Does Gravity Entangle, and Would That Settle It?

I · The proposal, and the phase it predicts

In 2017 two papers proposed the same test: hold two micron-size masses, each in a superposition of two positions, close enough to pull on each other through gravity and far enough that Casimir–Polder forces stay a tenth of the gravitational pull; if the two masses come back entangled, the field that connected them is, on the papers' argument, a quantum entity. The first paper gives a geometry: masses of 10⁻¹⁴ kg, traps 450 µm apart, each mass split by 250 µm, a hold of 2.5 s, a pressure of 10⁻¹⁵ Pa and an internal temperature of 0.15 K. The second gives the general argument for why entanglement would be sufficient evidence. Both are proposals, and both rows are labelled so.

Each of the four branches (left–left, left–right, right–left, right–right) accumulates the Newtonian phase φ = G m² t/(ħ d) for its own separation d; the two-mass state is a product exactly when Δφ = φ_LR + φ_RL − φ_LL − φ_RR is a multiple of 2π, and otherwise its concurrence is |sin(Δφ/2)|. The label is DERIVED: the concurrence formula is Wootters's for a pure two-qubit state, and the potential it is fed is the Newtonian one, the law the parent MEASURED row confirmed between millimetre gold spheres at a surface distance of 400 µm. For the 2017 geometry the kernel finds the RL and LR branch phases to exceed the LL phase by 0.44 rad and −0.13 rad (the paper's order-of-magnitude 0.7 and −0.2), an entanglement phase of 0.31 rad and a concurrence of 0.16; the verifier recomputes the concurrence from the reduced state's purity and requires agreement to one part in 10¹².

A local dephasing rate γ on each mass shrinks the off-diagonal terms by e^{−γt}; the Peres negativity of the resulting mixed state, computed from the partial transpose, equals half the concurrence when γ = 0 and reaches zero for the 2017 geometry at γt = 0.16, so the entanglement survives only while the dephasing phase budget stays below the entanglement phase's own order. The paper's own requirement reads the same way: at 10⁻¹⁵ Pa and 0.15 K it estimates a collisional decoherence time of the order of the 3.5 s the masses spend falling, and a witness value of about 1.16 against a threshold of 1. This page's dephasing model is a single rate, not a laboratory budget; the paper's witness and the kernel's negativity are different measures of the same state.

II · Three channels, equal weight

A classical channel can carry the force without carrying the entanglement: in the 2014 classical-channel model gravity acts as a continuous measurement of each mass's position fed back as a force on the other, which reproduces Newton's law on average, adds a minimum decoherence, and, being local operations with classical communication, can create no entanglement at all. That is the second channel of the instrument, drawn at the same stroke weight as the first. The kernel builds it as a mean field: each mass feels the average of the other's two positions, the four phases take the product form α_i + β_j, and the entanglement phase is zero to one part in 10¹² on a grid of twenty-seven geometries. In the 2017 argument this is the case a positive result would exclude.

Whether every classical account of gravity falls into that second channel is disputed: a 2025 paper argues that a classical gravitational interaction can transmit quantum information and generate entanglement through physically local processes when matter is treated by quantum field theory, and a 2025 letter finds that the Diósi–Penrose dynamics with classical gravity entangles two separate particles, so that 'GIE detection is not enough to validate quantum gravity'; two preprints and a 2026 paper reply that classical gravity cannot mediate entanglement by local means and that the perturbative result is precluded by an exact derivation, and a 2025 preprint by the witness's own proposers argues that the collapse-based models which entangle do so through nonlocal features and do not escape the witness. This is the third channel. The kernel gives it no curve of its own, only the interval between the two published answers, no entanglement at all and as much as the quantum channel gives, drawn as a hatched band with neither edge marked as the expected one; the page does not pick a side, and the row labels say why: the papers are young, two are preprints, and none is a measurement.

The band's upper edge can be computed. In the Diósi–Penrose model as the 2025 letter writes it, gravity is classical, matter is quantum, and the mass density is smeared over a width σ, the model's only free parameter and the same R0 that T.06 bounded from below; with the free motion switched off the model integrates in closed form, and the kernel reproduces the letter's result: the model predicts entanglement only when the masses' nearest separation is below 0.8508σ (the letter: 0.8501σ in its text, 0.850872σ in its appendix), that entanglement is transient, and for the letter's example geometry (10⁻¹⁵ kg, split 23 µm, 24 µm apart) nothing is entangled at σ = 50 µm while at σ = 75, 100 and 250 µm the negativity lasts about 5 hours, 16 hours and 20 days. On the same model the smearing softens gravity itself below about 2σ: at σ = 75 µm it puts the pull between those two masses at 24 µm at 18 % of Newton's. For the 2017 geometry, with a split of 250 µm, the model entangles only for σ above 525 µm and shows less than one part in a thousand of negativity at 2.5 s, because its entanglement takes hours to build. The page's planted defect for this kernel drops the model's own decoherence and keeps its phases; the critical separation then disappears, and the verifier rejects it. All of this is one model's arithmetic; the label is PROPOSED.

The kernel can also say where the model's entangling power sits. Its generator has two parts: an imaginary part, a commutator with the smeared pair potential −Gm₁m₂ erf(d/2σ)/d, and a real part, the model's own noise. With the noise switched off and only the potential kept, the evolution is unitary, the four-branch state stays pure, and the negativity is that of the smeared phase, exactly this page's quantum channel, 0.078 for the 2017 geometry, as σ → 0; with the potential switched off and only the noise kept, the negativity is zero wherever the kernel looks, as expected of a dephasing channel in the position basis. In full the model gives less than its potential alone: for the letter's geometry at σ = 75 µm, 0.024 against 0.152 after 10⁴ s, and at σ = 50 µm nothing against 0.363, the noise, stronger at smaller σ, having removed it all. In this construction the entangling term is the commutator with the smeared pair potential, the term left once the classical field is traced out; the noise term only takes away. Whether such a term counts as classical mediation is the dispute itself, and the page states the decomposition and no more.

The band's lower edge is computed from the classical-channel rate: in the 2014 model the measurement-and-feedback channel reproduces the linearised gravitational spring K = 2Gm²/d³ and adds position decoherence at KΔx²/(2ħ) per mass, 'minimal in the sense that, if it were any smaller, evolution ... would immediately entangle'; the kernel finds that the dephasing threshold at which this page's negativity vanishes is exactly that rate in the small-split limit (ratio 1.000 at a 2 µm split, 1.43 at the 2017 split of 250 µm, where the linearisation is 30 % low on the phase). The same threshold, 0.156 over 2.5 s or 0.062 per second, is within 1 % of the 1/τ_G of a 2021 decoherence study of these proposals, 0.0628 per second on its formula with the 2017 numbers, which it rounds to ≃0.06 s⁻¹ and above which it finds the pair separable at all times. So the two edges are computed, not chosen: the lower one is the least noise a classical channel of that kind must add on its own model, the upper one is what one named classical model predicts, and between them the page still picks no side.

