The divisor angle of an integer

To a positive integer n we attach the angle θ(n)∈[0,π/2) between the vector of its divisors (d)d n and the all-ones vector of the same length, so that cos²θ(n)=σ(n)²/(τ(n)σ₂(n)) and tanθ(n) is the coefficient of variation of the divisors. We study when two integers have the same angle. For coprime $m,n$ the angles satisfy the spherical Pythagorean theorem cosθ(mn)=cosθ(m)cosθ(n), and θ is strictly increasing along divisibility. Apart from $1$ and sixteen exceptions, the primes are exactly the integers with θ(n)<π/4; consequently no integer shares its angle with a prime, and the same holds for squares of primes. The exceptions come from the primes $2$ and $3$: for Beurling generalized primes the line π/4 separates the primes exactly if and only if every prime is at least 5+√212. No integer $n>1$ has θ(n)=π/4 or π/3, and θ(n)/π is always irrational. Read as a velocity, tanθ(p)=p-1/p+1 turns multiplication into relativistic addition of velocities, and the angle π/4 approached by the primes becomes the light cone. Two $2$-adic and $3$-adic parity laws restrict coincidences θ(m)=θ(n), and for primitive squarefree coincidences the largest prime q involved must have all prime factors of q²+1 below q. The mean of cos²θ(n) over n≤ x is asymptotic to C/√log x with an explicit constant C≈0.8654, and -log₂cos²θ(n) obeys an Erd{o}s--Kac law. Up to 10⁸ all coincidences come from $8$ primitive ones, such as θ(1495)=θ(4648), and a search by prime signatures finds larger ones.