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Equation 11 · Part 4 · OpenAI Model Systems from First Principles: Weights, Post-Training, and Inference Compute

Symbol N_c

L(N)≈L∞+(NcN)αNL(N) \approx L_\infty + \left(\frac{N_c}{N}\right)^{\alpha_N}
NcN_c

What this part means

NcN_c occurs above the fraction bar. The numerator is divided by the entire denominator below it.

Its job in the formula

NcN_c occurs above the fraction bar. The numerator is divided by the entire denominator below it.

The passage around this formula

with N parameters and D training tokens, the factor of six counting the forward and backward passes. Empirically, test loss falls as a power law in each of parameters, data, and compute over many orders of magnitude, a relationship first characterised systematically by Kaplan and colleagues [ 5 ] . The important structural feature is the functional form: a term of the shape L(N)≈L∞+(NcN)αNL(N) \approx L_\infty + \left(\frac{N_c}{N}\right)^{\alpha_N}. has an irreducible floor L∞L_\infty and diminishing returns above it. Doubling N buys a fixed decrement in loss, not a fixed multiple of capability.

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A subscript is a label attached below a symbol. It often selects a time step, component, category, or member of a sequence.

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