Equation 5 · Astra's 63%-Cheaper Claim Has to Beat a Price Disadvantage Nobody Mentions
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Symbol s
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What the article says around this equation
Run that ratio across a stated bracket rather than picking one number and presenting it as fact: . At zero cache reuse, the two models cost the same on input tokens — unsurprising, since the base rate ties. From there, every step toward a higher cache-hit share moves the ratio in exactly one direction: against Astra. There is no cache-hit share above 0% at which Astra’s own published rate produces a cheaper input-token bill than Fable 5.1’s, under the equal-token-count assumption. This is not a sensitivity finding that happens to favor one reading over another; it is the direct, unavoidable consequence of a 4x rate gap combined with identical base pricing, and it holds…
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Run that ratio across a stated bracket rather than picking one number and presenting it as fact: . At zero cache reuse, the two models cost the same on input tokens — unsurprising, since the base rate ties. From there, every step toward a higher cache-hit share moves the ratio in exactly one direction: against Astra. There is no cache-hit share above 0% at which Astra’s own published rate produces a cheaper input-token bill than Fable 5.1’s, under the equal-token-count assumption. This is not a sensitivity finding that happens to favor one reading over another; it is the direct, unavoidable consequence of a 4x rate gap combined with identical base pricing, and it holds regardless of which specific share the real benchmark run actually used, because the function is monotonic across the entire range.
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