Newton's inverse-square law has been tested in the laboratory at separations down to 52 µm, with any gravitational-strength Yukawa interaction limited to ranges below 38.6 µm at 95 % confidence, and shown by a second group to hold with |α| ≤ 1 down to a length scale of 48 µm. The smearing that lets the model entangle softens its pull below about 2σ, and the same pull must act between the test bodies of these experiments.The second group's 2020 apparatus is described in a 2026 review: tungsten plates 0.2 mm thick on the pendulum facing a 1.8 mm plate at a surface gap of 210 µm, with a rotating eight-plate source farther away; its modulation scheme is not in open text, and the site does not model it. At the pair level, between a test-mass element and a source element at that gap, the smeared potential removes erfc(210 µm/2σ) of Newton's pull, 4.8 % at σ = 75 µm, where the α = 1, λ = 48 µm Yukawa at that group's 95 % edge would remove 1.3 % of the same pull; the two match at σ ≈ 60 µm, so at this gap the shape sits under that Yukawa for smaller σ and above it beyond, and the 52 µm test is the more sensitive of the two for it.

The kernel can redo the comparison with the experiment's own method. Between two patterned plates each in-plane mode of the mass pattern with wavenumber q contributes to the pull across a gap s in proportion to e^(−qs) under Newton, and to ½[erfc(qσ − s/2σ) + e^(2qs) erfc(qσ + s/2σ)] of that under the smeared potential, a closed form checked against direct quadrature; the 2020 test bodies are two concentric sets of glue-filled wedges cut from platinum foils, 120 wedges between 13.00 and 23.00 mm and 18 between 23.51 and 26.00 mm at 50 % duty, 54 and 99 µm thick, measured to the micrometre in the 2020 dissertation, and the Fourier–Bessel expansion the collaboration published in 2021 turns the torque at each harmonic into one integral over wavenumber. The kernel rebuilds that integral with its own Bessel functions, checked against their integral representation, and gets the Newtonian 120-fold torque at 1.5 fN·m at 72 µm, where the letter's magnetic subtraction of 0.0165 fN·m, 'roughly 1 %' of the signal, implies by arithmetic about 1.6. On the same integral the smeared potential removes 17 % of the 120-fold torque at 52 µm for σ = 75 µm (4.3 % at 50 µm, 0.3 % at 30 µm, 6.3 % at 55 µm, the smallest σ at which that model entangles the letter's own geometry), 13 % at 72 µm, 3.8 % at 150 µm and 0.1 % at 300 µm, while a Yukawa at the published 95 % edge, α = −1 at λ = 38.6 µm, removes 2.5 % at 52 µm. That sets two predicted signal sizes side by side; it is not a bound on σ, because the experiment's fit, with its floating offset, roughness correction and calibrated scale, is not recomputed here. The one-mode lower bound at the pattern's lowest wavenumber had given 44.4 µm, and the Yukawa shape closest to the σ = 75 µm curve has a range near 49 µm and about three times gravity's strength, a curve match with no data behind it.The published limits can be recast onto the smeared pull. The 2020 experiment's dissertation tabulates, for 66 Yukawa ranges from 5 µm to 9 mm, the fitted strength with its uncertainty and the signed 95 % limits. On this page's full-pattern kernel the smeared pull is an excellent Yukawa mimic, within 2 to 4 % in shape at a range near 0.55 σ and a strength of several, and at those ranges the published limit is far below gravitational strength; the experiment's own best fit there leans slightly towards a stronger pull, so the signed limit the smeared pull faces is the tight one. Projecting the smeared signal onto each tabulated shape and comparing with the signed limit at the shape it mimics best, the published limits exclude σ above 33 µm at 95 %: 33.1 µm on three separation weightings, on an absolute-torque weighting and on both measured harmonics, 33.6 µm on a linear weighting. At 33 µm the effective strength is −7.3 against a limit of −7.4 at a 18 µm range; at 35 µm it is −10.3, excluded. This is a recast of published limits, not a refit of the data: it assumes the smeared shape shares its mimic's degeneracy with the fitted offset and scale, which the shape agreement supports; a direct refit of the raw torque data, requested from the authors, is the only check that closes this, and the weighting of the experiment's 95 runs is not public, hence the several weightings. The earlier figure on this page, 44 µm from matching the gravitational-strength edge at one separation, was a looser reading and is superseded; the change box keeps it. With σ below 33 µm the model reaches, in the collinear layout, only pairs closer than 27 µm at small splits, and for the letter's 23 µm split only separations below 6 µm, not its own geometry 24 µm apart. Below the limit the mimic range drops under 10 µm, where the published limit sits hundreds to thousands of times above the smeared pull's effective strength; that, not a gap in the data, is what keeps the region open. The proposal the letter took its geometry from screens the Casimir force with a conducting plate 1 µm wide and demands that the Casimir pull of each mass toward the plate stay below a tenth of the magnetic pull that closes the interferometer; on that condition, with its own numbers, the inner states must keep 8 µm from the plate at 10⁻¹⁵ kg, so the nearest separation cannot fall below 17 µm (27 µm at 10⁻¹⁴ kg; the proposal's own 24 µm clears the floor). Under that floor the window for the letter's 23 µm split, 15 µm, is closed; for splits of 10 µm and less it stays open, between 17 and 27 µm, with a negativity peaking at 0.01 to 0.04 after five hours to five weeks at 10⁻¹⁵ kg.Whether the published analysis, run on its own data, would reject the smeared shape is for the experiment to say, and the second group's bound at 48 µm is stated as published because its geometry is not in the open text; the letter calls the entangling range of σ 'previously untested' and its main text does not mention these tests. The page notes the question and leaves it open.

The proposed experiment is also a collapse test. If gravity is quantum and the Diósi–Penrose collapse exists besides, with Newton's law untouched, the collapse decoheres the two masses on the same two-body kernel the hybrid model uses for its noise, at a rate that grows with the mass squared and falls as the smearing grows; on this page's kernel the quantum channel's negativity of 0.078 for the 2017 geometry is reduced to zero at 2.5 s unless σ is above 130 µm, and to half unless σ is above 177 µm, while the letter's closer geometry keeps some unless σ is below 14 µm. The threshold is set by the geometry's size and barely by the mass or the hold time while the entangling phase stays small, since both the phases and the collapse damping grow as m²t: scaling the 2017 geometry by a half or by two moves it to 66 and 259 µm, while a tenfold change of mass at fixed geometry leaves it at 130 µm. The 2017 proposal named spontaneous collapse as a mechanism that 'will typically lead to a strong loss of coherence' and gave no number; these are the numbers this page computes on that model. A positive witness at the 2017 geometry would therefore set a lower bound of 130 µm on the collapse length, more than five orders of magnitude above the radiation detectors' 4.9 Å; in that scenario Newton's law is untouched, so the hybrid form's Eöt-Wash estimate does not apply to it. A null result would not by itself separate classical gravity from quantum gravity with collapse. A 2021 study of decoherence in these proposals found that the other collapse model, CSL, would hinder the entanglement for any value of its parameters; it did not treat the Diósi–Penrose case, which is this page's computation.

The two laboratory uses of σ sit on one geometric axis. For any geometry both thresholds scale with the nearest approach d and depend otherwise only on the ratio of split to nearest approach: the letter's hybrid dynamics entangles the pair only for σ above 1.2 to 38 times d as the split grows from a thirtieth of d to thirty-two times it, while quantum gravity with pure collapse keeps a surviving witness for σ above 0.37 to 1.0 times d over the same range. So a positive witness would read as a lower bound on σ in either model, and the two bounds differ by a factor of 3.3 at small splits, 3.8 at the letter's geometry, 4.1 at the 2017 geometry and about 11 at a split eight times the gap; the page computes the two curves and draws no conclusion from the gap between them, since no witness has been measured. At the T.05 recast limit of 33 µm the hybrid dynamics would entangle only geometries whose nearest approach is below 27 µm at tiny splits and below 14 µm at equal split and gap, against a Casimir floor of 17 µm for 10⁻¹⁵ kg; at equal split and gap the window is closed, it opens only for splits below about half the gap, and whether anything sits in it is not determined by this page.

