The core of the Quantum-Geometric Correspondence series
The Euclid move
When a field is stuck, one productive response is to stop building upward from speculative microphysics and ask instead: what is the smallest set of assumptions that could be true? Euclid did this for geometry. Einstein did it for relativity — two postulates, then the rest. The Quantum-Geometric Correspondence tries the same move at the quantum–gravity interface.
The current formulation rests on three primitive axioms. Everything else in the programme — holographic bounds, observer horizons, much of the cosmology — is treated as following from them.
This post maps those three axioms, the headline prediction they underwrite, and an honest account of what is still open.
Axiom I — Generalized entropy
Take a region of spacetime bounded by a horizon. Impose the gravitational constraints. The claim is that the physical content of that region has an entropy that combines geometry and quantum matter:
Area of the boundary, in Planck units, plus the entropy of the quantum matter, up to a constant.
This is the modern descendant of Bekenstein–Hawking black-hole entropy. Total information is conserved on a complete slice of the universe; the generalized second law governs open regions; the surface itself is selected by extremizing this entropy.
Honest caveat: the area law is an input, not something we derive. We postulate it. Only the sum, its differences, and its growth are treated as physical.
Axiom II — Entanglement equilibrium
Every small causal diamond — a bounded region defined by light rays, large compared with the Planck length but small compared with the curvature of spacetime — sits at entanglement equilibrium: the generalized entropy is stationary at fixed volume,
From that stationarity, together with the first law of entanglement, the Einstein equations follow. Gravity's field equation is an equation of state for entanglement.
What this axiom does not do is invent new quantum dynamics. Quantum states still evolve in the ordinary way. What it fixes is which backgrounds and states are self-consistent together.
Axiom III — Modular time
This is the load-bearing, falsifiable axiom.
For an observer associated with a causal diamond, a canonical "entanglement clock" that quantum theory assigns to any suitable state and region is physical time. At the level of operators:
In words: the modular Hamiltonian equals 2π times the physical energy in the region, up to corrections second-order in Newton's constant.
Why care? Because a mass in spatial superposition has different gravitational energy in each branch. If modular flow is physical time, those branches lose coherence at a rate set by that energy:
with C an order-one number between about 0.64 and 1, and natural value C = 1 — the rate proposed by Diósi and Penrose. For a one-microgram particle separated by one millimetre, that is a decoherence time of about 1.6 nanoseconds.
Newton's constant appears once. Ordinary perturbative quantum field theory puts it in squared — roughly 10³⁴ times slower for the same object, utterly unobservable. The entire experimental programme of the theory lives on this distinction.
Where we stand, stated plainly: the energy scale E_G is classical and solid. The operator equation that turns it into a rate is still a conjecture as a full statement — though it holds in expectation values and solvable models. Observation of the slow, field-theory-scale coherence for a preparable superposition would falsify this axiom and, with it, the theory's distinctive content.
What follows
From these axioms the programme develops, among other things:
- a gravitational decoherence floor for solid bodies that saturates with particle size;
- a shaped suppression of the tabletop entanglement witness meant to prove quantum gravity;
- a cosmic decoherence rate slow enough that the universe is allowed to exist;
- a noise spectrum with a characteristic frequency set by mass and separation;
- and cosmological readings in which dark energy and the arrow of time share one horizon-entropy budget.
We grade the theory honestly against Einstein's standard. We are not there yet. We say so on purpose.
How to read this programme
Three commitments worth making explicit:
- Honesty about status. Proven, derived, and conjectured are different words. A result that would be at home in a paper must earn the right one.
- Experiment decides the distinctive claim. Elegant axioms are cheap; the scaling with Newton's constant is not. Levitated nanoparticles in the coming decade are where the wager lands.
- We do not settle. Premature victory is a failure mode. Critical reviews are welcomed; gaps are named; the central equation stays conjectural until it isn't.
A companion essay on research philosophy and the origin of reality is here. It is explicitly not part of the physics case. The axioms above have to stand without it.
The one-sentence version
Quantum mechanics and general relativity are complementary descriptions of one informational structure; three axioms make that precise; and the first place nature can catch us is a dust grain that refuses — or agrees — to stay in two places at once.