Inset: the five sack oscillators as phasors of phase error (psi_j minus j times psi_1). Vertical and aligned is locked. The same phases drive the ripple on the sack's outline, the spacing of its cloth dots, and the spiral angle of its flour dots.

choice	score s	measured s

Measured from the rendered geometry in window.PAE_TRACE (one row per frame); onsets in window.PAE_CHECK. Tolerance 0.25 s.

A sack is a chord: knocked out of tune by the knife, it comes into tune as it decides, and it is full when every partial locks.

Made of. Every visible thing is a computed point of light on a black field: the sack is about 150 cloth dots on a closed curve whose radius is a sum of harmonics (sin and sin 3 terms whose weights and phase put gravity in the outline, so the lower third is always wider than the upper and the bottom flattens where it meets the table; cos 2 and cos 4 for width and shoulders, two Fejer sums of 12 cosines for the two corners, a short cosine sum for the dip of the crown) plus one flour dot inside for every unit of its volume, laid on a phyllotaxis spiral. The tower is 11 stacked levels of three concentric rings of dots, the knife is a dotted blade, the penny a ring of amber dots, and the flour is 996 individual dots (600 in the sack, 396 in the tower) that are counted every frame and never created or destroyed.

Moved by. The score's times are a conductor: at each one, oscillators get a new target, a new coupling, or an impulse. Every shape between comes from integrated damped harmonic oscillators (squash, lean, droop, tilt, table contact, gravity harmonic, corner height and angle, tower lean, knife rest, camera), plus a bank of five phase oscillators whose n:1 coupling is the sack's mood: detuned and kicked by each cut, locking as it decides, a pure lock when full. Its jitter amplitude is what scatters the cloth and flour dots, so the interior drifts continuously from stipple to lock. Two things are scheduled rather than integrated, and named here plainly: the gait is a fixed sequence of cycloid step blocks (1 hop at 0.96 s, 1 at 3.97 s, 7 between 9.0 and 14.0 s, a planted 0.08 m push step into the tower at 18.0 s, the leap at 21.4 s) whose phase gives the travel, so the base is still at every contact; the knife's strike is a scheduled window: one turn of a raised cosine to the 10th power, 0.5 s long, centred on each cut time, added to its rest spring; and every grain's fall, cloud throw and gather flight is a quarter or half turn of its own phasor over a fixed duration, released by the cut time, a fixed-period sift clock, or a sweeping beam. The penny falls by an integrated ballistic step (gravity plus air drag, Euler at 240 Hz) from release at 21.0 s, with the gravity constant solved once by bisection so that it meets the crown at 22.4 s; it never rises. The tower's lean is a driven oscillator clamped each sample by the sack's outline: it rests on the sack and is never pushed back more than 0.022 rad, and the sack's area is rescaled every frame to exactly (dots inside) x (area per dot).