Read together with the 2020 inverse-square data, the map says where the one explicit classical model on the page could still reach the witness. At the 2017 geometry the hybrid dynamics needs σ above 525 µm, a smearing that on this site's full-pattern kernel would remove the whole 120-fold torque the 2020 experiment measured at its Newtonian value at 52 µm; at the letter's own geometry it needs 55 µm, at which the smeared pull would take 6 % off that torque where the published exclusion edge corresponds to 2.5 %. At this site's recast limit of 33 µm, in the collinear layout of the 2017 proposal, the model reaches only pairs closer than 27 µm at a split a thirtieth of the gap and 14 µm at equal split and gap, against a Casimir floor of 17.1 µm for 10⁻¹⁵ kg and 26.5 µm for 10⁻¹⁴ kg. No hold time widens that reach and heavier masses only narrow it, since the hybrid threshold is geometric while the floor rises with mass: at 10⁻¹⁴ kg the reach clears the floor only at the smallest splits, and at 10⁻¹³ kg the floor of 41.5 µm lies above the 27 µm reach. What stays open on the recast is a region of masses up to about 10⁻¹⁴ kg, splits below about half the gap, nearest approaches between the floor and 27 µm, and σ between about 21 and 33 µm. So a witness at a nearest approach beyond that reach could be attributed to this model only with a smearing that contradicts the 2020 torque on this site's kernel. What such a witness would say about gravity remains the dispute above, since other classical accounts are argued in the literature and this page assesses none of them; and whether the 33 µm recast holds as a bound waits on a refit of the 2020 data with the smeared kernel.

Inside that region, in that layout, time does the separating. For a 10⁻¹⁵ kg pair at the 17 µm floor with a split a thirtieth of the gap, the quantum channel reaches full entanglement in 14 hours; at that moment the hybrid dynamics at σ = 33 µm shows a negativity of 0.0013, and its own peak of 0.037 comes after about five weeks. At the 10⁻¹⁴ kg floor, where 33 µm barely clears the hybrid threshold of 32 µm, the quantum channel takes 13 minutes and the hybrid dynamics shows 0.0004 then and never exceeds 0.001. Across the region the hybrid peak arrives roughly four to sixty times later than the quantum channel's full entanglement and stays well under a tenth of it. So the hybrid dynamics does not predict a witness of the quantum channel's size on the quantum channel's timescale, and the quantum channel does not predict a peak of a few hundredths after days of coherence; the page states the two predictions, and no witness has been measured.

The map and the data, applied to the geometries the literature has actually proposed. In the collinear layout the 2017 proposal needs σ above 525 µm from the hybrid dynamics and a 2020 refinement with the masses 100 µm apart at closest approach needs 406 µm, both erasing the 52 µm torque; the 2020 Casimir-screened proposal, a 23 µm split 47 µm from its partner behind a 1 µm plate, needs 82 µm, a 22 % deficit against a published edge at 2.5 %. That paper calls 47 µm an initial separation without saying whether it is the centre distance or the nearest approach; read as the centre distance, as the 2025 letter reads it, the nearest approach is 24 µm and the threshold 55 µm, a 6 % deficit, so the table carries both readings. Side by side the threshold drops: with the two superpositions parallel rather than in line, the hybrid dynamics entangles above 0.65 times the gap for any split, half the collinear value, and the pure-collapse witness survives above 0.44 to 0.47 times it. So the 2024 proposal of micrometre superpositions of 10⁻¹⁴ kg diamonds beside a screening plate, 1.7 µm splits 21 µm apart, needs only σ above 13.6 µm, a smearing that leaves the 2020 torque untouched: among the geometries mapped here it is the only one the hybrid dynamics could entangle without contradicting the inverse-square data. Its 21 µm gap sits below the 26.5 µm floor this page computes from the free-fall formula; the 2024 scheme traps and screens instead, and the page takes its geometry as published. Time still separates the two there. That geometry is designed for an entangling rate of 0.01 per second, so the quantum channel reaches full entanglement in about 160 s; at that moment the hybrid dynamics at σ = 33 µm shows a negativity of 0.0062 (0.0078 at 30 µm), and its peak of 0.059 comes after about 53 minutes (0.054 after half an hour at 30 µm). The 61 µm spacing needs 39 µm, above the recast limit, so it closes.

Decoherence enters the same way for both models. With the local dephasing the instrument applies, the quantum channel at that geometry keeps some entanglement for decoherence rates below 0.0098 per second, the proposal's own criterion reproduced to a part in a million; the hybrid dynamics keeps some only below 0.00008 per second at σ = 33 µm, 0.0001 at 30 µm and 0.0002 at 20 µm, forty-five to a hundred and twenty-five times lower. At a thousandth per second, ten times below the rate the proposal budgets for gas and black-body scattering, the hybrid dynamics shows no entanglement at any time in that geometry while the quantum channel still reaches 0.34. So within the one geometry the data leave it, the hybrid dynamics shows entanglement only at decoherence rates fifty to a hundred and thirty times below the proposal's own budget; the page computes the two thresholds and measures neither.

The single rate this page's instrument applies has two standard physical sources. Residual gas first. For a sphere far larger than the gas's thermal wavelength, split far wider than that wavelength, each hard collision already resolves the split, so the rate no longer depends on it: it is proportional to the pressure and to the square of the radius, and inversely to the square root of the gas atom's mass times the temperature, so that at a fixed pressure a colder gas dephases faster. That is a rate per mass, with the terms in which both masses differ decaying twice as fast: the convention of the 2024 proposal, of the 2021 decoherence analysis and of this page's instrument, so no factor of two separates them. The measured parent tested this theory on C₇₀ molecules, about ten orders of magnitude lighter and at room temperature; no decoherence measurement at micron scale or at a few kelvin is among this page's sources, so every number below is an extrapolation of the theory, not a measurement. Two saturated rates are carried. The rate at which helium atoms at 4 K strike a 10⁻¹⁴ kg diamond sphere is 6.4 × 10¹² per second for each pascal; it is the lowest defensible rate, so the ceiling it sets is the necessary one. The closed form the proposals cite is an interpolation that exceeds it by exactly 8π/3, about 8.4, and gives 5.4 × 10¹³; its ceilings are the proposals' own stricter working figures. At the 2017 proposal's 10⁻¹⁵ Pa and 0.15 K the cited form gives 2.8 s for helium, beside the 3.5 s fall that paper calls the same order; that reproduces the paper's own formula and tests no physics. Two measurements sit closer in scale. The damping of a silica nanoparticle levitated in a Paul trap follows the gas pressure in proportion from 10⁻⁴ down to 7 × 10⁻¹¹ mbar; and, in a separate experiment, the heating of a silica sphere of 71.5 nm radius by gas collisions has been measured in re-heating runs at 9 × 10⁻⁹ mbar. The first experiment's own heating rates exceeded what gas alone predicts, so only its damping is cited. Both measure the momentum the gas hands over, in particles hundreds to thousands of times lighter than the spheres here; neither prepares a spatial superposition, and its loss to gas at any size above a large molecule is not among the measurements this page found.

