Putting a Number on What You Haven't Proven

Modular-Identity Obstruction — Quantum-Geometric Correspondence series

An Obstruction Worth Taking Seriously

There is a relation at the centre of this programme:

the modular Hamiltonian of an observer's algebra equals $2\pi$ times the physical Hamiltonian. For a free field in a Rindler wedge it is a theorem — Bisognano–Wichmann. For gravity it underlies the thermal-time hypothesis, the crossed-product construction of subregion algebras, and the first-order-in-$G$ decoherence rate $\Gamma_{\mathrm{dec}} = E_G/\hbar$ that the whole programme stakes itself on.

For gravity it is not proven. And the reason it is not proven is specific, structural, and easy to state.

A mass in superposition drags its gravitational field along. That field is the Newtonian potential, which falls off as $1/r$ and is therefore supported outside every bounded region you might draw. There is no box you can put the experiment in that contains its own gravitational dressing. Worse, the flat wedge algebra is Type III$_1$ — a von Neumann algebra with no minimal projections and, crucially, no interior factor against which that exterior tail could be localised. The mathematical structure that would let you say "the part outside is small" does not exist for the wedge.

That is a real obstruction, not a technicality. The honest options are to prove it away, to ignore it, or to measure it. This paper takes the third.

Collapsing the Problem to One Functional

The first move is the one that makes everything else possible.

For a coherent — Gaussian — dressing, the displacement conjugation of the modular Hamiltonian is exact. Not approximate to some order; exact. And that exactness has a consequence: the entire deviation $K - 2\pi H_{\mathrm{phys}}$ is carried by a single object.

That object is the boost-weighted energy of the dressing field on the modes lying outside a finite causal diamond.

This is the paper's real contribution, and it is worth pausing on. An open conjecture with a vague obstruction is not something you can make progress against. An open conjecture whose entire failure is concentrated in one named, computable functional is a target. The question changes from "is the identity true?" to "how big is this specific quantity?" — which is a question with a number for an answer.

Three Cancellations

The bound comes in three steps, and each one is a physical statement rather than a technical estimate.

The monopole cancels — because of what decoherence actually measures. Decoherence lives in the off-diagonal, which-path matrix elements of the density matrix. Those elements see only the difference between the two branches' gravitational dressings, never either one alone. Both branches carry the same total mass, so the global $1/r$ monopole — the very tail that caused the obstruction — cancels identically. The thing that could not be localised turns out not to be there.

The dipole cancels — because of momentum conservation. With the monopole gone, the leading surviving term would be a dipole. Momentum conservation removes it.

What is left is a quadrupole, and it is tiny. The exterior weight of a quadrupole falls as $(d/h)^4$, where $h$ is the depth of the source inside the diamond. For a millimetre-scale superposition sitting one metre inside the laboratory light cone, that is

Twelve orders of magnitude of suppression, from geometry alone.

The Piece the Wedge Was Missing

The three cancellations bound the size of the exterior contribution. They do not by themselves supply what the Type III$_1$ obstruction denied: a finite localisation constant.

Here the finite diamond does something the infinite wedge cannot. Unlike the wedge, a finite causal diamond satisfies Buchholz–Wichmann energy nuclearity. Its field modular Hamiltonian is gapped — the gap comes from the curvature of the hyperbolic slice $\mathbb{H}^3$ — and its level density is polynomial rather than exponential. Together those give a finite nuclearity index,

and with it the split property, which supplies exactly the finite localisation constant the wedge lacks.

The geometry does the work. Choosing a finite region rather than an infinite one is not a technical convenience; it is what makes the estimate exist at all.

The Residual, and Where It Starts

One channel survives all of this: a bi-local term coupling the interior and exterior. Its leading part cancels exactly in the Gaussian covariance, so the residual first appears at order $G^2$ — the same order as the perturbative decoherence rate the programme is trying to distinguish itself from, and therefore harmless at first order.

What This Does Not Prove

It would be easy to over-read the preceding sections, so let me be blunt about the ledger.

The identity is not proven here. What is established is a composition estimate on a named functional in the Gaussian sector. Two constants are not computed:

1. the interacting localisation constant, and
2. the coefficient of the $\mathcal{O}(G^2)$ residual.

Those two numbers are not incidental gaps. They are exactly the open content of the conjecture — the whole of it, with nothing else hiding. A conjecture reduced to two named uncomputed constants is in a materially better position than one facing an unquantified structural obstruction, but it is not a theorem, and calling it one would be dishonest.

The Part That Makes This Testable

Here is what closes the loop, and it is the reason this paper matters beyond bookkeeping.

Those two uncomputed constants are the same quantities a laboratory would be measuring when it discriminates between first-order and second-order gravitational decoherence — rates separated by a factor of about $3\times10^{34}$ at the microgram–millimetre benchmark.

So the open theoretical content and the experimental discriminator are not two separate problems that happen to sit near each other. They are the same problem, approached from opposite ends. Either the mathematics gets there first, or the experiment does.

That is an unusual and rather satisfying position for a conjecture to be in. The obstruction was real; making it quantitative turned it from an objection into a measurement.

This is the modular-identity-obstruction paper of the Quantum-Geometric Correspondence series, making the Type III₁ obstruction to K = 2πH_phys quantitative: the deviation is collapsed to a single boost-weighted exterior functional and bounded through monopole cancellation, dipole cancellation, quadrupole suppression, and diamond nuclearity. The full paper carries the kinematics and the numerics.