= Universality of an invariant speed under boost composition
Suppose standard collinear boosts have the <Lorentz transformation> form, but initially assign speed scales $a$ and $b$ to two successive transformations with relative <velocities> $v$ and $u$. Their composition has spatial and temporal diagonal factors proportional to $1+uv/a^2$ and $1+uv/b^2$, respectively. Closure within the same calibrated transformation family requires these factors to agree, forcing $a^2=b^2$ for generic nonzero boosts.
This argument fixes a universal reciprocal squared speed; it does not establish that it is nonzero. The common limit $1/a^2=0$ gives the consistent <Galilean transformation> family. A finite invariant <speed of light> is an additional empirical property of spacetime and electromagnetism, expressed by the second of the <postulates of special relativity>. It cannot be deduced by excluding the Galilean limit through boost composition alone.
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