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Equation 6 · Part 7 · Context Window Size Versus What a Frontier Model Can Actually Recall From It

=

Leff(τ)=max⁡{ n≤W : S(n)≥τ⋅S0 }L_{\mathrm{eff}}(\tau) = \max\left\{\, n \le W \ :\ S(n) \ge \tau \cdot S_0 \,\right\}
=

What this part means

The expressions on both sides represent the same quantity under the stated assumptions.

Its job in the formula

The equals sign connects the complete expression on the left with the complete expression on the right. Both sides must have compatible units.

The passage around this formula

Put the documentation and the independent benchmarks side by side and a pattern emerges that is more informative than any single score. Formalise the distinction the whole comparison rests on: let W be the context window a vendor documents for a given model, a fixed engineering figure set by attention implementation, position encoding, and what the serving stack supports. Let S(n) be some benchmark’s measured accuracy at input length n ≤\le W , and S0S_0 the same benchmark’s accuracy at a short reference length. For a chosen retention threshold τ\tau , define the effective context length as Leff(τ)=max⁡{ n≤W : S(n)≥τ⋅S0 }L_{\mathrm{eff}}(\tau) = \max\left\{\, n \le W \ :\ S(n) \ge \tau \cdot S_0 \,\right\}. W is a documented constant, published on day one of a model’s release, identical no…

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Learn the underlying idea

An equals sign says that the expression on its left and the expression on its right have the same value under the stated definitions and assumptions.

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Sources cited in the article section

These citations provide research context; check each source for the exact claim it supports.