Thermal photons work the other way round. At a few kelvin their wavelength is a fraction of a millimetre, far longer than any split here, so a single photon resolves little and the rate grows as the square of the split; for emission it also grows as the cube of the radius and the sixth power of the sphere's internal temperature, and for photons scattered from the surroundings as the sixth power of the radius and the ninth of their temperature. The measured parent is again C₇₀, hot molecules whose interference faded as the theory of thermal emission predicts; a cold micron sphere of bulk diamond is an extrapolation of that theory. For the same sphere split by 1.7 µm, emission at 4 K dephases at 2 × 10⁻⁶ per second, on an assumed absorption: diamond's far-infrared loss is poorly known, the page takes the imaginary part of the dielectric constant that the 2021 analysis takes, 10⁻⁴, and carries a decade either side, which with cold surroundings moves a temperature ceiling by a factor of 1.5. On the same forms the 2021 analysis's own geometry reaches its separability rate at 3.5 K, where the paper says 'lower than 4 K'; the kernel is stricter there by about a factor of two in rate. The kernel takes the internal and ambient temperatures as given; how a trapped sphere reaches and holds an internal temperature is not computed here.

Set against the two thresholds of the 2024 geometry, those rates translate into ceilings on two environmental parameters. For the quantum channel to keep any entanglement, helium at 4 K must stay below 1.5 × 10⁻¹⁵ Pa, about 28 atoms in a cubic centimetre, and the sphere's internal temperature below about 16 K with the surroundings at 1 K. For the hybrid dynamics at σ = 33 µm the pressure ceiling is 1.2 × 10⁻¹⁷ Pa, about one atom in five cubic centimetres, and the internal-temperature ceiling 7.3 K. On the form the proposals cite the two pressure ceilings are 1.8 × 10⁻¹⁶ Pa and 1.5 × 10⁻¹⁸ Pa. With cold surroundings the pressure ceiling scales with the threshold itself and the internal-temperature ceiling with its sixth root, so the 125-fold stricter threshold of the hybrid dynamics appears as a 125-fold lower pressure ceiling and a 2.2-fold lower temperature ceiling. With surroundings as warm as the sphere, scattered photons join in, the sixth-root rule no longer holds, and the two temperature ceilings are 12 K and 6.3 K. The 2024 proposal budgets about 0.01 per second 'for the ambient temperatures of ∼1 K' and states no pressure; that rate is helium at 8 × 10⁻¹⁶ Pa on the kinetic rate and 9 × 10⁻¹⁷ Pa on the cited form. Each ceiling lets one source use the whole threshold, with the full split held throughout; together the sources must share it, hydrogen would add to helium, with a collision rate 1.4 times higher at the same pressure, and dipole and Casimir couplings, gravity gradients, phonons and the trap are not counted. These are necessary conditions, not a budget, and the page pronounces on no experiment's feasibility. One published figure for a cryogenic vacuum can be set beside them: from how long antiprotons survived in a cryogenic Penning trap, its collaboration infers that the hydrogen and helium partial pressures there were below 1.2 × 10⁻¹⁶ Pa and 2.7 × 10⁻¹⁶ Pa at 68 % confidence. These are model-dependent upper limits for that trap at its own temperature, not measured pressures and not a statement about an interferometer's chamber. Read as helium at 4 K, that helium limit is about a fifth of the quantum channel's necessary ceiling and one and a half times its cited one; the hybrid dynamics' two ceilings lie about 20 and about 180 times below the limit.

The dispute's other classical route can be put on the same map. The 2025 Nature paper that argues classical gravity entangles quantum fields gives its entangling term in closed form: it grows as the sixth power of the object's mass and the fourth power of Newton's constant (its square root as the cube and the square), and its authors write that at 10⁻¹⁴ kg it is 'substantially smaller' than the quantum phase and becomes significant for masses 'approaching the Planck mass and beyond'. Evaluated by this page on their formula for diamond spheres, the proposals' material, the ratio of that term to the quantum phase is 2 × 10⁻²⁹ at the 2017 geometry, 5 × 10⁻³⁰ at the 2020 refinement, 7 × 10⁻³⁴ at the screened 2020 geometry and 8 × 10⁻²⁹ at the 2024 side-by-side one; ytterbium, the paper's material, gives about ten thousand times more. For a Planck-mass ytterbium sphere ten radii apart, the paper's own setting, the term lies far above the 0.1 the paper calls significant. The formula is a leading-order one that assumes a small quantum phase and a nearest approach well inside the split, conditions these geometries meet only partly, so the ratios are order-of-magnitude readings, not a finding about any experiment; whether the mechanism is right is the dispute above.

III · How far the laboratories are

Gravity has been measured on the table with milligram-scale masses by two groups: in 2021 between two gold spheres of about 1 mm radius and 90 mg, on a torsion pendulum at 12.7 mHz over 350 hours, down to a surface distance of 400 µm; and in 2024 by a levitated 0.4 mg magnetic particle at 26.7 Hz with a force noise of 0.5 fN/√Hz, sensing kilogram source masses about half a metre away. Both are classical measurements of a classical pull, by different collaborations, which is what MEASURED requires. Neither mass was in two places; the proposal's masses are between seven and ten orders of magnitude lighter still.

In 2026 a one-milligram torsional pendulum at 18 Hz was laser-cooled from room temperature to 240 µK, more than twenty times colder than earlier oscillators from micrograms to kilograms; the paper presents it as a platform for the quantum–gravity interface, not as a measurement of gravity. The same year two librational modes of a levitated silica nanorotor were cooled to occupation numbers of 0.21 ± 0.03 and 0.54 ± 0.32 one at a time, and 1.02 ± 0.08 and 0.73 ± 0.22 together, with the rotor aligned to better than 20 µrad. Both rows are young and each is one laboratory's result, so each carries SINGLE RESULT; neither put a mass in two places, and the proposal's mass sits between them. The page lists them as distance covered, not as the experiment.

A 2026 preprint proposes to replace the second mass by an array of quantum clocks and ask whether a levitated particle's gravity entangles with them, noting that the quantised active gravitational mass has not been tested in any prior experiment. A preprint stays PROPOSED whatever it proposes. The page records it because it ties this question back to T.04: the second mass becomes a clock.

IV · What would count, and what this page does not say

What a positive result would settle depends on assumptions the dispute is about: the 2017 sufficiency argument needs the field to be a mediator acting locally, the 2026 paper grants both and concludes that an observed entanglement would then have to come from something other than classical gravity, and the 2025 letter answers that a specific classical model entangles anyway, so that the experiment would test that model rather than the quantum nature of gravity. The page does not say gravity is quantum, or that it is not; it does not say the experiment will run, or what it will show. It says which published statement each channel rests on, and that no channel has a measurement behind it.

Grounding ledger rows cited on this page

Every row below resolves in the site's offline Crossref cache. The lint derives what each set of rows can carry; a person decided the label on each claim above. Rows pass a review date and the build fails until someone rereads the source.

Scite column: [observed: Scite ML citation classification, accessed 2026-11-09, 2026-11-10] via Scite's public tallies endpoint. The counts are a classifier's output over indexed citation statements, an evidence-synthesis habit borrowed from medicine; they never replace reading the source, and a row with more contrasting than supporting statements is flagged for a human re-read in the label checklist.

