Entropic / emergent gravity: genuine derivation, or suggestive analogy?

Status: Active research note — disaggregated verdict on four programs Last updated: 2026-06-08 Iteration: 2

Question

Reach a sharp, program-by-program verdict on what each entropic / emergent-gravity proposal actually establishes versus what it assumes, and decide which is a genuine result, which is a structural rewriting, which is analogy, and what would convert any of them into testable physics. The four programs (bundled in ../UNIFYING_PRINCIPLES.md §6 and ../UNIFICATION_LANDSCAPE.md §7) are:

Sakharov induced gravity (1967) — the Einstein–Hilbert term as a residual effect of matter loops.
Jacobson 1995 — the Einstein equation as a local horizon equation of state.
Padmanabhan — equipartition and holographic equipartition of spacetime degrees of freedom.
Verlinde — the 2011 entropic force and the 2016 emergent-gravity / apparent-dark-matter program.

We also assess the major objections: the Kobakhidze coherence/reversibility argument from neutron interferometry (COW) and ultracold-neutron bound states (GRANIT/qBounce), and the observational tests of Verlinde's 2016 emergent gravity against galaxy–galaxy lensing and the radial-acceleration relation.

The central methodological move, per ../EPISTEMICS.md, is to refuse the slogan "gravity is thermodynamics" and ask of each program: what is derived, what is assumed, and is the assumption load-bearing in a way that relocates rather than dissolves the mystery?

State of the art

1. Sakharov induced gravity (1967) — a mechanism, not a derivation of GR's form

ESTABLISHED, as a mechanism Integrating out quantum matter fields on a curved background generates, in the one-loop effective action, terms $\propto \Lambda_{\rm cut}^4 \sqrt{-g}$ (a cosmological constant), $\propto \Lambda_{\rm cut}^2 \int R\sqrt{-g}$ (an induced Einstein–Hilbert term), plus higher-curvature pieces. Thus $G^{-1} \sim \Lambda_{\rm cut}^2$ is generated by vacuum fluctuations: gravity's stiffness can be a residual effect of matter loops, with no fundamental graviton kinetic term postulated. [A. D. Sakharov, Dokl. Akad. Nauk SSSR 177, 70 (1967) = Sov. Phys. Dokl. 12, 1040.]

What it establishes: the EH action is natural as an induced low-energy term — gravity need not be inserted by hand. INFERENCE What it does NOT establish: it presupposes a metric, a curved-space QFT, and a cutoff; it does not derive Lorentzian geometry, diffeomorphism invariance, or the equation of motion — only that, given a metric, an EH term appears. The induced $G$ is cutoff-sensitive (the cosmological-constant problem in its most acute form). ESTABLISHED Verdict: a genuine field-theoretic effect, the conceptual ancestor of all "gravity is emergent" claims, but it relocates rather than removes — it assumes the geometric arena.

Note for the registry (per referee NI1): Sakharov is a counterexample to any universal statement that "every emergent-gravity derivation takes the area law as input." Here the gravitational action — and hence the horizon area entropy — is an output of integrating out matter, not a premise. ESTABLISHED within that framework

2. Jacobson 1995 — the strongest result, with the sharpest reading of what it costs

ESTABLISHED, theorem-level given its premises Thermodynamics of Spacetime: The Einstein Equation of State, PRL 75, 1260 (1995); gr-qc/9504004. Through each spacetime point $p$ and each null direction, construct the local Rindler horizon — the past horizon of a uniformly accelerated observer. Impose the Clausius relation $\delta Q = T\,dS$ on every such horizon, with

$S = \eta A$ (entropy proportional to horizon cross-sectional area, $\eta = 1/4G\hbar$ fixed to match Bekenstein–Hawking),
$T = \dfrac{\hbar a}{2\pi c k_B}$ the Unruh temperature of the accelerated frame,
$\delta Q = \int T_{\mu\nu}\,\chi^\mu\, d\Sigma^\nu$ the boost-energy flux of matter across the horizon ( $\chi^\mu$ the approximate boost Killing vector).

The horizon area change is computed from the Raychaudhuri equation for the null congruence, $\frac{d\theta}{d\lambda} = -\tfrac{1}{2}\theta^2 - \sigma_{\mu\nu}\sigma^{\mu\nu} - R_{\mu\nu}k^\mu k^\nu,$ and demanding $\delta Q = T\,dS$ for all such horizons forces $R_{\mu\nu} - \tfrac{1}{2} R\, g_{\mu\nu} + \Lambda\, g_{\mu\nu} = \frac{8\pi G}{c^4}\, T_{\mu\nu},$ the full nonlinear Einstein equations, with $\Lambda$ an undetermined integration constant.

