$P_i$

the set of dots in square i

$q$

each dot, from 1 up to the current count

$N_i(k)$

how many dots have arrived so far (grows as it fills)

$c$

the centre of the square

$\hat{d}$

direction of this square's random line

$\hat{n}$

the perpendicular to that line

$a_q\sim U$

position along the line — uniform random

$\zeta_q\sim\mathcal N$

offset across the line — Gaussian (bell-curve) random

$\sigma$

band width — wide (noise) early, near zero (a line) by the ninth interlude

Each dot is a random draw: uniform ALONG a random line, a narrow Gaussian ACROSS it. The pile-up is the line — order from accumulation, not placement. Over the nine interludes σ shrinks, so the band tightens to a single random line.