A timing hidden-variable construction for the singlet correlation, and a thin-crystal test of it

Toner and Bacon showed that the singlet correlation E = -cos(Delta) can be reproduced exactly by two shared uniform random unit vectors and one bit of communication. We ask which unobserved quantities of a down-conversion source could supply the four uniform numbers that fix those vectors. For a collinear, nondegenerate source, the emission time and the signal-idler delay (which depends on the position of emission in the crystal and is shifted but not removed by compensation optics), together with the signal and idler optical phases as two clocks, give four such numbers. A Fourier argument then gives the departure of the correlation from -cos(Delta) in closed form: it vanishes at aligned and opposite settings, appears at intermediate angles, and is controlled by |sinc[(f_s - f_i) T]|, where T is the delay spread across the crystal. For type-I BBO at 763/849 nm the predicted deviation at a relative angle of 90 degrees reaches up to 14% at 1 mm and 42% at 0.3 mm, against 0.9% at 15.76 mm, while the CHSH value stays near 2.83. A thickness series with fixed optics would test this, and a null result at 0.5 mm would rule the construction out. The construction still needs either a communicated bit or setting choices that depend on the source. We give an exact measurement-dependent version and state what a physical cause would have to supply; we do not identify one.