What's Proven, What Isn't, and What an Experiment Would Settle

Status and Outlook — Quantum-Geometric Correspondence series

The Uncomfortable Question

Every research programme eventually has to answer a question it would rather defer: what, exactly, do you have?

Not what you hope to have. Not what the framework suggests. What is proven, what is conjectured, what is merely consistent — and what would have to happen in a laboratory for the whole thing to be wrong.

This is that accounting for the Quantum-Geometric Correspondence. It is worth stating the conclusion up front, because the interesting part is not that the programme survives the audit but where it turns out to be load-bearing. One number does almost all the work, and that number is not proven.

What the Axioms Buy

The programme rests on three primitive axioms — generalized entropy, entanglement equilibrium, and modular time. From them follow, without additional input:

• the Bekenstein–Hawking entropy of a horizon,
• the holographic bound on information in a region,
• the Unruh temperature seen by an accelerating observer,
• the Einstein field equations,
• and a thermodynamic dark-energy density $\rho_{\mathrm{DE}} = \alpha c^2 H^2/G$.

That list looks impressive, and it is worth being precise about why it is not evidence. Every item on it is known physics. Recovering the Einstein equations from thermodynamic axioms is an achievement in the Jacobson tradition — it shows the axioms are not obviously wrong and that they connect to gravity in the right way — but a framework that reproduces what we already know has, so far, made no risky claim. Recovery is a licence to continue, not a result.

The risky claim is elsewhere.

The One New Prediction

A mass $M$ in spatial superposition, its two branches separated by $d$, decoheres through its own gravitational field at a rate

with the coefficient bounded to $C \in [2/\pi,\,1]$. For a microgram over a millimetre this gives a decoherence time of about 1.6 nanoseconds.

The textbook calculation gives something else entirely. Treat gravity as a perturbative quantum field, compute the same decoherence, and the rate comes out proportional to $G^2$ rather than $G$ — smaller by roughly thirty-four orders of magnitude. The same superposition survives for longer than the age of the universe.

There is no interpretive room between these. One is right.

That gap is the programme's whole value. A theory whose distinctive prediction differs from the standard one by a factor of two is a theory you argue about for decades. A theory that differs by $10^{34}$ is a theory an experiment closes.

Where the Load Actually Sits

Take the prediction apart and it has two pieces, with very different standing.

The energy scale $E_G = GM^2/d$ is the gravitational self-energy of the difference between the two branches. It is classical, it is calculable, and it is not in doubt. Diósi and Penrose identified the same scale decades ago. This part is proven.

The conversion of that energy into a rate is the conjecture. It runs through an identity between the modular Hamiltonian of an observer's algebra and the physical Hamiltonian,

which is a theorem for a free field in a wedge — the Bisognano–Wichmann theorem — and is posited for gravity. Given it, the energy scale converts to a rate by the most elementary step in quantum mechanics, energy over $\hbar$. Without it, nothing forces the conversion.

So the honest statement is: the scale is established, the rate is conjectural, and the entire distinctive content of the programme lives in the gap between them.

How Far the Conjecture Has Been Pushed

Not proven is not the same as unexamined. The finite causal-diamond construction now has a complete kinematic and admissibility floor: within a controlled model, the interacting dressed state sits some thirty-three orders of magnitude inside the wall where the construction would break down, under a canonical one-mode observer clock.

That is a strong result and it is worth being careful about what it means. It means the obvious failure mode does not occur — the construction is not sitting marginally on the edge of validity, hoping. It does not mean the identity is proven. What remains open is its interacting field-theoretic completion, and the rate coefficient $C$ is bounded rather than pinned: $[2/\pi,\,1]$, with $C = 1$ the natural value and $2/\pi$ a floor set by the Margolus–Levitin bound.

A factor of $\pi/2$ of remaining ambiguity, against a $10^{34}$ discriminator, is not the problem. The problem is the identity itself.

The Experiment

The falsification is concrete, and it is close.

A picogram silica sphere, held in superposition for one second in cryogenic extreme-high vacuum, gives a visibility of 0.18 if the programme is right, against ≈ 1 for standard quantum gravity. That separation is decided in roughly thirty interference runs.

Thirty runs. Not thirty years of theory.

The bottleneck is not statistics and not the modulation — it is holding coherence and isolation long enough at that mass scale, which is exactly where levitated optomechanics is currently pushing. The plausible window is this decade.

If the experiment sees coherence surviving to the slow $G^2$ timescale, gravity is just another quantum field, and the programme's distinctive content is wrong. Not weakened — wrong. If it sees decoherence on the fast timescale, the conjecture at the centre has been vindicated by measurement rather than proof, and the modular identity becomes something to explain rather than something to hope for.

The Cosmological Sector

One further thing is worth reporting, because it is the kind of structure that is hard to arrange by accident.

Three apparently unrelated cosmological quantities turn out to be readings of a single de Sitter frequency: the MOND acceleration scale $a_0 = cH_0/(2\pi)$, the holographic dark-energy density, and a cosmic decoherence rate $\Gamma_{\mathrm{total}} = g(1)H/(4\pi)$. The horizon supplies one clock, and these are three of its hands.

This is suggestive rather than decisive. The MOND scale in particular comes out with no free parameter and lands on the observed value, which is more than most modified-gravity proposals manage. But a numerical coincidence recovered from a framework is weaker evidence than a prediction made before the measurement, and it should be weighted accordingly.

Against the Neighbours

The programme sits in a crowded field, and the comparison matters.

Against spontaneous-collapse models (CSL and relatives): those carry free parameters — a collapse rate, a smearing length — which can be tuned to evade bounds. This programme has no such dial. Its rate is fixed by $G$, $M$, $d$ and an $\mathcal{O}(1)$ coefficient.

Against Diósi–Penrose: the rate is numerically identical, which is a genuine and underappreciated problem. The two frameworks disagree about what is happening — unitary branching versus physical collapse — while agreeing on every number a decoherence experiment reads. Separating them requires going past the rate to the shape of the noise spectrum, which is a harder measurement.

Against stochastic gravity: that approach keeps gravity classical and adds noise; here gravity is quantum throughout and the decoherence is informational.

The Balance

The physics is intact. The falsifiable claim is sharp and near-term. The central identity carries the status of a strong conjecture with no experimental confirmation, and the programme's grade reflects that honestly rather than optimistically.

What makes this worth pursuing is not confidence that it is right. It is that the claim is arranged so that being wrong is cheap to discover. Thirty interference runs, this decade, and the question closes one way or the other.

That is a better position than most quantum-gravity programmes can claim, and it is the only property the framework has that its competitors mostly lack.

This is the status-and-outlook paper of the Quantum-Geometric Correspondence series: a review of what the three axioms recover, what the one new prediction claims, what stands between them, and what an experiment would settle. The full paper carries the falsifiability map, the comparison table, and the open-problems roadmap.