 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.