The medium-bound model as instant maths. The two labs are bound by one shared medium, not by a signal. Each trial: H and the sync of each particle are fixed at emission, and the lab whose start angle (the pair leaves at Lab A 0° and Lab B 45°, or Lab A 90° and Lab B 135°, chosen 50:50) is closest to the pair's hidden plane is the primary; the medium makes one draw, a
half-cosine lobe centred on the primary's own angle (E, the reference angle); each lab flips its sign if the draw lands more than 90° from its own setting.
The primary therefore never flips, and the secondary flips with chance (1 − cos gap)/2: 0° 0%, 45° 14.6%, 90° 50%, 135° 85.4%. The herald is a sync check: a run is kept only if both particles were in step with the frame (set the percentage below 100 to see runs set aside). CHSH is computed from the kept results.
Setting dependence λ is how often the draw is centred on the primary's angle: λ = 1 is the medium-bound model (|S| = 2.83); λ = 0 centres it on a fixed angle that ignores the settings (fully independent). The axis-cloud and Hall options below stay for comparison.
| Setting pair | Angles | n | Raw E |
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How much does the shared draw's distribution have to change when the settings change? M is the largest total distance between that distribution for any two of the four setting pairs (0 means the settings never matter, 2 means completely different). F = 1 − M/2 is the fraction of measurement independence kept. From Hall (2010), arXiv:1007.5518. Computed from the exact distribution for the settings and draw chosen above, so it updates as you change them and needs no run.
Half-cosine distribution: density ∝ cos(x) on ±90° around the centre (a cosine-weighted lobe, not a full circle). It gives the single-spin cos² law and the two-lab −cos Δ exactly, so there is nothing to tune and |S| lands on 2.83 exactly.
Axis cloud: density ∝ 1 + 2c₂ cos 2x, a weak preference for the axis through the centre, both ends alike. With c₂ = (π/4)(√2 − 1) = 0.3253 it gives |S| = 2.828 exactly, a correlation within 0.029 of −cos Δ at every angle, and M = 0.586. Its single-lab curve is flat, as for one unpolarised half of an entangled pair, so it is a cloud for the pair and not for a prepared spin. The pair correlation depends only on a cloud's even harmonics and the single-spin curve only on its odd ones.
Hall's density (a benchmark, not a mechanism): the shared draw is flat inside the sectors where the two settings' outcomes agree and where they differ, with total weights (1 ± cos Δ)/2. It needs both settings at once to place its sectors and set its weights, so the quantum answer is written into it. It shows what is possible, not how a source could do it.
Least dependence possible for the standard settings 0°/90°/45°/135°, matching all four quantum correlations, found by linear programming: M = 0.276 if the draw may use both angles at once (this equals Hall's model); M = 0.414 if each trial's draw uses only one lab's angle (the coin-flip structure used here, with the cloud shape free); M = 0.828 if it depends on one lab's angle only (F = 59%, the published one-sided optimum). These bounds are for those four angles only.
A direction is emitted into the hemisphere facing one lab's setting, with density ∝ cos θ (the cosine law of a flat emitter). Each lab reports which side of the plane perpendicular to its own setting the direction lands on. Seen from above, the facing lab's plane is the edge of the shaded half. Amber dots land on the same side of both labs' planes, grey dots on opposite sides. The share of amber dots is (1 + cos Δ)/2 for the angle Δ between the two settings, with no fitted number. Directions below the page project onto the same disc. Uses the four angles set above.
No current angle is shared. The draw is centred on a historic setting, one that already existed before this trial's coin flips. With a fair coin each lab has a 50% chance of repeating its previous setting, so both repeat 25% of the time. The table splits the results by whether the settings happened to repeat. Uses the four angles set above. q is each lab's chance of repeating its previous setting (0.50 = fair coin); k is how many past trials the draw can look back over.
| Which trials | Share of trials | S | |S| |
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