The Quarter Circle

We investigate the geometric properties of prime numbers in a multiplicative embedding where each integer n is represented as a vector of its prime exponents. In this space, primes are provably unit basis vectors — the irreducible anchors of the multiplicative geometry. We show analytically that primes have L2 norm exactly 1 while all composites have norm strictly greater than 1, and confirm numerically that primes exhibit significantly lower path-tension than composites (p ≈ 0). We further identify a structural even/odd heartbeat: the tension signal carries 3.44× excess power at frequency 0.5 (the even/odd alternation frequency), confirming that even numbers act as a periodic equilibrium structure. These results support a coherent geometric picture: primes are fixed minimal-energy reference points; composites oscillate around them; even numbers provide the dominant periodic rhythm. The multiplicative embedding is the correct instrument for these questions — a result confirmed by showing that the previously studied log-spiral embedding is blind to divisibility structure.