This is a real, reproducible result. The honest accounting of inputs (the reason it is "relocation, not closure"):

The area–entropy law $S \propto A$ is assumed, not derived. This is the single most load-bearing input; it imports Bekenstein–Hawking, whose microscopic origin is exactly the open problem. ESTABLISHED that it is assumed
The Unruh temperature is assumed, hence QFT-in-curved-spacetime and a thermal vacuum response, presupposing quantum fields on a metric. ESTABLISHED
The "equation of state, do not quantize it" reading (Jacobson's own): if $G_{\mu\nu} = 8\pi G\, T_{\mu\nu}$ is to gravity as $PV = N k_B T$ is to a gas, then quantizing the metric to find "quantum gravity" would be a category error, like quantizing sound waves and calling phonons fundamental. This is a suggestive interpretive stance, NOT a theorem; nothing in the derivation forbids a fundamental metric whose long-wavelength behavior happens to satisfy a Clausius relation. INFERENCE/CONTESTED
Local equilibrium is assumed. Eling, Guedens & Jacobson (Nonequilibrium Thermodynamics of Spacetime, PRL 96, 121301, 2006) showed that for $f(R)$ and higher-curvature gravity the equilibrium Clausius relation fails: one must add an internal entropy-production term $dS_i$ and treat spacetime as a non-equilibrium thermodynamic system, with the Wald entropy replacing the area. ESTABLISHED This both extends the program and exposes that the equilibrium assumption was doing hidden work; the derivation is clean only for the EH term. (Whether this non-equilibrium term is a sign that the thermodynamic reading is fundamental or an artifact that only works cleanly for EH is itself an OPEN interpretive question.)

The modern refinement — Jacobson 2016, Entanglement Equilibrium (PRL 116, 201101; arXiv:1505.04753) — replaces thermodynamic entropy with entanglement entropy: the hypothesis that in a small geodesic ball the vacuum entanglement entropy is maximal at fixed volume yields the semiclassical Einstein equation iff the geometry varies to compensate matter-entropy changes. INFERENCE This is conceptually deeper (it ties §6 to §7 "spacetime from entanglement") and removes reliance on a literal horizon temperature, but it inputs an area-proportional vacuum entanglement entropy whose coefficient is precisely the UV-divergent, cutoff-dependent quantity that is the unsolved problem; for non-conformal fields it rests on a conjecture about the entropy variation. The closely related AdS/CFT entanglement-first-law route (Faulkner et al.; $\delta S = \delta\langle K\rangle$ ) recovers only the linearized Einstein equations, and there the area = entropy (Ryu–Takayanagi) relation is itself derived from the duality rather than postulated. ESTABLISHED, within AdS/CFT

Net verdict on Jacobson. A genuine structural result — the most rigorous in the program — establishing that the Einstein equations are equivalent to a local horizon Clausius relation given (i) the area-entropy law and (ii) the Unruh temperature. It is a derivation-given-premises, not an unconditioned biconditional. It identifies no microscopic degrees of freedom, makes no prediction distinguishing it from ordinary GR, and the "don't quantize it" moral is an interpretation. It reframes the mystery (why $S \propto A$ ?) rather than closing it — consistent with the non-renormalizability-as-emergence reading of ../UNIFYING_PRINCIPLES.md §3/§9: a non-renormalizable EFT is exactly what one expects from a collective/hydrodynamic variable.

3. Padmanabhan — equipartition and holographic equipartition

INFERENCE, structural Padmanabhan's program (Surface Density of Spacetime Degrees of Freedom from Equipartition Law in theories of Gravity, arXiv:1003.5665, 2010; and Rep. Prog. Phys. reviews) rewrites gravitational dynamics thermodynamically. Two signature results:

Equipartition of horizon DOF: $E = \tfrac{1}{2}\int_{\partial V} \dfrac{k_B T_{\rm loc}\, dn}{1}$ with surface DOF density $dn = dA/\ell_P^2$ and local Davies–Unruh temperature; this reproduces the correct mass/energy and, generalized with the Wald entropy, holds for all Lanczos–Lovelock theories.
Holographic equipartition: for static spacetimes $N_{\rm surface} = N_{\rm bulk}$ (surface and bulk DOF counts balance); cosmic expansion is driven by the departure from balance, $\dot V \propto (N_{\rm bulk} - N_{\rm surface})$ , giving a thermodynamic route to the Friedmann equation.

What it establishes: that GR (and Lovelock gravity) can be rewritten in an equipartition / DOF-counting language with remarkable internal consistency — strong evidence that the thermodynamic structure is not coincidental. INFERENCE What it does NOT establish: any microscopic theory; the "degrees of freedom" are defined by the area law, so the counting is a consistent re-encoding, not an independent derivation. Verdict: the most systematic demonstration that gravitational dynamics carries thermodynamic structure across theories — but a structural rewriting that is predictively identical to GR.

