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Equation 1 · Part 10 · The Jagged Frontier Fallacy: Deconstructing Harvard's Canonical AI Productivity Study

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P(Y∣X)=∏t=1TP(yt∣y<t,X).P(Y \mid X) = \prod_{t=1}^{T} P(y_t \mid y_{<t}, X).
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What this part means

A raised number can be a power. When it is a label or bound, it selects a case or the upper limit of a sum; the formula’s structure distinguishes these uses.

Its job in the formula

A raised mark can be a power or an index. Its position and the surrounding notation determine which.

The passage around this formula

When a knowledge worker asks a bare conversational model to analyze a multi-variable spreadsheet with interdependent business constraints, the model must perform several disjoint cognitive operations within a single autoregressive forward pass: P(Y∣X)=∏t=1TP(yt∣y<t,X)P(Y \mid X) = \prod_{t=1}^{T} P(y_t \mid y_{<t}, X). Because token generation is strictly left-to-right and causal, any early latent error in intermediate arithmetic or structural interpretation compounds multiplicatively across subsequent tokens. If the prompt lacks an explicit scratchpad, state harness, or programmatic tool-execution loop, the model cannot pause, verify an invariant, execute a deterministic calculation, or revise an earlier premise [ 6 , 7 ] .

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Learn the underlying idea

An exponent tells how a base is used in multiplication. In x³, x is the base and 3 is the exponent: x³ = x × x × x.

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Sources cited in the article section

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