AI contribution to the cited results: none. No AI system contributed to any proposal, measurement, derivation or rebuttal cited on this page. The page text, instrument and verifier were drafted with AI assistance and checked by the site maintainer against the primary sources listed below.

---
name: timegravity-t05-entanglement-channels
description: Rebuild and verify the Time & Gravity T.05 instrument - the Bose-Marletto-Vedral four-branch entanglement phase (phi_ij = G m^2 t / hbar d_ij, concurrence |sin(delta phi / 2)|, Peres negativity under local dephasing) with three channels at equal weight (quantum, classical LOCC, disputed classical non-LOCC) - with a planted defect that lets a LOCC channel entangle.
license: MIT
compatibility: Claude Code, GitHub Copilot, Cursor, Google Antigravity, OpenAI Codex, Ollama (Qwen 3.8 / Gemma 4)
---
# Does Gravity Entangle, and Would That Settle It? (T.05 · Time & Gravity)

> **Fixed Boundary (`EN`):** Nothing on this page is a theory of everything, and this site proposes none. Every claim carries one of six labels, each tied to a ledger row: MEASURED, DERIVED, BOUND, PROPOSED, OPEN, or SINGLE RESULT. Preprints never rise above PROPOSED. Each page ends with one open question that the next page takes up.
>
> **Frontera Fija (`ES-MX`):** Nada en esta página es una teoría del todo, y este sitio no propone ninguna. Cada afirmación lleva una de seis etiquetas, cada una ligada a una fila del libro de registro: MEDIDO, DERIVADO, COTA, PROPUESTO, ABIERTO o RESULTADO ÚNICO. Las prepublicaciones nunca suben de PROPUESTO. Cada página termina con una pregunta abierta que la siguiente página retoma.