Caveat (per referee NI3): the holographic-equipartition / emergent-cosmic-space cosmology does advance distinguishing-flavored but contested, unconfirmed claims (e.g. relating $\Lambda$ and the primordial perturbation amplitude to a single parameter). So "no distinguishing prediction" is true for the bare GR-reproducing core but should be softened to "no agreed, confirmed distinguishing prediction" for the Padmanabhan branch as a whole. CONTESTED

4. Verlinde 2011 entropic force — heuristic, widely judged flawed

SPECULATIVE/CONTESTED On the Origin of Gravity and the Laws of Newton, JHEP 04 (2011) 029; arXiv:1001.0785. A test mass $m$ at distance $\Delta x$ from a holographic screen changes the screen entropy by $\Delta S = 2\pi k_B \frac{mc}{\hbar}\,\Delta x \qquad (\text{one Compton wavelength} \leftrightarrow \text{one bit}).$ With screen temperature from equipartition $E = \tfrac{1}{2} N k_B T$ , bit count $N = A c^3/(G\hbar)$ , and $E = Mc^2$ , the entropic force $F = T\,\Delta S/\Delta x$ reproduces $F = GMm/r^2$ . Newton's law (and, with a relativistic screen, the Einstein equations) drops out.

The objections are serious and largely decisive against the literal reading:

Circularity / reverse-engineering. The screen bit count $N \propto A/\ell_P^2$ and the equipartition energy already encode Bekenstein–Hawking and the Unruh relation; Visser (Conservative entropic forces, arXiv:1108.5240, JHEP) shows the construction requires a pre-existing temperature field and Newtonian potential to define the screens, so it presupposes what it derives. CONTESTED
The Kobakhidze coherence objection. Kobakhidze (Gravity is not an entropic force, PRD 83, 021502, 2011, arXiv:1009.5414; and arXiv:1108.4161) argues that a genuinely entropic force is irreversible and thermal, so it should decohere a particle's center-of-mass wavefunction. Yet ultracold neutrons in Earth's field form discrete quantum gravitational bound states (GRANIT/qBounce; Nesvizhevsky et al., Nature 415, 297, 2002) and the COW neutron-interferometry phase shift is observed with full coherence. Kobakhidze's actual argument is a conceptual reversibility/coherence inconsistency claim — he does not compute a decoherence rate. CONTESTED
The Chaichian–Oksanen–Tureanu loophole (On gravity as an entropic force, PLB 702, 162, 2011, arXiv:1104.4650; and arXiv:1109.2794). They locate a misidentification in Kobakhidze's argument: the entropy of the holographic screen is not the entropy of the particle, so screen-entropy changes need not decohere the particle in the emergent space; and weak (Earth-bound) fields are exactly the regime where the entropic picture is least pinned down. Net: Kobakhidze identifies a genuine conceptual tension but not a clean experimental refutation — it is a domain-of-validity / interpretation dispute. CONTESTED

A crucial update (per referee NI2): the decoherence number Kobakhidze's argument invokes has since been computed from a specified open-system model. Schimmoller, McCaul, Abele & Bondar, Decoherence-free entropic gravity: Model and experimental tests, Phys. Rev. Research 3, 033065 (2021), arXiv:2012.10626, derive a Lindblad master equation with a single dimensionless coupling $\sigma$ , compute the induced center-of-mass decoherence and energy increase, and bound it with qBounce ultracold-neutron data ( $\sigma \gtrsim 250$ at 90% CL), comparing to Diósi–Penrose. Sung, Campos, Abele & Bondar (arXiv:2307.00170, 2023) extend this to Dirac fermions. ESTABLISHED Critically, in the strong-coupling ( $\sigma \to \infty$ ) limit the model recovers ordinary conservative Newtonian gravity with no anomalous decoherence — "decoherence-free entropic gravity." This refutes the framing "this number has never been computed" and undercuts the assumption that a generic entropic screen must predict observably large decoherence. The genuinely open residue is narrower (see Open subquestions).

Net verdict on Verlinde 2011: an evocative analogy / heuristic, not a derivation; it relocates the mystery into the unexplained screen-entropy postulate and is widely regarded as circular. The Kobakhidze objection sharpens the unease without delivering a kill; the existing decoherence-free model shows entropic gravity is not forced to be decoherent.

