Minus One Eighth

In “Simulating Physics with Computers” (1982), Richard Feynman asked whether a local hidden-variable program could imitate two entangled photons. Because identical polarizer angles always agree (cos² 0° = 1) and orthogonal angles always disagree (cos² 90° = 0), a deterministic state evaluated across six angles 30° apart (0°–150°) reduces to one of eight six-dot cards with opposite colors 90° apart (plotted here at 2θ so orthogonal angles sit across the ring). Any card must switch color at least twice around the circle, bounding classical agreement at 4/6 = 2/3 (Figure 5a), whereas quantum mechanics predicts cos² 30° = 3/4.

Feynman noted that matching quantum correlations with local hidden variables requires negative probabilities. Each of the six half cards matches 2/3 of the time at 30° and 1/3 at 60°; the two alternating stripe cards match 0 at 30° and 1 at 60°. Giving the six half cards weight +3 and the two stripes weight −1 (+9/8 and −1/8 of the net total) reproduces cos² θ at all four angle steps (1, 3/4, 1/4, 0). Sampling by absolute weight (20 for a net sum of 16) and subtracting negatively signed trials inflates estimator variance by (20/16)² = 1.56× per pair—the sign problem behind Feynman’s case for quantum computers.

Each alternate deal demonstrates one classical Bell loophole. Talk exploits the locality loophole by forwarding East’s dial over the mesh before West answers (closed with fast-switching analyzers by Aspect et al., 1982, and random settings at 400 m by Weihs et al., 1998). Go dark exploits the fair-sampling loophole by answering only along a card’s preferred axis (1/3 of trials), matching 3/4 on post-selected pairs while falling below 2/3 when silences are scored honestly (closed for ions in 2001, photons in 2013, and jointly with locality in 2015). Fixed dials violates measurement independence by choosing both dials at the source before printing the card (tested with human and cosmic random number generators in 2018; see Already Sealed).

One Rust binary runs across GKE us-west1 (The Dalles), GKE us-east4 (Ashburn), and EKS us-east-1 (Northern Virginia), linked by an Istio ambient mesh over mutual-TLS east–west gateways. Between Oregon and Ashburn (≥3,605 km after campus radii), one-way light travel takes 12.0 ms. Before each 2,400-pair run (one trial every 0.5 ms over 1.2 s), stations query their cloud’s link-local SNTP server (169.254.169.254 on GCP, 169.254.169.123 on AWS) and draw dials from getrandom; a trial is certified spacelike only when its answer timestamp precedes light arrival from the remote dial minus both clocks’ error bounds. Same-region pairs, mid-run disconnects, or unmeasured clocks are flagged automatically.

Hensen et al. entangled two NV-centre electron spins 1,280 m apart (light: 4.27 µs) and ran 245 event-ready trials over 220 hours. Each lab drew its basis in 160 ns, began readout within 0.48 µs and finished a 3.7 µs readout 90 ns before light from the other lab could arrive. Their axes sit 22.5° apart, so each trial reads two neighbors of an eight-dot card: classical cards match at most 6/8 = 3/4, a perfect pair cos² 22.5° ≈ 0.854. The spins matched in 196/245 = 0.800 of trials (S = 2.42 ± 0.20, p ≤ 0.039 even allowing memory), which as a deck needs at least 1/15 negative weight. The replay deals the published per-setting counts (Gill, 2022, consistent with k = 196 in Hensen’s Figure 4) in shuffled order. The trial-by-trial file on 4TU.ResearchData was unreachable when this was built, so the diagram shows the published timing budget rather than each trial’s clock.

When the dealer can’t be reached, or fewer than two stations answer, the page replays runs recorded from the live clusters: Feynman’s card, one of each, minus one eighth, and the three cheats, each 2,400 pairs between The Dalles and Ashburn with their measured clocks and timestamps. A deck that wasn’t recorded is re-dealt from recorded trials that carried the same cards, keeping each trial’s real dials, answers and timing and signing it by your counts; the status line says so. A run is only archived if both clocks were measured, no station went dark and the stations sat a light-cone apart. With no archive at all, the page shuffles in your browser and says it has no light cone.

• Feynman (1982), Simulating Physics with Computers, International Journal of Theoretical Physics 21, 467
• Mermin (1985), Is the moon there when nobody looks? Reality and the quantum theory, Physics Today 38, 38
• Aspect, Dalibard and Roger (1982), Experimental Test of Bell’s Inequalities Using Time-Varying Analyzers
• Weihs, Jennewein, Simon, Weinfurter and Zeilinger (1998), Violation of Bell’s Inequality under Strict Einstein Locality Conditions
• Rowe and colleagues (2001), Experimental violation of a Bell’s inequality with efficient detection
• Giustina and colleagues (2013), Bell violation using entangled photons without the fair-sampling assumption
• Hensen and colleagues (2015), Loophole-free Bell inequality violation using electron spins separated by 1.3 kilometres
• Hensen and colleagues (2015), dataset on 4TU.ResearchData
• Gill (2022), Optimal statistical analyses of Bell experiments, with the Delft counts per setting in Appendix A
• Giustina and colleagues (2015), Significant-Loophole-Free Test of Bell’s Theorem with Entangled Photons
• Shalm and colleagues (2015), Strong Loophole-Free Test of Local Realism
• The BIG Bell Test Collaboration (2018), Challenging local realism with human choices
• Rauch and colleagues (2018), Cosmic Bell Test Using Random Measurement Settings from High-Redshift Quasars
• Istio, Ambient multicluster