## 1. Page, canonical URL and governing equations
- **Track page**: `T.05` (Time & Gravity, reading order 6 of 9, dated `2026-11-09`). Page label **OPEN**: the experiment has not run, and the literature disputes what a positive result would prove.
- **Canonical URL**: https://informatico.ai/timegravity/t05/ (Spanish: https://informatico.ai/es-mx/timegravity/t05/)
- **Branch phases**: `φ_ij = G m² t / (ħ d_ij)` with `d_LL = d_RR = d`, `d_LR = d + Δx`, `d_RL = d − Δx`.
- **Entanglement phase**: `Δφ = φ_LR + φ_RL − φ_LL − φ_RR`; the four-branch state is a product exactly when `Δφ ≡ 0 (mod 2π)`.
- **Measures**: concurrence `C = 2|ad − bc| = |sin(Δφ/2)|` (Wootters 1998); negativity from the partial transpose (Peres 1996), `N = C/2` for the pure state; local dephasing `e^{−γt}` on each mass.
- **Channels** (equal weight): `quantum` (the four phases as written); `locc` (each mass feels the mean field of the other: `φ_ij = α_i + β_j`, `Δφ = 0`); `disputed` (classical, non-LOCC: no single prediction; carried as the interval `[0, C_quantum]`).
- **Central numbers (2017 geometry: 10⁻¹⁴ kg, d = 450 µm, Δx = 250 µm, t = 2.5 s)**: `Δφ = 0.314 rad`, `C = 0.156`, `N = 0.078`; `C = 1` at `t = 25.0 s`; negativity vanishes at `γt = 0.156`; LOCC channel `< 10⁻¹²`.
- **Generator**: `python skills/timegravity/t05/make_t05.py`
- **Verifier**: `python skills/timegravity/t05/verify_t05_entanglement_channels.py`
- **Recast kernel** (`tools/timegravity_kernel_recast.py`, data `data/eotwash2020_table11_4.json`): the dissertation's Table 11.4 (66 λ, α* ± σ_α, ±α95) recast onto the smeared pull: the 120ω deficit over 52 µm–3 mm projected onto the unit Yukawa shape at each λ, best-mimic λ ≈ 0.55 σ with 2–4 % shape residual, excluded where α_eff < −α95; σ_95 = 33.1 µm (33.1 on three separation grids, on an absolute-torque weighting w = T_N² and on both harmonics, 33.6 on a linear grid; the data's own α* is positive across 5–100 µm, so the signed −α95 is the operative limit); the experiment's own |α95| = 1 crossing recovered at 38.6 µm (it states 39); the retired single-separation edge match (44 µm) is the planted defect; below the rung the mimic range falls under 10 µm (σ = 25 / 18 / 13.6 µm → λ = 11 / 7.1 / 5 µm, about 7 / 140 / 5,400 times too weak for the published limits there, `reach_below_limit`), so the room below 33 µm is the absence of near-gravitational sensitivity at 5–10 µm. The other published table below the balance's separations, Geraci et al. 2008 (micro-cantilever, about 25 µm; Table 5), is weaker at every common range from 10 to 66 µm (|α| < 14,000 against the balance's 3,100 at 10 µm) and stronger at 6 µm (460,000 against the balance's interpolated 730,000), where it is still about two thousand times the strength the smeared pull would need (`short_range_comparison`). Verifier `python skills/timegravity/t05/verify_t05_recast.py` (14 s; `--full` runs all eight variants); `--planted-defect` must exit 1. `SIGMA_ESTIMATE_M` / `EW_MATCH_SIGMA_M` = 33.1 µm.
- **Decoherence-budget kernel** (`tools/timegravity_kernel_decoherence.py`): the page's one dephasing rate (per mass, e^{−γt}; doubly off-diagonal terms e^{−2γt}; the convention of Schut 2024 Eq. 23 and Rijavec 2021 Eq. 13) as residual gas and thermal photons. Gas, saturated (helium at 1–4 K has λ_th under a nanometre): the kinetic rate `n v̄ πR² = √(8π) P R²/√(m k_B T)` is the lowest defensible one and sets the necessary ceiling; the form the proposals cite (Romero-Isart 2011; Rijavec 2021 Table 2; van de Kamp 2020 Eq. 33), `Γ₀ = λ_th²Λ = (16π√(2π)/3) P R²/√(m k_B T)`, is an interpolation larger by exactly 8π/3. Photons, long wavelength: `Λ Δx²` with scattering ∝ R⁶T⁹ and emission ∝ R³T⁶ Im[(ε−1)/(ε+2)]; diamond's Im ε = 10⁻⁴ is assumed, with a decade either side. Measured parents are C₇₀ experiments (Hornberger 2003, Hackermüller 2004): every number is an extrapolation. Closer in scale, gas damping (Dania 2024, Paul trap: linear in pressure from 10⁻⁴ to 7 × 10⁻¹¹ mbar; its own heating rates exceed the gas prediction, so only the damping is cited) and gas heating (Magrini 2021: 71.5 nm sphere, 9 × 10⁻⁹ mbar) of levitated silica are measured; the loss of a spatial superposition to gas is not among the page's sources. Hydrogen's collision rate is 1.41 times helium's at the same pressure. Transcription checks (they reproduce the papers' formulas, they validate no physics): Bose 2017 (10⁻¹⁵ Pa, 0.15 K → 2.8 s on the cited form beside the 3.5 s fall), Rijavec 2021 (1/τ_G 0.063 /s; photons reach it at 3.5 K against 'lower than 4 K', a factor 2.4 stricter in rate at 4 K; cited gas at 10⁻¹⁶ Pa an order below), van de Kamp 2020 (Eq. 33 identical; 3.6 K); Schut 2024 states 10⁻² Hz at ~1 K and no pressure (implied: 8 × 10⁻¹⁶ Pa kinetic, 9 × 10⁻¹⁷ Pa cited). At the 2024 z = 10 µm geometry, helium at 4 K: quantum channel (9.8 × 10⁻³ /s) below 1.5 × 10⁻¹⁵ Pa (28 atoms/cm³; cited 1.8 × 10⁻¹⁶) and internal temperature below 16 K (surroundings at 1 K); hybrid at σ = 33 µm (7.9 × 10⁻⁵ /s) below 1.2 × 10⁻¹⁷ Pa (0.22 atoms/cm³; cited 1.5 × 10⁻¹⁸) and 7.3 K: pressure by the full ratio (125), temperature by its sixth root with cold surroundings (12 K and 6.3 K with warm ones). Beside the antiproton-trap helium limit (2.7 × 10⁻¹⁶ Pa, Sellner 2017, inferred, that trap): 0.18 of the quantum kinetic ceiling, 1.5 of its cited one; the hybrid ceilings 22 and 184 times below. One source per ceiling, hydrogen and the other noise sources not counted: necessary conditions, not a budget. Verifier `python skills/timegravity/t05/verify_t05_decoherence_budget.py` (5 s); `--planted-defect` leaves the gas rate in its long-wavelength form and must exit 1.
- **Dispute kernel** (`tools/timegravity_kernel_dispute.py`): the band's computed edges. Lower: Kafri–Taylor–Milburn channel at `K Δx²/2ħ` per mass (`K = 2Gm²/d³`) equals this page's dephasing threshold in the small-split limit (ratio 1.000 at 2 µm). Upper: the Diósi–Penrose model as Trillo & Navascués write it (σ-smeared, `H = 0`), `⟨x|ρ(t)|y⟩ = exp[(G/2ħ) g(x,y) t]⟨x|ρ(0)|y⟩`, predicts entanglement only below `d_c = 0.8508σ` (letter 0.8501σ, appendix 0.850872σ); Fig. 3 geometry (10⁻¹⁵ kg, 23 µm, 24 µm): none at σ = 50 µm, about 5 h / 16 h / 20 d at 75 / 100 / 250 µm; 2017 geometry needs σ > 525 µm and shows < 10⁻³ at 2.5 s; the smeared pull is `erf(d/2σ)` of Newton's. Decomposition: `tn_state(..., part="interaction" | "noise")`; the noise part alone entangles nothing, the interaction part alone (the smeared pair potential as a Hamiltonian) gives at least the full negativity and equals the quantum channel as σ → 0 (0.078 for the 2017 geometry); letter geometry at σ = 75 µm: 0.024 full vs 0.152 interaction-only; at 50 µm: 0 vs 0.363. Pure collapse (`dp_collapse_state`, quantum phases plus Diósi's two-body damping, Newton intact): critical σ 130 µm for the 2017 geometry, 14.3 µm for the letter's; `dp_collapse_scaling` shows the threshold scales with the geometry's size (×0.5 → 65.6 µm, ×2 → 259 µm) and moves under 1 % for a tenfold mass change or 1 s vs 2.5 s while the phase is small. `rijavec_tau_g`: τ_G = ħd[(d/Δx)² − 1]/(Gm²) of Rijavec et al. 2021; this page's dephasing threshold 0.0623 /s is 0.992 of their 1/τ_G. `threshold_map`: over r = split/nearest from 0.03 to 32, the hybrid dynamics entangles above G(r)·d with G from 1.21 to 37.8 and the witness survives pure collapse above F(r)·d with F from 0.37 to 1.0, both mass-independent (Plate T.05-M, `data-tg-figure="threshold-map"`). `map_with_isl`: the map read with the 2020 data: the smeared pull at the 2017 geometry's hybrid threshold (525 µm) removes 100 % of the 52 µm torque, at the letter's (55 µm) 6.2 % against the published edge's 2.5 %; at the 33 µm recast limit the model reaches nearest approaches of 27 µm (r = 0.03) and 14 µm (r = 1) against Casimir floors of 17.1 / 26.5 / 41.5 µm at 10⁻¹⁵ / 10⁻¹⁴ / 10⁻¹³ kg; a pair at the 10⁻¹⁵ kg floor needs σ > 20.7 µm (r = 0.03). `open_region_timing`: inside that region the hybrid negativity at the quantum channel's full-entanglement time is below 2 × 10⁻³ (quantum: 0.5), its peak (≤ 0.037) comes 4 to 60 times later (10⁻¹⁵ kg at the floor, r = 0.03, σ = 33 µm: t_full 14.3 h, peak after about five weeks). Planar branches (`branches_collinear`, `branches_parallel`, `hybrid_critical_sigma_points`, `collapse_critical_sigma_points`) reproduce the 1-D kernels to 1e-12; side by side the hybrid threshold is about 0.65 of the gap for any split and the collapse one 0.44–0.47 (`parallel_map`). `proposals_table` (Table T.05-P): Bose 2017 → hybrid 525 µm (100 % of the 52 µm torque), Chevalier 2020 d = 350 µm → 406 µm (99.9 %), van de Kamp 2020 → 82 µm (21.6 %), Schut 2024 z = 30 / 10 / 5 µm → 39.4 / 13.6 / 7.1 µm (1.5 / 0.0 / 0.0 %); at σ = 33 µm the Schut z = 10 geometry shows hybrid negativity 6.2e-3 at the quantum channel's full-entanglement time (160 s) and peaks at 0.059 after about 53 min. `open_window_decoherence` (local dephasing e^{−γt} per mass, as the instrument): at that geometry the quantum channel keeps entanglement only for γ < 9.8e-3 /s (the proposal's |ω_ent|, reproduced to a part in a million), the hybrid dynamics only for γ < 3.7e-5 / 1.0e-4 / 2.2e-4 /s at σ = 44 / 30 / 20 µm; at γ = 1e-3 /s the hybrid shows nothing while the quantum channel peaks at 0.34. `aziz_howl_table`: Aziz & Howl 2025 Eq. 85, √ϑ = (6/25) G² m² M³ R t/(ħ³ d), against their Eq. 42 phase; diamond spheres at each proposal's time: ϑ/φ = 2e-29 (2017), 5e-30 (Chevalier), 7e-34 (van de Kamp, 47 µm), 8e-29 (Schut z = 10, 1 s); ytterbium 2017 at 2 s 4e-25; Planck-mass ytterbium sphere at 10R, 1 s: ϑ = 6e15 (their 'significant' regime). Verifier `python skills/timegravity/t05/verify_t05_dispute_kernel.py`; `--planted-defect` (decoherence dropped, no critical separation) must exit non-zero.
- **Smeared-gravity kernel** (`tools/timegravity_kernel_isl.py`): the model's pair potential `G m₁m₂ erf(d/2σ)/d` set beside the sub-millimetre inverse-square-law tests. Between two patterned plates each in-plane mode `q` pulls as `R(q, s; σ) = ½[erfc(qσ − s/2σ) + e^(2qs) erfc(qσ + s/2σ)]` of Newton's `e^(−qs)` (closed form vs direct quadrature 1e-14; limits σ → 0, s ≫ σ, q → 0 give Newton, s = 0 gives erfc(qσ); monotone fall with `q`, so the pattern's lowest wavenumber bounds each harmonic's deficit from below up to the Bessel tail). Eöt-Wash 2020 (Pt plates 54 / 99 µm, patterns 52 mm across, 18- and 120-fold, s = 52 µm … 3.0 mm, λ < 38.6 µm at 95 %): 120-fold deficit at 52 µm 10.4 % (σ = 75 µm), 2.5 % (50 µm), 0.2 % (30 µm), 3.7 % at σ_c = 55 µm of the letter's geometry; Yukawa at the edge (α = −1, λ = 38.6 µm) 1.5 %; the retired single-separation match σ 44 µm; best-mimic Yukawa λ ≈ 49 µm, α ≈ −3.2 (a curve match). A one-mode estimate of predicted signal sizes, not a bound on σ (the recast kernel carries the limit); HUST 2020 (λ = 48 µm) quoted as published. Verifier `python skills/timegravity/t05/verify_t05_smeared_isl.py`; `--planted-defect` (the closed form's `e^(2qs) erfc` term dropped) must exit non-zero.
- **Wedge kernel** (`tools/timegravity_kernel_wedge.py`): the full Eöt-Wash 2020 geometry from Lee's 2020 dissertation (hdl.handle.net/1773/45542; 120 wedges 13.004–22.998 mm at 1.490°, 18 wedges 23.505–26.000 mm at 9.984°, foils 54 / 99 µm, glue 1.162 g/cm³) under the collaboration's Fourier–Bessel expansion (Lee et al. 2021, Eqs. 7–19): `T_N(s) = 8 N G ρc² sin²(Nβ) ∫dk R(N,k)² Z(k,s)`, stdlib Bessel functions (Miller recurrence; checked against the integral representation to 1e-9), radial series with a truncation that grows with `k·r₁` (checked against quadrature to 1e-6). Newtonian 120-fold torque 1.51 fN·m at 72 µm (the letter's 'roughly 1 %' magnetic subtraction implies about 1.6). Smeared deficits at 52 µm: 17.0 % (σ = 75 µm), 4.3 % (50), 0.3 % (30), 6.3 % (55 µm, σ_c of the letter's geometry); Yukawa at the edge 2.5 %; the single-separation edge match, now retired, gave σ 44.1 µm (one-mode bound: 44.4 µm) with mimic Yukawa λ ≈ 49 µm, α ≈ −3.1. Window at the recast limit σ ≤ 33 µm: d_c 27.2 µm (small split), 5.7 µm (23 µm split), the letter's geometry never. Same verifier; `--planted-defect` also plants a fixed 250-term truncation, which the quadrature double entry rejects.
- **Knockout check**: `python skills/timegravity/t05/verify_t05_entanglement_channels.py --planted-defect` (must exit non-zero: a LOCC channel that entangles)
- **Label lint**: `python tools/check_timegravity_labels.py` (and `--planted-defect`, `--mutants`)