5. Verlinde 2016 emergent gravity / apparent dark matter — testable, with a corrected reading of the lensing data

CONTESTED → disfavored overall Emergent Gravity and the Dark Universe, arXiv:1611.02269 (2016). The new ingredient: in de Sitter space the cosmological horizon carries a volume-law entropy (an "elastic medium" of long-range entanglement) absent in AdS. Baryonic matter displaces this dark-energy entropy, and the medium's elastic memory response produces an extra "apparent dark matter" acceleration. The headline relation: $g_D^2 = \frac{a_0}{6}\, g_B \quad (a_0 = c H_0), \qquad M_D(r) = \sqrt{\frac{c H_0\, r^2\, M_B(r)}{6 G}},$ recovering a MOND-like radial-acceleration relation with $a_0 \sim c H_0$ from first principles rather than as a fitted constant. (The "parameter-free" framing is the program's own; $a_0$ is fixed to $\sim cH_0$ by an ansatz, so the label warrants CONTESTED.)

The observational record (this is the one falsifiable piece):

Brouwer et al. 2017 (MNRAS 466, 2547; arXiv:1612.03034): galaxy–galaxy weak lensing of 33,613 isolated KiDS/GAMA centrals. EG's parameter-free prediction matched the lensing profiles about as well as an NFW/MOND fit — an early apparent success. ESTABLISHED
Brouwer et al. 2021 (A&A 650, A113; arXiv:2106.11677, KiDS-1000 weak-lensing radial-acceleration relation). CORRECTION to the iteration-1 framing (per the track-verdict referee, who flagged a factual inversion): the paper's headline result is that the measured RAR — extended ~2 decades in $g_{\rm bar}$ into the low-acceleration regime — agrees well with both MOND and Verlinde EG. It does not disfavor EG on its primary measurement. Separately, the paper finds a $\geq 6\sigma$ difference between the RARs of early- vs late-type galaxies at the same stellar mass — a galaxy-type dependence that no parameter-free, baryon-only modification (EG and MOND alike) can accommodate, while it is consistent with $\Lambda$ CDM via formation-history-dependent halo/circumgalactic-gas content. The paper presents this as a class-level challenge to baryon-only modified gravity (and a mixed test for $\Lambda$ CDM: MICE simulations reproduce it, BAHAMAS does not), left to future surveys — not as a clean Verlinde-specific kill. ESTABLISHED, verified from abstract
Cluster / cosmological scale: EG fails the Bullet Cluster mass–gas offset, the CMB acoustic-peak structure, and structure formation, all of which strongly favor particle dark matter (see ../domains/cosmology.md). EG also currently exists only for spherically symmetric, static mass distributions — no full covariant / dynamical / cosmological theory — so it cannot even be tested on the CMB on its own terms. ESTABLISHED Contrast relativistic MOND / the Skordis–Złośnik AeST model (PRL 127, 161302, 2021), which does fit the CMB acoustic peaks — a covariant completeness EG lacks.
Dwarf spheroidals (Diez-Tejedor, Gonzalez-Morales & Profumo, MNRAS 477, 1285, 2018): EG fares poorly, comparably to MOND's known dwarf-scale problems. ESTABLISHED

Net verdict on Verlinde 2016: the one part of the program that made a sharp, falsifiable prediction. Its weak-lensing tests are not clean falsifications — on their headline measurements they favored EG (2017) and agreed with EG (2021). EG is nonetheless CONTESTED → disfavored OVERALL on independent grounds: cluster-scale lensing/gas mismatches (Bullet Cluster), the absence of any covariant/CMB-capable dynamical formulation, structure formation, and dwarf-scale failures. Testable — and the cosmological-scale tests are going against it, even as the galaxy-lensing tests are not.

6. The recent revival — concrete toy substrates, still no independent evidence

SPECULATIVE Recent work (the "2025 LBNL qubit" attribution to Carney, Karydas, Scharnhorst, Singh & Taylor, reported in Quanta, is unverified per the referee) builds explicit microscopic models — a "crystalline qubit" model (oriented qubits aligning near mass, lowering entropy) and a "non-local qubit" model — that reproduce Newtonian attraction from entropy extremization. Significance: the first attempts to name candidate missing microscopic DOF rather than postulate screens. Limitations (acknowledged by the authors): no independent evidence for the qubits; the force must be fine-tuned; and the models reproduce only weak-field gravity — "precisely what we already understand" (Brustein); Van Raamsdonk: such constructions "don't really have anything to do with gravity." This confirms the structural verdict: absent identified, independently-evidenced microscopic DOF, naming them does not convert analogy into physics.

