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Equation 26 · Part 4 · Dense, Sparse, and Distilled: Comparing Approaches to Frontier Model Capacity

Symbol M_kv

bmax⁡≈Mdevice−MweightsMkv(s),b_{\max} \approx \frac{M_{\mathrm{device}} - M_{\mathrm{weights}}}{M_{\mathrm{kv}}(s)},
MkvM_{\mathrm{kv}}

What this part means

MkM_kv occurs below the fraction bar. The quantity above the bar is divided by this expression; zero is excluded as a denominator.

Its job in the formula

MkM_kv occurs below the fraction bar. The quantity above the bar is divided by this expression; zero is excluded as a denominator.

The passage around this formula

Trace one design decision through that structure to see why the tiers behave as they do. Suppose the middle tier’s key–value cache is halved by moving from multi-head to grouped-query attention. Achievable batch size is roughly bmax⁡≈Mdevice−MweightsMkv(s)b_{\max} \approx \frac{M_{\mathrm{device}} - M_{\mathrm{weights}}}{M_{\mathrm{kv}}(s)}. so halving the denominator roughly doubles concurrency at a given context length, which roughly halves cost per request at fixed hardware. The quality cost of that change, per the uptraining results, is small and concentrated on tasks that stress long-range attention [ 8 ] . A buyer sees a price cut. What actually happened is a targeted quality trade whose effects appear on a specific class of input.

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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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