Equation 12 · Codex vs Claude Code: A Systems Comparison, Not a Benchmark Beauty Contest

What does this equation mean?

G(n)=\frac{W_{\mathrm{useful}}(n)} {C_{\mathrm{inference}}(n)+C_{\mathrm{merge}}(n)+C_{\mathrm{review}}(n)}.

Read the formula alongside the article passage below. Each part has a deeper page with its role in the equation, the supporting passage and nearby citations.

Start withW_useful(n)

Divide byC_inference(n)+C_merge(n)+C_review(n)

This relates toG(n)

How to read the two sides of this formula. Follow the article passage for the meaning of each quantity.

This equation states an equality: the expressions on both sides have the same value under the article’s assumptions. Read the equation part by part below; each part has a contextual explanation and a link to its mathematical background.

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G

Symbol G

G is part of the quantity the equation computes from the expression on the right.

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n

Symbol n

n is an argument of the function-like quantity on the left; its role is set by that function’s stated inputs.

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W_{\mathrm{useful}}

Symbol W_useful

$W_u$ seful occurs above the fraction bar. The numerator is divided by the entire denominator below it.

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C_{\mathrm{inference}}

Symbol C_inference

$C_i$ nference occurs below the fraction bar. The quantity above the bar is divided by this expression; zero is excluded as a denominator.

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C_{\mathrm{merge}}

Symbol C_merge

$C_m$ erge occurs below the fraction bar. The quantity above the bar is divided by this expression; zero is excluded as a denominator.

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C_{\mathrm{review}}

Symbol C_review

$C_r$ eview occurs below the fraction bar. The quantity above the bar is divided by this expression; zero is excluded as a denominator.

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=

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

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fraction

Divide the expression above the line by the one below it.

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subscript

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.

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W_{\mathrm{useful}}(n)

Numerator: W_useful(n)

The complete quantity above the fraction bar.

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C_{\mathrm{inference}}(n)+C_{\mathrm{merge}}(n)+C_{\mathrm{review}}(n)

Denominator: C_inference(n)+C_merge(n)+C_review(n)

The complete quantity below the fraction bar; it must be nonzero for this division.

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How to interpret it

With a fixed numerator, increasing a nonzero denominator reduces the fraction. Read it with the definitions, units, and assumptions supplied by the article.

What the article says around this equation

The important metric is not maximum concurrent workers. It is useful independent work per coordination token. For n workers, $G(n)=\frac{W_{\mathrm{useful}}(n)} {C_{\mathrm{inference}}(n)+C_{\mathrm{merge}}(n)+C_{\mathrm{review}}(n)}$ . G(n) can decline when workers rediscover shared context, return verbose summaries, edit overlapping files, or rely on inconsistent assumptions. Start parallelism with read-heavy mapping, test analysis, security review, and documentation verification. Isolate writers in worktrees and keep a single owner for global invariants.

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Sources cited in the article section

These citations give research context. Read each source to check which claims it supports.

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