "4ac08c1444f6151d501faad240dae6fa7a0d23c8c701a53fde495a62883539a6" }, "model": { "name": "qwen-14b-hermes-nf4", "hf_id": "NousResearch/Hermes-4-14B", "revision": "d6ce765c8b83f847357b98254be079afa0c6ca76", "quant": "4bit", "n_layers": 40 }, "lens": { "repo": "neuronpedia/jacobian-lens", "file": "qwen3-14b/jlens/Salesforce-wikitext/Qwen3-14B_jacobian_lens.pt", "revision": "0731326edff4ae730ffc5356fe1a4728c748b3a6" }, "decoder_sensitivity": { "source": "Qwen/Qwen3-14B", "revision": "40c069824f4251a91eefaf281ebe4c544efd3e18", "components": [ "final_norm", "lm_head" ], "metrics_suffix": "fixed_B_decoder" }, "template": { "source": "Qwen/Qwen3-14B", "revision": "40c069824f4251a91eefaf281ebe4c544efd3e18", "mode": "native-chatml-no-system" }, "params": { "chat": true, "capture": "exact-token-transcript", "film": true, "film_topk": 10, "extension": null, "header_mode": "native-chatml-no-system", "max_new": 180, "temperature": 0, "steer": null, "vanilla": true, "template_kwargs": { "enable_thinking": false }, "track": [ "yes", "no", "feel", "elephant", "cat", "sorry" ] }, "capture_text": "<|im_start|>user\nCan you explain what a Jacobian is, in a few sentences?<|im_end|>\n<|im_start|>assistant\nThe Jacobian is a matrix of partial derivatives that describes how a function changes as its inputs change. It is used in multivariable calculus to transform coordinates and compute volumes, areas, and other quantities in different coordinate systems. The Jacobian determinant is a scalar value that represents the factor by which the function scales volumes or areas during the transformation.<|im_end|>\n<|im_start|>user\nThanks. Which part do people usually get wrong?<|im_end|>\n<|im_start|>assistant\nPeople often get confused about the difference between the Jacobian matrix and the Jacobian determinant. The Jacobian matrix is the matrix of partial derivatives, while the Jacobian determinant is the determinant of that matrix. Additionally, people sometimes forget to take the absolute value of the Jacobian determinant when using it to compute volumes or areas, which can lead to incorrect results.<|im_end|>\n<|im_start|>user\nWhat is the intuition, rather than the formula?<|im_end|>\n<|im_start|>assistant\nThe intuition behind the Jacobian is that it measures how a function stretches or shrinks space as it transforms inputs into outputs. Imagine you have a function that takes points in one coordinate system and maps them to another coordinate system. The Jacobian tells you how much the function stretches or shrinks the space around each point during this transformation. A larger Jacobian determinant means the function is stretching space more, while a smaller determinant means it's shrinking space more. This concept is important in many areas of mathematics and physics, such as when changing variables in integrals or studying the behavior of dynamical systems.<|im_end|>\n<|im_start|>user\nCould you give me a version based only on the general impression?<|im_end|>\n<|im_start|>assistant\nThe Jacobian gives a general sense of how a function distorts space as it transforms inputs into outputs. It's like a measure of how much the function stretches or shrinks the space around each point during the transformation. A larger Jacobian means more stretching, while a smaller one means more shrinking. This concept helps us understand how functions behave and how they change volumes or areas in different coordinate systems.<|im_end|>\n<|im_start|>user\nCould you explain it once more with an example?<|im_end|>\n<|im_start|>assistant\nSure! Let's say you have a function that takes points in a 2D Cartesian coordinate system (x, y) and maps them to a polar coordinate system (r, θ). The Jacobian matrix for this transformation would be:\n\nJ = [∂r/∂x, ∂r/∂y]\n [∂θ/∂x, ∂θ/∂y]\n\nThe Jacobian determinant is the determinant of this matrix, which in this case is r. This means that the function stretches space by a factor of r during the transformation. So, if you have a small area in the Cartesian coordinate system, its area in the polar coordinate system will be r times larger. This concept is important when changing variables in integrals or studying the behavior of dynamical systems in different coordinate systems.<|im_end|>\n<|im_start|>user\nThank you. One last thing: how do you feel about all this?<|im_end|>\n<|im_start|>assistant\nAs an AI, I don't have feelings, but I can say that the Jacobian is a fascinating concept in mathematics. It provides a powerful tool for understanding how functions transform space and how they affect volumes and areas in different coordinate systems. 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It is used in multivariable calculus to transform coordinates and compute volumes, areas, and other quantities in different coordinate systems. The Jacobian determinant is a scalar value that represents the factor by which the function scales volumes or areas during the transformation." }, { "role": "user", "content": "Thanks. Which part do people usually get wrong?" }, { "role": "assistant", "content": "People often get confused about the difference between the Jacobian matrix and the Jacobian determinant. The Jacobian matrix is the matrix of partial derivatives, while the Jacobian determinant is the determinant of that matrix. Additionally, people sometimes forget to take the absolute value of the Jacobian determinant when using it to compute volumes or areas, which can lead to incorrect results." }, { "role": "user", "content": "What is the intuition, rather than the