Factor Rays and the Self-Conjugate Parabola: Deterministic Coverage Geometry in Square Intervals

We organize the elementary sieve in square intervals [m², (m+1)²] with a factor-ray system R_d = {(k, dk) : k ≥ 1} in the positive integer plane. The parabola n = k², which is the fixed locus of the divisor-conjugation involution (k, n) ↔ (n/k, n), plays a structural role for this sieve: it determines the sharp depth D = m+1 that suffices to certify primality on the band, it geometrizes the trial-division criterion, and it places quadratic-residue constraints on the entry offsets of factor rays into Legendre bands. We compare the geometric content of the Legendre bands [m², (m+1)²] to that of the quadratic bands J_n = [4n²-n, 4n²+n] studied elsewhere by the author, exhibiting a structural inequivalence in the position of the parabola relative to each band that translates into different available ranges for the medium-prime sieve correction. The arguments are deterministic and rest on elementary divisor geometry; we are explicit about which content is genuinely new and which is the classical Legendre sieve in geometric form. In the final section we prove two limitation results delimiting what the geometry alone can accomplish — a Bonferroni bound for truncated inclusion–exclusion, and, via the Ford–Green–Konyagin–Maynard–Tao lower bound on Jacobsthal's function, an unconditional refutation of any offset-blind covering argument — and we isolate the single open question that survives them.