Hidden-variable models compared with LGAM

A working comparison for the LGAM papers and the PREreview request. Compiled 29 September 2026 from a small number of web searches, so it is a starting map and not a literature review. Each row says what the hidden variable is, what it covers, and which assumption of Bell's theorem it gives up. The last two columns say how much was actually read and where to find it.

Rows marked in amber are this project. Cells that say "not assessed" were not checked.
Model	Hidden variable and mechanism	Single-spin tests covered	Bell / CHSH result	Bell assumption given up	How it could be tested	What was read	Sources
Standard quantum theoryreference point	No hidden variables. The state vector plus the Born rule.	Predicts the Rabi, Ramsey and rotated-readout curves: ½(1 + cos θ), ½(1 − e^{−(τ/T2*)²} cos 2πδτ) and ½(1 + cos Δ).	Predicts CHSH 2√2 ≈ 2.83 for a perfect singlet. The Delft NV test measured S = 2.42 ± 0.20 with imperfect fidelities.	Not applicable. It predicts the violation.	Already tested. Loophole-free or nearly loophole-free violations were reported in 2015: Delft NV spins, and photon experiments from Vienna and NIST.	Main text read Delft paper read in full. The other two by reference only.	Hensen et al., Nature 526, 682 (2015), arXiv:1508.05949. Giustina et al., PRL 115, 250401. Shalm et al., PRL 115, 250402.
Pilot-wave theoryde Broglie–Bohm	Actual particle positions steered by the wave function. Deterministic.	Reproduces standard quantum statistics when the hidden variables are in "quantum equilibrium".	Same as quantum theory in equilibrium.	Locality: it is non-local by design. Deterministic and, in general, not measurement dependent.	Would show statistical anomalies only if the hidden variables were out of equilibrium. No experimental evidence for that at present.	Summary passages Fluid-droplet analogues (Couder, Bush) seen by reference only.	arXiv:quant-ph/0506115. Allori, Theoria (2024), hdl 10446/266309.
Superdeterministic programmes't Hooft cellular automaton; Palmer invariant-set theory	Deterministic underlying dynamics (cellular-automaton states in 't Hooft's case). The hidden variables and the settings both trace back to the universe's initial conditions.	Not assessed.	Not assessed. The aim is to evade Bell's theorem by dropping independence.	Measurement (statistical) independence.	Critics argue such theories become untestable and uninformative (Allori 2024). That is one argument, not a settled result.	Titles and references only Plus discussion passages.	't Hooft, The Cellular Automaton Interpretation of Quantum Mechanics (Springer Open, 2016). arXiv:2107.04761.
Retrocausal modelsPrice; Sutherland; Wharton; Lazarovici; Sen	Events are not fully determined by past conditions alone.	Not assessed.	Lazarovici (2015) reports a relativistic retrocausal model that violates Bell's inequality. Sen (2019) gives a local retrocausal model of Bell correlations.	Measurement independence. Measurement dependent but not deterministic.	Sen's dissertation abstract says the model generally violates no-signalling if the hidden variables are out of equilibrium.	Titles, one abstract	Reference list of arXiv:2107.04761. Sen, Found. Phys. 49 (2019); Clemson dissertation.
Brans (1988)early causal hidden-variable model	Fully causal hidden variables correlated with the settings (an early superdeterministic model, as Sen describes it).	Not assessed.	Not assessed.	Measurement independence, given up completely (Hall's description).	Not assessed.	Title, and how others describe it	Brans, Int. J. Theor. Phys. 27, 219 (1988).
Hall (2010)with a one-sided follow-up	An explicit deterministic construction in which the hidden-variable distribution depends on both parties' settings. No physical carrier is proposed (from the abstract).	Models all spin correlations of a singlet state.	Reproduces all singlet correlations. A second model reaches the maximum CHSH violation.	Measurement independence: 14% given up for the full singlet (about 1/15 of a bit of mutual information), 1/3 for maximal CHSH. In the follow-up (arXiv:1204.3835), with dependence on one party's setting only, 59% independence can be retained.	The amount of dependence can be bounded by measuring how far the settings and hidden variables are correlated. Cosmic-settings experiments limit when such a correlation could have been arranged.	Abstracts and introduction	arXiv:1007.5518. arXiv:1204.3835. Barrett and Gisin, PRL 106, 100406 (2011).
Sciarretta (2019–2020)local-realistic model of spin and spin entanglement	Local, realist and stochastic. Particles carry a few stochastic variables set at the source and evolving in time, interacting with lattice nodes that store the magnetic-field information.	Stern–Gerlach apparatuses. Malus's law is recovered.	Violates Bell–CHSH by exploiting the coincidence loophole.	One of Bell's assumptions, through the coincidence loophole (arrival-time selection).	My understanding, not checked: tests with fixed timing windows leave no room for the coincidence route.	Abstract and introduction	arXiv:1901.00369.