## 2. Ledger rows this page may cite (timegravity/ledger.json)
| Row | Label | Source | DOI |
| --- | --- | --- | --- |
| `T05-BOSE2017` | PROPOSED | Bose et al. 2017, spin entanglement witness | https://doi.org/10.1103/PhysRevLett.119.240401 |
| `T05-MARLETTO2017` | PROPOSED | Marletto & Vedral 2017, sufficiency argument | https://doi.org/10.1103/PhysRevLett.119.240402 |
| `T05-WESTPHAL2021` | MEASURED | Westphal et al. 2021, gravity between 90 mg spheres | https://doi.org/10.1038/s41586-021-03250-7 |
| `T05-FUCHS2024` | MEASURED | Fuchs et al. 2024, 0.4 mg levitated test mass | https://doi.org/10.1126/sciadv.adk2949 |
| `T05-AGAFONOVA2026` | SINGLE RESULT (young) | Agafonova et al. 2026, 1 mg pendulum at 240 µK | https://doi.org/10.1038/s42005-026-02514-w |
| `T05-TROYER2026` | SINGLE RESULT (young) | Troyer et al. 2026, librational ground state | https://doi.org/10.1038/s41567-026-03219-1 |
| `T05-KTM2014` | PROPOSED | Kafri, Taylor & Milburn 2014, classical channel model | https://doi.org/10.1088/1367-2630/16/6/065020 |
| `T05-PERES1996` | DERIVED | Peres 1996, separability criterion | https://doi.org/10.1103/PhysRevLett.77.1413 |
| `T05-WOOTTERS1998` | DERIVED | Wootters 1998, two-qubit concurrence | https://doi.org/10.1103/PhysRevLett.80.2245 |
| `T05-AZIZ2025` | PROPOSED (young, disputed) | Aziz & Howl 2025, Nature | https://doi.org/10.1038/s41586-025-09595-7 |
| `T05-TRILLO2025` | PROPOSED (young) | Trillo & Navascués 2025, PRD Letter | https://doi.org/10.1103/PhysRevD.111.L121101 |
| `T05-SCHNEIDER2026` | PROPOSED (young) | Schneider, Huggett & Linnemann 2026, CQG | https://doi.org/10.1088/1361-6382/ae6f62 |
| `T05-MARLETTO2025` | PROPOSED (preprint) | Marletto & Vedral 2025, rebuttal | https://doi.org/10.48550/arXiv.2510.19969 |
| `T05-DIOSI2025` | PROPOSED (preprint) | Diósi 2025, rebuttal | https://doi.org/10.48550/arXiv.2511.00852 |
| `T05-TOBAR2026` | PROPOSED (preprint) | Tobar, Aspelmeyer & Zych 2026, clock proposal | https://doi.org/10.48550/arXiv.2611.08313 |
| `T05-TILLOY2016` | PROPOSED | Tilloy & Diósi 2016, hybrid construction | https://doi.org/10.1103/PhysRevD.93.024026 |
| `T05-VANDEKAMP2020` | PROPOSED | Van De Kamp et al. 2020, the Fig. 3 geometry | https://doi.org/10.1103/PhysRevA.102.062807 |
| `T05-LEE2020` | MEASURED | Lee et al. 2020, 1/r² down to 52 µm | https://doi.org/10.1103/PhysRevLett.124.101101 |
| `T05-TAN2020` | MEASURED | Tan et al. 2020, |α| ≤ 1 down to 48 µm | https://doi.org/10.1103/PhysRevLett.124.051301 |

## 3. Tool-neutral agent packet and security boundary (`security-runtime`)
1. **Rebuild locally**: `python skills/timegravity/t05/make_t05.py`; standard library only; writes only under `src/research/timegravity/t05/files/`.
2. **Acceptance gate**: `python skills/timegravity/t05/verify_t05_entanglement_channels.py` must print `"status": "PASS"` with `locc_channel_entangles_nothing: PASS`; `--planted-defect` must exit non-zero. Then `python tools/check_timegravity_labels.py`.
3. **Security boundary (`security-runtime`)**: no shell outside the repository, no home-directory access, no outbound network for this task; the cache refresher is the track's only network script, run by hand by a maintainer.

