Typesetting Test: What 'Doubling Every 4 Months' Actually Means

This short note doubles as a typesetting test for this site (KaTeX math is rendered to static HTML at build time — zero client-side JS), using the arithmetic behind the headline numbers in Is AI Still Accelerating?.

A quantity that doubles every $T$ months grows in $t$ months by a factor of $2^{t/T}$ . So “capability doubles every ~4 months” compounds to

2^{12/4} = 2^3 = 8\times \text{ per year.}

Doubling time and annual growth factor $k$ convert via

T = \frac{12 \ln 2}{\ln k},

so a 200× annual price decline corresponds to a “halving time” of $12\ln 2/\ln 200 \approx 1.57$ months — prices halve roughly every 7 weeks.

A literal dollar sign also needs to survive the math pipeline: $10 per million tokens should render as plain text, not as a formula.