Synthesis: a tiered verdict

Program	What it genuinely establishes	Load-bearing assumption	Status
Sakharov (1967)	EH term induced by matter loops	metric, curved-space QFT, cutoff	ESTABLISHED mechanism; relocates
Jacobson (1995)	Einstein eqns $\Leftrightarrow$ local Clausius relation on Rindler horizons (nonlinear), given $S\propto A$ , Unruh $T$	area-entropy, Unruh $T$ , local equilibrium	ESTABLISHED result; "don't quantize" is INFERENCE/CONTESTED
Jacobson (2016)	semiclassical Einstein eqn $\Leftrightarrow$ stationary vacuum entanglement	maximal-entanglement hypothesis, UV cutoff	INFERENCE; ties to §7
Padmanabhan	GR/Lovelock = equipartition + holographic DOF balance	area-entropy defines the DOF	INFERENCE structural rewriting
Verlinde (2011)	(heuristic) Newton from screen entropy	screen-entropy postulate (circular)	SPECULATIVE/CONTESTED analogy
Verlinde (2016)	RAR with $a_0\sim cH_0$	dS volume-law entropy + elastic medium	CONTESTED → disfavored but testable

Key arguments

Jacobson 1995 is a genuine theorem-level result, not an analogy. ESTABLISHED Given $S = A/4G\hbar$ and the Unruh $T$ , imposing $\delta Q = T\,dS$ on all local Rindler horizons, combined with the Raychaudhuri equation, forces the full nonlinear Einstein equations. The logic is forward and reproducible; it is an equivalence between a local horizon thermodynamic identity and the field equations.
But Jacobson derives the FORM of GR given thermodynamic inputs — it relocates rather than closes the mystery. INFERENCE The area-entropy law and Unruh temperature are inputs. The microscopic origin of $S \propto A$ is precisely the open question; assuming it and recovering GR shows consistency, not microscopic explanation. The Eling–Guedens–Jacobson 2006 result that $f(R)$ needs a non-equilibrium entropy-production term shows the equilibrium assumption was load-bearing and only the EH term comes out cleanly.
The "equation of state, do not quantize it" reading is an interpretation, not a theorem. CONTESTED That GR is an equation of state is suggestive (one does not quantize Navier–Stokes to find molecules), and it elegantly rationalizes non-renormalizability (§3/§9). But nothing in the derivation forbids a fundamental metric whose long-wavelength dynamics happens to obey a Clausius relation; the inference "therefore gravity must not be quantized" does not follow with theorem-level force. (Note: Weinberg–Witten constrains only emergent same-spacetime gravitons, not the equation-of-state interpretation.)
Verlinde 2011 is circular / heuristic, not a derivation. CONTESTED The screen bit count $N = Ac^3/G\hbar$ and equipartition energy already encode Bekenstein–Hawking and Unruh; Visser (arXiv:1108.5240) shows the screens require a pre-existing temperature field and Newtonian potential to be defined. The construction reverse-engineers the answer it seeks.
Kobakhidze's neutron argument is a real conceptual tension but not a clean experimental refutation. CONTESTED An irreversible thermal entropic force should decohere a center-of-mass wavefunction, yet GRANIT discrete bound states and COW interferometry show full coherence in a conservative potential. However, Chaichian–Oksanen–Tureanu show screen entropy $\neq$ particle entropy, and weak fields are exactly where the screens are underspecified. Moreover, the Schimmoller et al. 2021 master-equation model exhibits a decoherence-free (strong-coupling) limit, so entropic gravity is not forced to decohere. It is a domain/interpretation dispute, not a derived contradiction with data.
Verlinde 2016 is the only falsifiable piece — but the lensing data do not falsify it; the cosmological-scale data do. ESTABLISHED, corrected Its parameter-free RAR matched 2017 KiDS/GAMA galaxy-galaxy lensing and agreed with the 2021 KiDS-1000 RAR (headline result). The $\geq 6\sigma$ early/late-type dependence challenges both EG/MOND and (in some simulations) $\Lambda$ CDM, left to future surveys. EG is disfavored OVERALL by clusters (Bullet), the CMB acoustic peaks, structure formation, and its lack of any covariant dynamical formulation — NOT by Brouwer 2021, which on its primary result cuts the other way.
Naming microscopic DOF does not yet convert analogy into physics. INFERENCE The recent qubit models reproduce only fine-tuned weak-field Newtonian gravity, with no independent evidence for the qubits; critics note this reproduces only what is already understood. The missing ingredient across all programs is an independently-evidenced microscopic substrate.

