A Divisor-Wall Recurrence for the Prime-Counting Function and the Harmonic Composite Shadow

We introduce a divisor-defined sequence M(x) counting integers n in [2, x] with g(n)³ < n, where g(n) is the largest divisor of n at or below √n (OEIS A033676). The prime-counting function π(x) satisfies the exact triangular identity π(x) = M(x) − Σ ( π(⌊x/c⌋) − π(c²) ), summed over c = 2 to ⌊x^(1/3)⌋. Every π-argument on the right is strictly less than x, so the identity determines π(x) bottom-up from divisor data alone, with no prime sieve used as input. The proof rests on the equivalence g(n)