## 4. Deterministic verification and falsification rules
1. Concurrence from the closed form `|sin(Δφ/2)|` and from the reduced-state purity `sqrt(2(1 − Tr ρ_A²))` agree within `10⁻¹²`.
2. The Peres negativity of the pure state equals `C/2` within `10⁻⁹` (partial transpose, Hermitian 4×4 through the real 8×8 embedding, Jacobi solver).
3. The recomputed phases `Δφ_RL = 0.44`, `Δφ_LR = −0.13` and `Δφ = 0.31 rad` lie within a factor of four of the proposal's order-of-magnitude values (0.7, −0.2, ≈ 0.5).
4. `C = 1` at `Δφ = π` (`t = 25.0 s` for the 2017 geometry); `C = 0` with `G = 0`.
5. The LOCC channel entangles nothing on a 3 × 3 × 3 grid of separations, splits and hold times (`< 10⁻¹²`) while reproducing the mean Newtonian phase; the disputed channel is carried as `[0, C_quantum]`.
6. Planted defect: the LOCC channel carries the quantum phases; rule 5 fails and `--planted-defect` exits non-zero.
7. `t05_reference.json` matches `manifest.json` and regenerates byte for byte.

## 5. What the instrument shows (reader side)
- Left: two masses each in two places and the four branch lines with their separations. Right: entanglement against hold time; the three channels drawn at equal weight (quantum curve, LOCC flat line, disputed hatched band).
- Knockout: switch the channel. Planted defect: a classical LOCC channel that entangles.
- Predict before you reveal: the concurrence after 2.5 s in the 2017 geometry.

## 6. Exact T0–T3 truth boundary
### In the repository today (T0/T1)
- Kernel `tools/timegravity_kernel_entangle.py` (`branch_phases`, `entanglement_phase`, `concurrence_closed`, `four_branch_state`, `negativity`, `negativity_threshold_gamma_t`, `disputed_interval`, `check_bmv`), generator, verifier, ledger rows above.

### Not on this page (T2/T3)
- No gravitationally induced entanglement has been measured; no claim is made about which of the 2025 positions is right.
- Casimir–Polder, magnetic and electrostatic forces, the witness statistics of the proposal, and the decoherence budgets of specific laboratories are not modelled beyond a single local dephasing rate.

#!/usr/bin/env python3
"""Standalone stdlib verifier and planted-defect knockout for Time & Gravity · T.05 (Does Gravity Entangle, and Would That Settle It?).
Canonical URL: https://informatico.ai/timegravity/t05/
Central number: for the 2017 proposal's own geometry (1e-14 kg, 450 µm apart, each split by 250 µm, held 2.5 s) the four
Newtonian branch phases give an entanglement phase Δφ = 0.314 rad, concurrence |sin(Δφ/2)| = 0.156, recomputed from the
reduced-state purity within 1e-12, and a Peres negativity of 0.078 (= C/2) from the partial transpose through the track's Jacobi
solver; full entanglement at 25.0 s; a classical mean-field (LOCC) channel entangles nothing (< 1e-12) on a grid of geometries,
and the disputed classical non-LOCC channel is carried as an interval, never a point. Planted defect: a LOCC channel that
entangles (its phases copied from the quantum channel); `--planted-defect` must exit non-zero with FAIL on stderr.
Boundary: Nothing on this page is a theory of everything, and this site proposes none.
"""
from __future__ import annotations

import hashlib
import json
import sys
from pathlib import Path

ROOT = Path(__file__).resolve().parents[3]
sys.path.insert(0, str(ROOT / "tools"))

from timegravity_kernel_entangle import ENT_TOL, NEG_TOL, check_bmv  # noqa: E402

FILES_DIR = ROOT / "src" / "research" / "timegravity" / "t05" / "files"


def verify_reference_files() -> dict:
    manifest = json.loads((FILES_DIR / "manifest.json").read_text(encoding="utf-8"))
    out = {}
    for entry in manifest["files"]:
        raw = (FILES_DIR / entry["filename"]).read_bytes()
        assert len(raw) == entry["bytes"], f"{entry['filename']}: byte count mismatch"
        digest = hashlib.sha256(raw).hexdigest()
        assert digest == entry["sha256"], f"{entry['filename']}: sha256 mismatch"
        out[entry["filename"]] = digest[:12]
    ref = json.loads((FILES_DIR / "t05_reference.json").read_text(encoding="utf-8"))
    assert abs(ref["quantum_channel"]["concurrence"] - 0.1563) < 1e-3, "reference concurrence off"
    assert ref["locc_channel"]["concurrence"] < ENT_TOL, "reference LOCC channel entangles"
    return out


def main(argv: list[str] | None = None) -> int:
    args = argv if argv is not None else sys.argv[1:]
    planted = "--planted-defect" in args
    try:
        res = check_bmv(planted_defect=planted)
        assert res["double_entry_ok"], f"concurrence closed form {res['concurrence_closed']:.12f} vs purity {res['concurrence_from_purity']:.12f} differ by more than {ENT_TOL}"
        assert res["negativity_ok"], f"negativity {res['negativity']:.9f} is not C/2 within {NEG_TOL}"
        assert res["paper_consistent"], "the recomputed phases are not within a factor of 4 of the proposal's order-of-magnitude values (0.7, -0.2)"
        assert res["full_ok"], f"concurrence at Δφ = π is {res['concurrence_at_pi']}, not 1"
        assert res["knockout_ok"], "with G = 0 the state is still entangled"
        assert res["locc_ok"], (
            "the classical LOCC channel entangles: worst concurrence/negativity "
            f"{res['worst_locc_entanglement']:.3e} (> {ENT_TOL}); a channel of local operations and classical communication cannot create entanglement"
        )
        assert res["mean_ok"], "the LOCC channel does not reproduce the mean Newtonian phase"
        assert res["budget_ok"], "no finite dephasing budget found"
        assert res["disputed_ok"], "the disputed channel is not carried as the interval [0, C_quantum]"
        assert res["defect_rejected"], "the planted entangling LOCC channel was not detectable"
        files = verify_reference_files()
    except AssertionError as exc:
        print(json.dumps({"status": "FAIL", "folio_id": "T.05", "error": str(exc)}), file=sys.stderr)
        return 1
    print(json.dumps({
        "status": "PASS",
        "folio_id": "T.05",
        "canonical_url": "https://informatico.ai/timegravity/t05/",
        "central_number": {"entanglement_phase_rad": res["entanglement_phase_rad"], "concurrence": res["concurrence_closed"],
                           "negativity": res["negativity"], "time_to_full_entanglement_s": res["time_to_full_entanglement_s"],
                           "gamma_t_negativity_zero": res["gamma_t_negativity_zero"], "worst_locc_entanglement": res["worst_locc_entanglement"],
                           "disputed_interval": res["disputed_interval"], "phase_rl_rad": res["paper_phase_rl_rad"], "phase_lr_rad": res["paper_phase_lr_rad"]},
        "geometry_checks": {
            "concurrence_closed_form_vs_purity": "PASS",
            "negativity_is_half_concurrence": "PASS",
            "locc_channel_entangles_nothing": "PASS",
            "disputed_channel_carried_as_interval": "PASS",
            "planted_defect_knockout": "REJECTED_AS_EXPECTED",
        },
        "reference_files_sha256_12": files,
    }, indent=2))
    return 0


if __name__ == "__main__":
    raise SystemExit(main())

Primary sources

1. Murata, J., Fujiie, T., & Suzuki, S. (2026). Short-range tests of the gravitational inverse-square law. arXiv:2605.18212. DOI DOI 10.48550/arXiv.2605.18212 [Scite ↗]
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