Sharpest obstructions

The area-entropy law $S \propto A$ is an INPUT to (almost) every program, and its microscopic origin is exactly the unsolved problem. Jacobson, Padmanabhan, and Verlinde all assume Bekenstein–Hawking proportionality (directly, via screen bit-counts, or via entanglement entropy). Deriving GR from it is a consistency statement; it cannot explain the area law because it presupposes it. No program identifies microscopic DOF whose count is $A/4\ell_P^2$ for a generic (non-extremal, non-BPS, asymptotically flat) horizon. (Exceptions where the count is derived — Strominger–Vafa for BPS black holes, Ryu–Takayanagi in AdS — cover only special cases; Sakharov instead outputs the area law.) ESTABLISHED
No new prediction distinguishes the derivation-versions from ordinary GR. By construction Jacobson/Padmanabhan reproduce the Einstein/Lovelock equations exactly, so there is no experimental discriminator. "Gravity is emergent" is, in these versions, empirically equivalent to "gravity is fundamental" — making the foundational claim metaphysical rather than physical, absent a substrate that predicts deviations. INFERENCE
Entanglement entropy is UV-cutoff-dependent and QFT has no tensor factorization of local regions (Type III von Neumann algebras). The Jacobson-2016 entanglement route and the it-from-qubit framing both presuppose a well-defined finite entanglement entropy and a subsystem/qubit structure that algebraic QFT says does not strictly exist for local regions. This is the same Type-III obstruction flagged in §7/§8 and undercuts the literal "qubits sewing spacetime" picture. ESTABLISHED
The coherence/reversibility tension is sharpened but not closed at the level of a derived rate. An entropic force is generically irreversible/dissipative, yet gravity is observed to act as a conservative, coherence-preserving potential (COW, GRANIT). A phenomenological master equation now bounds the implied decoherence (qBounce: $\sigma \gtrsim 250$ ) and admits a decoherence-free limit — but that model is an open-system ansatz, not derived from Verlinde's specific screen entropy. So the conceptual objection is reframed, not settled. CONTESTED
Verlinde EG lacks a covariant, dynamical formulation, and the galaxy-scale data show a galaxy-type dependence no baryon-only modification can accommodate. EG exists only for static spherically symmetric mass models, so it cannot be tested against the CMB or structure formation on its own terms; and the $6\sigma$ early/late-type RAR difference (Brouwer 2021) is incompatible with any gravity modification that depends only on baryonic acceleration. ESTABLISHED
The de Sitter setting (our universe) is exactly where holography / area-laws are least understood. Verlinde 2016 relies on a de Sitter volume-law entropy with no controlled holographic dual (no established dS/CFT), and the entanglement-equilibrium and AdS-derived intuitions hold only for AdS — the wrong cosmological-constant sign, the obstruction shared with §4/§7. ESTABLISHED

What would count as decisive progress

Identify the microscopic degrees of freedom independently: a model whose DOF count reproduces $S = A/4\ell_P^2$ for generic (non-extremal, non-supersymmetric, asymptotically flat) horizons AND predicts an independently-measurable quantity (a deviation from GR, a specific Lorentz-violation pattern, or a decoherence rate), not merely re-deriving the area law it assumes.
A genuinely new falsifiable prediction from a derivation-version (Jacobson/Padmanabhan) — e.g. a calculable higher-curvature correction with a fixed coefficient, or a specific quantum-gravity-induced correction to the Clausius relation — distinguishing "emergent GR" from "fundamental GR."
A first-principles (non-ansatz) derivation of the entropic-gravity center-of-mass decoherence rate from Verlinde's holographic-screen entropy and equipartition postulates (rather than the existing phenomenological open-system model), with the strong-coupling/decoherence-free regime shown to be natural rather than fine-tuned, confronted with COW / GRANIT-qBounce / atom-interferometer gravimetry beyond the existing $\sigma \gtrsim 250$ bound.
A covariant, dynamical formulation of Verlinde 2016 EG (beyond static spherical symmetry) confrontable with the CMB acoustic peaks, the Bullet Cluster offset, and structure formation — as relativistic MOND / AeST already are.
A de Sitter analog of the entanglement-equilibrium / area-law derivation that does not borrow AdS-specific structure, addressing the wrong- $\Lambda$ -sign obstruction common to §4/§6/§7.
A resolution of the Type-III / no-tensor-factorization problem giving a rigorous, cutoff-independent meaning to the entanglement entropy used in the Jacobson-2016 route (e.g. via recent crossed-product / Type-II algebra constructions in gravity).

Proposed registry items

Each item below carries the referee verdict (keep / severity / refined statement).

Referee verdict: keep — severity minor. Recommended tag: INFERENCE. Refined statement (incorporating referee NI0 corrections): Split the bundled "entropic/emergent gravity" entry (§6 / §7) into three distinctly tagged claims:

(a) ESTABLISHED GR carries genuine thermodynamic structure. Jacobson (1995) derives the Einstein field equations as a local equation of state from $\delta Q = T\,dS$ on all local Rindler horizons, given the area-entropy law and Unruh temperature as inputs (a derivation-given-premises, not an unconditioned biconditional; the AdS entanglement-first-law route yields only the linearized equations). Padmanabhan's equipartition / holographic-DOF reformulation is a related INFERENCE-level structural rewriting, NOT a co-equal theorem (do not bundle it under ESTABLISHED).
(b) SPECULATIVE Gravity is foundationally emergent from identified microscopic DOF — no substrate identified, no new prediction.
(c) CONTESTED → disfavored Verlinde-type entropic gravity as a replacement for galactic/cosmological dark matter. The weak-lensing RAR is fit comparably well by EG and MOND (Brouwer 2017, 2021); a $\geq 6\sigma$ early/late-type RAR dependence defeats any parameter-free baryon-only modification (a class-level challenge), and the decisive disfavoring is at cluster / CMB / Bullet-Cluster scales, not the RAR itself.