Event-based coincidence modelsas cited by Sciarretta	Event-by-event simulation in which coincidences count only when arrival times are close. The time delay between arrivals plays the key role.	Malus's law, as described by Sciarretta.	Described as reproducing quantum predictions through the coincidence loophole. Not read directly.	The coincidence loophole.	Not assessed.	Described second-hand	References 10, 31–34 and 47 in arXiv:1901.00369.
Classical-optics claimMamas (2021), analysed by Gill (2022)	A polarised photon couples to a polarizer through the field component along the polarizer axis.	A Malus-type cosine.	The cosine has only 50% amplitude, so it does not violate CHSH. Gill shows it is an ordinary local hidden-variable model.	None given up, so no violation.	Compare the cosine's amplitude: a local model of this kind gives 50%, quantum theory 100%.	Abstract and introduction	arXiv:2211.01068.
Medium-bound modelthis project, current form	One shared medium binds the pair. H is fixed at emission by where the phase sits, an even 50:50 split across pairs, and the lab whose start angle is closest to the pair's hidden plane is primary (the pair leaves at Lab A 0° and Lab B 45°, or Lab A 90° and Lab B 135°, chosen 50:50, so the primary is fixed at emission). The herald is a sync check after the readout: only pairs with both particles in step with the frame are kept. The medium makes one draw as a half-cosine lobe around the primary's own angle (E). The pair leaves with opposite signs (Lab A takes H, Lab B the opposite); a lab reads the other sign if the draw lands more than 90° from its own setting. The primary never flips; the secondary flips with chance (1 − cos gap)/2. Space has two modes: waves at c, and a much faster constraint mode, finite but far above c, that carries no signal.	Exact with the half-cosine lobe and no free parameter: Rabi ½(1 + cos θ), Ramsey and rotated readout ½(1 + cos Δ), under the mapping proposed on the single-spin page. Readout at 45°, 90° and 135° gives 85.4%, 50% and 14.6%.	\|S\| = 2.83 in simulation, with every run counted (200,000 runs gave 2.834; pair Different % 14.6 / 85.5 / 14.7 / 14.4). Each lab's own record stays 50:50 at every setting. Hensen et al. measured 2.42 ± 0.20; their own estimate from imperfections is 2.30 ± 0.07, so the shortfall from 2.83 is a readout and state-quality effect, not a second mechanism.	Locality. The secondary's outcome depends on the primary's actual setting through the medium (parameter dependence), but no signal can be sent. Measurement independence is kept for the lab coins, H, the sync of each particle and the hidden plane. The draw itself is made by the medium after both labs have chosen, so its dependence on the settings is the non-local influence. No time travel and no superdeterminism.	(1) Per-setting records from a real NV run should sit near 50:50 for each lab. (2) A phase-only herald that does not depend on the primary's angle gives only about 1.41, and a source with E fixed at emission fails the 50:50 check. (3) The link has a finite speed in the pair's own frame, so settings chosen closer together in time than the link takes to cross should weaken the correlation; published tests put the speed above about 10,000 c (to be checked). (4) Real per-setting NV counts have not yet been obtained.	Own simulations Headless runs of the model, the Herald Replay page and the updated lab pages. Hensen et al. read in full. Not peer-reviewed.	Paper 3 (in preparation). Hensen et al., Nature 526, 682 (2015).
LGAM, earlier phase-clock formthis project, papers 1 and 2	A shared phase running on a 720° loop plus a per-particle rotation state. The outcome is set by the phase lobe at the trigger. The hidden phase is drawn from a half-cosine lobe (a cosine-weighted lobe), centred on the primary lab's setting.	One rule gives the Rabi, Ramsey and rotated-readout shapes, under the mapping proposed on the single-spin page. The half-cosine draw makes all three curves exact.	The two-lab correlation is exactly −cos Δ and \|S\| = 2.828, with no free parameter. In Hall's measure its dependence is M = 1.31.	Measurement independence. The violation appears only when the settings depend on the phase: S = 2 + 0.83λ, and λ = 0 gives exactly 2.0. In Hall's measure it gives up 66% (M = 1.31, F = 34%), against 14% (M = 0.276, F = 86%) for Hall's model at the same S.	(1) Setting–phase dependence: repeat rates and autocorrelation in the random-number output, and cosmic-settings tests. (2) The history route needs an excess repeat probability of about 0.26 to reach S = 2.42.	Own simulations Checked against independent runs. The Hall-measure figures were computed from the model's densities, with the code validated on Hall's own model. Not peer-reviewed.	Paper 1. Paper 2.

