Exercise 3: Scaling at a Non-Doubling Font-Size — Possible Solution
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THE TEST
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    w16 = measure_word('Welcome', 16.0)
    w24 = measure_word('Welcome', 24.0)

RESULT
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    w16 -> 61.12
    w24 -> 91.68

91.68 is exactly 1.5 times 61.12 (61.12 * 1.5 = 91.68).

WHY THIS IS EXACTLY WHAT THE EM-BASED MODEL PREDICTS
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measure_word sums, for every character in the word, char_width_px(ch,
font_size_px) = CHAR_WIDTHS_EM[ch] * font_size_px. Summing across all
seven characters of 'Welcome':

    sum_of_em_values = W(0.94) + e(0.44) + l(0.28) + c(0.44) + o(0.50)
                      + m(0.78) + e(0.44)
                      = 3.82   (a fixed number, independent of font-size)

measure_word('Welcome', font_size_px) is therefore always exactly
3.82 * font_size_px, for ANY font_size_px:

    at 16px: 3.82 * 16 = 61.12
    at 24px: 3.82 * 24 = 91.68

Since 24 is exactly 1.5 times 16, and the whole expression is a
straight multiplication by font_size_px with no other font-size-
dependent term anywhere, 3.82*24 is necessarily exactly 1.5 times
3.82*16 -- this isn't a coincidence discovered by testing, it's a
direct algebraic consequence of the formula's own shape.

WHY THIS CONFIRMS GENUINE LINEARITY, NOT A 2X SPECIAL CASE
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The chapter's own main demonstration used 16px vs. 32px -- a clean
doubling, which could in principle be produced by a model that only
handles font-size correctly for round-number multiples (e.g., a lookup
table keyed by a small set of common sizes, or code with a special
case for "2x" specifically). Testing at 24px -- a genuinely different,
non-doubling multiple (1.5x) -- and finding the SAME proportional
relationship holds rules that possibility out. Because
sum_of_em_values is computed once, independent of font_size_px, and
then multiplied by whatever font_size_px happens to be, the model is
provably linear in font-size for ANY value, not just the specific ones
already tested -- a font-size of, say, 20px or 13px would work exactly
the same way, following directly from the same formula rather than
needing to be separately verified case by case.

WHY THIS WORKS AS AN ANSWER
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This is the mathematical guarantee that makes the em-based approach
genuinely superior to a naive lookup table keyed by absolute pixel
sizes: defining widths as a RATIO of the font's own size, once,
automatically produces correct measurements at every font-size a real
page might use, without needing a separate table entry -- or a
separate test -- for each one.