Every pair starts in the singlet state |Ψ⁻⟩ = (|↑↓⟩ − |↓↑⟩)/√2. If Alice measures spin along an axis at angle θA and Bob along θB, quantum mechanics gives:
Δθ = θ_B − θ_A
P(A=+1) = 1/2 (Alice's own marginal — always fair)
P(B=+1 | A=+1) = sin²(Δθ/2)
P(B=+1 | A=−1) = cos²(Δθ/2)
E(A,B) = ⟨AB⟩ = −cos(Δθ)
This view samples exactly that distribution — Alice's outcome is a fair coin flip, then Bob's is drawn from the conditional above — and renders it two ways instead of as particles flying between labs. The left plot bins every sampled pair by Δθ and scatters the running average of A·B against the theoretical −cos(Δθ) curve: watch the dots snap onto the curve as more pairs accumulate in each bin. The right strip chart scrolls Alice's analyzer setting (top lane) beside Bob's raw ±1 outcome stream (bottom lane) in real time.
P(B=+1) = ½·sin²(Δθ/2) + ½·cos²(Δθ/2) = ½ — for every Δθ
That identity is the no-signaling theorem: press "try to send 1011" and Alice's lane visibly snaps between 0° and 90° in that bit pattern, but Bob's tick lane keeps looking like flat noise — his marginal statistics never depend on θA. The correlation between the two lanes is real (stronger than any classical hidden-variable model, the heart of a Bell test) and is exactly what the left-hand curve is measuring — but reading it out requires comparing both lanes side by side over an ordinary, light-speed-limited classical channel. That is why "entanglement across universes" can't work as a messaging channel on its own.
- θA / θB sliders — set each lab's analyzer orientation independently; a vertical marker on the correlation plot tracks the current Δθ.
- "Try to send 1011" — Alice snaps her angle between 0° and 90° following that bit pattern, as fast as pairs are emitted.
- Correlation plot — scatter dots are the live per-bin average of A·B; the dashed curve is the QM prediction −cos(Δθ).