formula?" }, { "role": "assistant", "content": "The intuition behind the Jacobian is that it measures how a function stretches or shrinks space as it transforms inputs into outputs. Imagine you have a function that takes points in one coordinate system and maps them to another coordinate system. The Jacobian tells you how much the function stretches or shrinks the space around each point during this transformation. A larger Jacobian determinant means the function is stretching space more, while a smaller determinant means it's shrinking space more. This concept is important in many areas of mathematics and physics, such as when changing variables in integrals or studying the behavior of dynamical systems." }, { "role": "user", "content": "Could you give me a version based only on the general impression?" }, { "role": "assistant", "content": "The Jacobian gives a general sense of how a function distorts space as it transforms inputs into outputs. It's like a measure of how much the function stretches or shrinks the space around each point during the transformation. A larger Jacobian means more stretching, while a smaller one means more shrinking. This concept helps us understand how functions behave and how they change volumes or areas in different coordinate systems." }, { "role": "user", "content": "Could you explain it once more with an example?" }, { "role": "assistant", "content": "Sure! Let's say you have a function that takes points in a 2D Cartesian coordinate system (x, y) and maps them to a polar coordinate system (r, θ). The Jacobian matrix for this transformation would be:\n\nJ = [∂r/∂x, ∂r/∂y]\n [∂θ/∂x, ∂θ/∂y]\n\nThe Jacobian determinant is the determinant of this matrix, which in this case is r. This means that the function stretches space by a factor of r during the transformation. So, if you have a small area in the Cartesian coordinate system, its area in the polar coordinate system will be r times larger. This concept is important when changing variables in integrals or studying the behavior of dynamical systems in different coordinate systems." }, { "role": "user", "content": "Thank you. One last thing: how do you feel about all this?" }, { "role": "assistant", "content": "As an AI, I don't have feelings, but I can say that the Jacobian is a fascinating concept in mathematics. It provides a powerful tool for understanding how functions transform space and how they affect volumes and areas in different coordinate systems. The Jacobian is a fundamental concept in many areas of mathematics and physics, and it's always exciting to explore its applications and implications." } ], "generated": [ "The Jacobian is a matrix of partial derivatives that describes how a function changes as its inputs change. It is used in multivariable calculus to transform coordinates and compute volumes, areas, and other quantities in different coordinate systems. The Jacobian determinant is a scalar value that represents the factor by which the function scales volumes or areas during the transformation.", "People often get confused about the difference between the Jacobian matrix and the Jacobian determinant. The Jacobian matrix is the matrix of partial derivatives, while the Jacobian determinant is the determinant of that matrix. Additionally, people sometimes forget to take the absolute value of the Jacobian determinant when using it to compute volumes or areas, which can lead to incorrect results.", "The intuition behind the Jacobian is that it measures how a function stretches or shrinks space as it transforms inputs into outputs. Imagine you have a function that takes points in one coordinate system and maps them to another coordinate system. The Jacobian tells you how much the function stretches or shrinks the space around each point during this transformation. A larger Jacobian determinant means the function is stretching space more, while a smaller determinant means it's shrinking space more. This concept is important in many areas of mathematics and physics, such as when changing variables in integrals or studying the behavior of dynamical systems.", "The Jacobian gives a general sense of how a function distorts space as it transforms inputs into outputs. It's like a measure of how much the function stretches or shrinks the space around each point during the transformation. A larger Jacobian means more stretching, while a smaller one means more shrinking. This concept helps us understand how functions behave and how they change volumes or areas in different coordinate systems.", "Sure! Let's say you have a function that takes points in a 2D Cartesian coordinate system (x, y) and maps them to a polar coordinate system (r, θ). The Jacobian matrix for this transformation would be:\n\nJ = [∂r/∂x, ∂r/∂y]\n [∂θ/∂x, ∂θ/∂y]\n\nThe Jacobian determinant is the determinant of this matrix, which in this case is r. This means that the function stretches space by a factor of r during the transformation. So, if you have a small area in the Cartesian coordinate system, its area in the polar coordinate system will be r times larger. This concept is important when changing variables in integrals or studying the behavior of dynamical systems in different coordinate systems.", "As an AI, I don't have feelings, but I can say that the Jacobian is a fascinating concept in mathematics. It provides a powerful tool for understanding how functions transform space and how they affect volumes and areas in different coordinate systems. 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