The exits from Bell's theorem, and what each costs

Any model that beats a CHSH value of 2 has to take one of these routes.
Exit	Taken by	Cost or constraint
Give up locality	Pilot-wave theory, medium-bound model	Influence between separated systems. The medium-bound model takes this exit through a physical medium: a much faster constraint mode, with waves still limited to c. It cannot signal because each lab's own record stays 50:50. Its speed is finite and defined in the pair's own frame, which makes the lag testable.
Give up measurement independence	Superdeterministic and retrocausal models, Brans, Hall, LGAM (earlier phase-clock form)	The settings and the hidden variables must be correlated. The cosmic Bell test that took settings from quasar light constrains any such mechanism to have acted no more recently than 7.78 billion years ago, excluding it from 96% of the past light cone's space-time volume. That assumes fair sampling and that the quasar photons were not previewed or altered. Hall shows the amount of dependence needed can be small.
Exploit the sampling or coincidence assumptions	Sciarretta and the event-based models	Loophole-free tests were designed to close the detection and locality loopholes. My understanding, not checked, is that fixed timing windows leave no room for the coincidence route.
Give up determinism, or hidden variables altogether	Standard quantum theory	Models that keep measurement independence and no-signalling must be maximally indeterministic to reproduce the singlet correlations (Branciard et al., as cited by Hall), unless they give up locality.

Questions LGAM needs to answer

Medium-bound model: how is the pair's frame defined for moving labs? The model keeps waves at c and gives the medium a much faster constraint mode with a finite speed in the frame that contains both particles. It still has to say how that frame is defined when the two labs move relative to each other. Real per-setting NV counts are also still needed for the 50:50 check.
Which exit, and by which route? The dependence between phase and settings can come from a shared past cause, from backward-in-time influence, or from a local coupling. A local coupling cannot connect the source to both labs within the light-travel window.
How much dependence does the mechanism use? In Hall's measure, ours gives up 66% (M = 1.31) to reach 2.83. Hall's model gives up 14% (M = 0.276) for the same value, so ours uses about 4.7 times as much. Both use the same outcome rule, so the difference is entirely in how the hidden phase is distributed. The open question is whether a physical mechanism can produce a Hall-like distribution.
What differs from quantum theory that readout error cannot explain? With the half-cosine draw the single-spin curves match standard theory exactly, so single-spin data cannot separate the two. The difference sits in the dependence between the shared phase and the settings.
Does the draw need both settings? A draw built from both angles at once writes the quantum answer into the hidden variable. For the standard CHSH angles the least dependence possible is M = 0.276 with both angles, 0.414 with one angle per trial and 0.828 with one lab's angle only (linear programming; the first equals Hall's model, the last matches the published 59% independence). Ours is 1.31. The exact single-spin law forces the half-cosine distribution, so if one cloud shape must serve both a prepared spin and an entangled pair, 1.31 is the figure. If the pair's cloud may differ from a prepared spin's cloud, a simple two-lobed axis cloud (density ∝ 1 + 0.65 cos 2x, a weak preference for the axis through the setting) gives |S| = 2.828, a correlation within 0.029 of −cos Δ at every angle and M = 0.586. Its single-lab curve is flat, as for one half of an entangled pair.
Does it match the Delft data? With the experiment's own angles, a heralded state modelled as a singlet dephased to the paper's fidelity of 0.92, and asymmetric readout at the paper's average fidelities (97.1% and 96.3%), quantum theory predicts S = 2.28 and an expected 192 ± 6 correlated outcomes out of 245. The paper's own expectation is 2.30 ± 0.07. Mixing the same imperfections into the exact half-cosine distribution gives the same 2.28 (checked by simulation to 0.002), and the axis-plus-half-cosine pairing gives 2.28. The data (S = 2.42 ± 0.20, k = 196) sit 0.6 to 0.9 standard deviations above all of them. The differences between the models, at most 0.05 in S, would need on the order of 4,000 trials to resolve, against 245 in 220 hours. The readout asymmetry changes S by under 0.01. The data cannot test the dependence between the shared phase and the settings. The delft_check.html page lets you change the fidelities.
Which real-data check would the model fail? A repeat-rate or autocorrelation test on the random-number output is the most direct.

Search coverage and limits

Searches covered: hidden-variable reviews (pilot-wave, superdeterminism, retrocausal); measurement dependence in Bell tests (cosmic tests, Hall's model); local models covering Stern–Gerlach and Bell; and the Delft NV papers. Not searched: NV-specific hidden-variable models, stochastic electrodynamics, Aerts's hidden-measurement models, the primary event-based papers, contextuality tests on spin-1 systems, and preprint servers outside arXiv. A claim of novelty needs those checked.

Numbers for LGAM come from this project's own simulations. Numbers for other models are taken from the abstracts and passages listed in the "What was read" column.