ASSUM-AREA-ENTROPY-INPUT (assumption)

Referee verdict: keep — severity minor. Recommended tag: INFERENCE (NOT ESTABLISHED as originally proposed — it is a sound meta-characterization, not established physics). Refined statement (incorporating referee NI1): The thermodynamic and entropic-force derivations in the Jacobson–Padmanabhan–Verlinde lineage, together with Jacobson's entanglement-equilibrium route, take the horizon area-entropy law $S = A/4\ell_P^2$ — or an area-proportional vacuum entanglement entropy / screen bit-count — as an INPUT ESTABLISHED for these specific programs. Its microscopic origin (the generic-horizon microstate count) is an OPEN problem they presuppose rather than solve; recovering the Einstein equations from it is a consistency / equation-of-state result, not a microscopic explanation. Caveats: (a) NOT universal — Sakharov-style induced gravity instead derives the gravitational action (and area entropy) from integrating out matter, so there the area law is an output; (b) in AdS/CFT the area = entropy (Ryu–Takayanagi) relation is derivable from the duality, not postulated; (c) area-entropy is the most prominent shared input but not the sole one — Jacobson also assumes the Unruh temperature and the Clausius relation on all local horizons, and Verlinde additionally assumes holographic-screen equipartition.

OPEN-ENTROPIC-DECOHERENCE-TEST (open-problem)

Referee verdict: keep — severity major. Recommended tag: CONTESTED (the original "open" framing rested on a false premise). Refined statement (incorporating referee NI2): The original framing — "the entropic-gravity decoherence number has never been computed from a specified screen model" — is incorrect. Schimmoller, McCaul, Abele & Bondar (Phys. Rev. Research 3, 033065, 2021; arXiv:2012.10626) derived an open-system master equation, computed the center-of-mass decoherence, and bounded it with qBounce data ( $\sigma \gtrsim 250$ at 90% CL); Sung et al. (arXiv:2307.00170, 2023) extended it to Dirac fermions, and the model has a decoherence-free strong-coupling limit. The actually-open problem is narrower: (i) can such a master equation be derived from Verlinde's holographic-screen entropy / equipartition postulates rather than from a phenomenological ansatz; (ii) is the required strong-coupling (decoherence-free) regime natural rather than fine-tuned; and (iii) does the resulting emergent-but-effectively-unitary dynamics genuinely answer Kobakhidze's reversibility objection rather than merely fit COW / GRANIT-qBounce / atom-gravimeter data?

Referee verdict: keep — severity minor. Recommended tag: INFERENCE (meta-claim), resting on ESTABLISHED facts. Refined statement (incorporating referee NI3): For Jacobson's equilibrium equation-of-state derivation ( $\delta Q = T\,dS$ on local Rindler horizons, $S \propto$ area), "gravity is emergent" and "gravity is fundamental" are currently under-determined rather than empirically adjudicable: the derivation reproduces the classical Einstein equations with no agreed, confirmed prediction distinct from standard GR. This is a conceptual-tension / domain-of-validity question (which description is deeper), NOT a logical contradiction and NOT (yet) experimentally settled. Two load-bearing qualifications: (a) higher-curvature/Lovelock recovery is not on the same footing — it needs a non-area (Wald) entropy plus a non-equilibrium entropy-production term (Eling–Guedens–Jacobson 2006), so the original "Einstein/Lovelock" bundling is corrected to "Einstein equations only" for the clean case; (b) the empirical equivalence holds only within the GR-reproducing regime — any concrete microscopic substrate generically predicts in-principle-testable deviations (Lorentz/dispersion violation, residual dissipation, finite- $N$ corrections), and Padmanabhan's emergent-cosmic-space cosmology already advances contested/unconfirmed distinguishing claims, so "no distinguishing prediction" softens to "no agreed, confirmed distinguishing prediction." Parallels the geometry-vs-information tension already catalogued in ../GAPS_AND_CONTRADICTIONS.md.

Verdict

INFERENCE overall, a well-supported synthesis; dark-matter sub-verdict at CONTESTED → disfavored

Disaggregate the program.

