The Prime-Square Chamber Lift: Short-Interval Occupancy via Quadratic Boundary Windows

This note gives an expository chamber-coordinate presentation of the prime-between-squares problem family. Legendre's conjecture asks for a prime in every interval $(n^2,(n+1)^2)$, Oppermann's conjecture asks for primes in both subintervals split by the pronic number $n(n+1)$, and Brocard's conjecture asks for multiple-prime occupancy between consecutive prime-square boundaries. The Chamber Lift vocabulary used here is not offered as a proof mechanism or as a new analytic method. It is an atlas: a visual and notational organization of classical short-interval prime conjectures over the multiplication-grid geometry already present in the square diagonal and its adjacent rectangular seam.