What gravitationally induced entanglement certifies: mediation semantics and a Bell-style theorem

Summary

A formal, theory-independent definition of “the entanglement was mediated by the gravitational field”, a proof that circuit topology alone cannot supply it, and a Bell-style theorem (Theorem W) stating what a certified positive result of the Bose–Marletto–Vedral protocol refutes under six priced assumptions, which alternatives survive, and what a null result certifies.

Abstract

Proposals to entangle two masses through gravity (Bose et al.; Marletto and Vedral) are widely described as tests of the quantum nature of gravity, but the field has no theorem stating what a positive result would certify, from which assumptions, with which alternatives surviving. We supply one. The central missing ingredient is semantic: a formal, theory-independent definition of “the entanglement was mediated by the gravitational field.” We prove that circuit topology alone cannot provide it, every bipartite gate decomposes exactly into mediator-local rounds, and construct a five-clause definition (interposition, autonomy, locally factorized coupling with a cut criterion, causal propagation, carriage) with three description-invariant operational signatures (retardation, interposed blockability, autonomy response). The definition is adequacy-tested against every configuration in the recent classical-gravity-produces-entanglement debate, is provably non-circular (an explicitly classical theory satisfies every clause and provably cannot entangle), and yields a slaving dichotomy: a candidate mediator is either constraint-slaved (no independent data, no dead time, fixed operator-Schmidt-rank-2 gate) or autonomous (retarded), with the two indistinguishable in all mass-side data. Around this definition we assemble a Bell-style certification theorem from three lemmas with stated scopes: a mediation definition adequacy lemma; a per-run Bell/steering inertness theorem for charge-transfer coherences, presented as a constrained-field-theory instantiation and strengthening of Baldijão et al., including a multi-copy activation boundary we exhibit exactly (joint processing of two runs restores Tsirelson-value violations, which forces an i.i.d. per-run clause into the certified statistic); and a continuum classical-mediator no-go, presented as a dynamical, instrument-level strengthening of Ludescher et al., covering back-reacting instruments with feed-forward over uncountable outcome spaces, continuous-time limits, and the Oppenheim classical–quantum framework, for which the anticipated “small” entanglement is shown to be exactly zero in the locally coupled and Diósi–Penrose classes. The resulting theorem: under six explicitly priced assumptions, a certified positive result implies that the degree of freedom coupling the two labs is non-classical in the state-space sense, either a non-classical mediator, or a nonseparable direct quantum gate with no carrier, the two separated in principle only by retardation. Both disjuncts are non-classical; every classical-gravity model in the current literature is refuted on its own formulation, while prequantum hybrids (van Hove) are correctly classified rather than excluded. We state the null-result converse, price its linear-additivity caveat, compare the theorem honestly with Bell’s, and give experimental guidance, including a visibility-scan certificate for the closed-lab assumption. All model-level claims are verified in deterministic ancillary scripts.

Cite

Vávra, D. (2026). What gravitationally induced entanglement certifies: mediation semantics and a Bell-style theorem. vavradev.com/research, first published 2026-11-04.

@misc{vavra2026b,
  author       = {V{\'a}vra, David},
  title        = {{What gravitationally induced entanglement certifies: mediation semantics and a Bell-style theorem}},
  year         = {2026},
  month        = aug,
  howpublished = {\url{https://vavradev.com/research/vavra-2026-B-what-gie-certifies-theorem-W.pdf}},
  note         = {Manuscript, Article. First published 2026-11-04 at vavradev.com/research; SHA-256 4a6b4ea2710427a93d563675dad8b06e6e51ac70d4914c77a984a9cb12af5470},
}