Jacobson 1995 is a GENUINE RESULT ESTABLISHED: imposing $\delta Q = T\,dS$ on all local Rindler horizons, with $S \propto A$ and the Unruh temperature, forces the full nonlinear Einstein equations — a real equivalence. But it derives the form of GR given thermodynamic inputs (the area-entropy law and Unruh $T$ ), identifies no microscopic substrate, and makes no new prediction; the "equation of state, do not quantize it" moral is a suggestive interpretation INFERENCE/CONTESTED, not a theorem, and the Eling–Guedens–Jacobson 2006 non-equilibrium extension shows the clean derivation works only for the Einstein–Hilbert term. It relocates the mystery (why $S \propto A$ ?) rather than closing it — consistent with the non-renormalizability-as-emergence reading.
Padmanabhan's equipartition / holographic-equipartition program is a CONSISTENT STRUCTURAL REWRITING INFERENCE of GR/Lovelock dynamics, strong evidence the thermodynamic structure is not accidental, but predictively identical to GR in its core and defining its DOF by the area law it assumes (its cosmological branch makes only contested, unconfirmed distinguishing claims).
Sakharov induced gravity is an ESTABLISHED field-theoretic MECHANISM (EH term from matter loops) that still presupposes the geometric arena — and notably outputs rather than inputs the area law.
Verlinde 2011 is an ANALOGY / HEURISTIC SPECULATIVE/CONTESTED: the entropic-force derivation is widely judged circular. The Kobakhidze neutron-coherence objection (COW + GRANIT) is a real conceptual tension but NOT a clean refutation — Chaichian–Oksanen–Tureanu correctly note screen entropy $\neq$ particle entropy and weak-field underspecification; and the Schimmoller et al. 2021 master equation now bounds the decoherence ( $\sigma \gtrsim 250$ ) and admits a decoherence-free limit, so entropic gravity is not forced to decohere. It is a domain/interpretation dispute, not a derived contradiction with data.
Verlinde 2016 is the ONE FALSIFIABLE piece. Its parameter-free relation ( $a_0 \sim cH_0$ ) scored an early galaxy-galaxy weak-lensing success (Brouwer 2017) and agrees with the 2021 KiDS-1000 RAR on its headline measurement — the weak-lensing tests are NOT clean falsifications (an iteration-1 inversion, corrected here). EG is nonetheless CONTESTED → disfavored OVERALL on independent grounds: cluster-scale lensing/gas mismatches (Bullet Cluster), no covariant / CMB-capable dynamical formulation (contrast Skordis–Złośnik AeST 2021), structure formation, and dwarf-scale failures. The $\geq 6\sigma$ early/late-type RAR dependence is a class-level problem for all baryon-only modified gravity (and a mixed test for $\Lambda$ CDM). The recent qubit revival names candidate microscopic DOF but reproduces only fine-tuned weak-field gravity with no independent evidence [the "2025 LBNL" label is unverified].

NET: exactly one genuine theorem (Jacobson's equivalence, given its premises) plus a structural rewriting (Padmanabhan) and a mechanism (Sakharov); the foundational-emergence claim is SPECULATIVE and the dark-matter claim is CONTESTED → disfavored. All of them relocate the mystery onto the unexplained area / entanglement-entropy law, and none identifies the microscopic substrate. The tally survives the referee's correction intact. What would convert any into physics: identify the microscopic DOF independently (reproduce $S = A/4$ AND predict a deviation); derive the entropic-gravity decoherence rate from a specified screen model to test Kobakhidze quantitatively; and give Verlinde EG a covariant dynamical form testable on the CMB.

Open subquestions

OPEN What microscopic degrees of freedom have count $A/4\ell_P^2$ for a generic (non-extremal, non-supersymmetric, asymptotically flat) horizon, and can any candidate predict an independently measurable deviation from GR rather than re-deriving the area law?
OPEN Can a center-of-mass decoherence rate be derived from Verlinde's specific screen postulates (not a phenomenological ansatz), is its decoherence-free regime natural, and does it answer Kobakhidze's reversibility objection — or merely fit COW / GRANIT-qBounce / atom-interferometer bounds?
OPEN Does the Jacobson-2016 entanglement-equilibrium derivation survive a rigorous, cutoff-independent definition of local entanglement entropy given the Type-III von Neumann algebra structure of QFT (e.g. via crossed-product / Type-II constructions)?
OPEN Is there a de Sitter analog of the area-law / entanglement derivations that does not import AdS-specific structure, addressing the wrong- $\Lambda$ -sign obstruction shared with §4 and §7?
OPEN Can Verlinde 2016 EG be promoted to a covariant, dynamical theory (beyond static spherical symmetry) that confronts the CMB acoustic peaks and the Bullet Cluster — and does it then survive?
OPEN Is the non-equilibrium entropy-production term required for $f(R)$ gravity (Eling–Guedens–Jacobson) a sign that the thermodynamic derivation is fundamental, or an artifact that only works cleanly for the Einstein–Hilbert term?