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Equation 11 · Part 7 · Measuring Tool Protocols and the Model Context Protocol: Evidence, Benchmarks, and Uncertainty

subscript

nmax⁡≈(B−T0)/tˉn_{\max} \approx (B - T_0)/\bar{t}
subscript

What this part means

The lower label selects a particular version, component, or indexed member of the quantity. For example, x₀ and xₜ can be values at different positions.

Its job in the formula

A subscript distinguishes a version, component, step, or member of a quantity. It does not automatically mean multiplication.

The passage around this formula

Under a fixed context budget B , the largest catalogue that fits is roughly nmax⁡n_{\max} ≈\approx (B - T0T_0)/tˉ\bar{t} . The model is a reasonable first approximation, and it is also the assumption a tenfold per-tool variance directly undermines: tˉ\bar t is not a constant of the tool, it is a property of how verbosely that tool’s schema and description happen to be written, which is an editorial choice rather than a physical fact about the function. A recent controlled study of exactly this trade-off tested fourteen models from 1.5B to 32B parameters plus one frontier API model across 6,566 controlled tool calls at three context budgets, and found that at an 8K-token budget uncompressed JSON schemas…

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Learn the underlying idea

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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Sources cited in the surrounding passage

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