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Equation 1 · Part 8 · 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).
=

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

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 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.