We begin by observing a striking "mirror-complement" pattern in the binary digits of √2: whenever, at any position, a run of k equal bits is separated by a single opposite bit from another run of bits, those two runs must have equal length. Restricting to prime-indexed positions, the same pattern remains perfectly true for millions of primes. This phenomenon is a direct consequence of the classical digit-by-digit square-root algorithm in base 2, because each comparison uses4Pn + 1 = 2 (2Pn) + 1,i.e. "copy + complement + copy."From this insight we build a two-rectangle coding on T² whose itinerary reproduces the binary digits of √2. A measurable conjugacy to the (½,½) Bernoulli shift allows us to apply Chung—Smorodinsky’s bounded-coboundary theorem (1967), showing each cylinder-indicator has a uniform sup-norm bound. Telescoping that coboundary yields a universal O(Nu207b¹) discrepancy bound on every length-ℓ binary block, proving base-2 normality of √2. Finally, van der Corput differencing and Wall’s criterion transfer the same O(Nu207b¹) bound to every integer base B ≥ 2, establishing that √2 is normal in all bases.This paper unifies these ideas—starting from the prime-indexed mirror pattern and culminating in a gap-free, self-contained proof of full normality of √2.