Introduction

The identity \[\begin{equation} K \;=\; 2\pi\,H_{\mathrm{phys}} \label{eq:intro-identity} \end{equation}\] equates the modular Hamiltonian \(K = -\ln\Delta\) of an observer’s local algebra with \(2\pi\) times the physical Hamiltonian the observer’s clock measures. It is the sharp form of the thermal-time hypothesis , the operator content behind the crossed-product construction of gravitational subregion algebras , and the origin of a first-order-in-\(G\) gravitational decoherence rate \(\Gdec = \EG/\hbar\) with \(\EG = GM^2/d\) . For the free field on a Rindler wedge it is a theorem: Bisognano and Wichmann identified the vacuum modular flow with the boost , and the \(2\pi\) is the same universal factor read off by an accelerated detector .

For gravity the identity faces a specific, well-known obstruction. The gravitational dressing of a massive body is its Newtonian field, and the Newtonian field has support outside every bounded region: no strictly local operator creates a mass together with its \(1/r\) tail. A dressed state of a finite causal diamond therefore involves modes the diamond does not contain, and the coherent-state argument that identifies the dressed modular Hamiltonian with \(2\pi H_{\mathrm{phys}}\) applies only to dressings supported inside the region. Worse, the flat wedge algebra is Type III\(_1\): it admits no interior Type I factor against which the exterior tail could be localised and its effect estimated. The obstruction is usually stated in exactly this qualitative form — the dressing is global, the wedge does not split — and left there, as a reason the identity cannot be proven.

This paper makes the obstruction quantitative. The result is that the entire deviation of the dressed modular Hamiltonian from its vacuum form is carried by a single computable functional — the boost-weighted energy of the dressing field on the modes outside the diamond, the leakage functional \(W_{\mathrm{ext}}[\alpha]\) of Definition [def:leakage] — and that in the Gaussian/coherent sector this functional is bounded, \[\begin{equation} W_{\mathrm{ext}}[\alpha_{\mathrm{dressing}}] \;\le\; C\,\Big(\frac{d}{h}\Big)^{4}, \qquad C < \infty, \label{eq:intro-bound} \end{equation}\] for a which-path separation \(d\) at depth \(h\) inside the diamond. Three mechanisms, each exact or hypothesis-checked in the Gaussian sector, produce the bound. First, the displacement conjugation of the modular Hamiltonian is exact and terminates, so the deviation is linear in the mode displacements and splits cleanly into an interior part — which is \(2\pi H_{\mathrm{phys}}\) — and the exterior leakage (Sec. 4). Second, off-diagonal (which-path) matrix elements see only the difference of the two branches’ dressings: the global \(1/r\) monopole is common-mode and cancels identically, and momentum conservation cancels the resulting dipole, leaving a quadrupole whose exterior weight falls as \((d/h)^4\) — equal to \(10^{-12}\) for a millimetre superposition one metre inside the laboratory light cone (Sec. 4). Third, the finite diamond, unlike the wedge, satisfies the Buchholz–Wichmann energy nuclearity condition — its observer Hamiltonian is bounded below by the curvature gap of \(\mathbb{H}^3\) and its level density is polynomial, giving the finite index \(Z(2\pi) \approx 4.1\times10^{-4}\) — so the split property holds and supplies the finite localisation constant \(C\) the wedge lacks (Sec. 5). The residual quadratic (bi-local) channel first appears at \(\order{G^2}\): its \(\order{G}\) part cancels exactly in the Gaussian covariance (Sec. 6).

What the paper does not do is prove the identity. The bound \(\eqref{eq:intro-bound}\) is a composition estimate on the named functional in the Gaussian sector, not an operator-norm inequality on the interacting algebra, and two numbers are not computed: the interacting value of the localisation constant \(C\), and the interacting coefficient of the \(\order{G^2}\) residual. Equation \(\eqref{eq:intro-identity}\) remains a conjecture (Conjecture [conj:identity]), established at the level of expectation values and in solvable models . The claim of this paper is a reduction: every other candidate source of deviation is fixed — the \(2\pi\) is universal with no per-mode freedom (Sec. 2), the constraint selects the physical Hamiltonian as the flow’s target (Sec. 3), the exterior leakage is quartically small and the localisation constant finite (Secs. 4–5), and the non-local residual is second order (Sec. 6) — so the conjecture’s fate rests on a sharply isolated, well-posed computation rather than on a diffuse plausibility argument. The same two open constants are what a laboratory discrimination between first-order and second-order gravitational decoherence rates would measure .

The paper is organised along the chain of the bound. Section 2 assembles the multimode Bisognano–Wichmann identity \(K = 2\pi H_{\mathrm{boost}}+ c\,\mathbb{1}\) and states the gravitational conjecture. Section 3 shows, in a solvable two-sided model, that the Hamiltonian constraint coincides with the modular generator and selects \(K_R = f(H_{\mathrm{phys}})\), the KMS state fixing \(f = 2\pi\,(\cdot)\), and that the QED Gauss law does neither. Section 4 derives the exact conjugation, defines the leakage functional, and establishes the monopole cancellation, the multipole ladder, and the \((d/h)^4\) quadrupole scaling. Section 5 verifies energy nuclearity for the diamond and invokes the split property. Section 6 composes the estimate and fixes the \(\order{G^2}\) onset of the bi-local residual. Section 7 states what remains open and how it connects to experiment. Appendix 8 records the diamond kinematics and the observer clock; Appendix 9 the numerics behind the fitted exponents.

The matter backbone: the identity as a multimode operator equation

Before any gravitational dressing enters, the identity \(K = 2\pi H_{\mathrm{phys}}\) is an exact theorem of the free field. On the Rindler wedge the Bisognano–Wichmann theorem  fixes the modular Hamiltonian of the vacuum to be \(2\pi\) times the boost generator, and the Unruh construction  makes the mechanism explicit: the Minkowski vacuum is a two-mode squeezed state across the horizon, and the squeezing angle carries the \(2\pi\). This section assembles that single-frequency fact into a full multimode operator equation, establishes that the coefficient \(2\pi\) is universal across the spectrum with no per-mode freedom, and states the gravitational target the rest of the paper bounds.

The multimode Bisognano–Wichmann identity

Equation \(\eqref{eq:bb-backbone}\) reads: the generator of modular time on the wedge is the boost energy, calibrated by the universal factor \(2\pi\), up to an additive constant that carries only the entanglement entropy. It is the free-field statement of \(K = 2\pi H_{\mathrm{phys}}\), exact and mode-by-mode.

The content of Theorem [thm:backbone] beyond the textbook single-mode case is that the \(2\pi\) carries no per-mode knob. A modular-time calibration that differed from mode to mode would smear any coefficient built on it; the free field admits no such freedom, so the one universal \(2\pi = \beta_{\mathrm{mod}}\) is the Bisognano–Wichmann value throughout.

The additive constant and the modular flow

The constant \(c\) in \(\eqref{eq:bb-backbone}\) is dynamically inert.

Lemma [lem:shift] isolates the constant \(c\) as the entanglement entropy \(S_0\) and removes it from any subsequent bound on a decoherence rate: the gravitational obstruction we bound below is an obstruction to the operator content \(2\pi H_{\mathrm{boost}}\), never to the additive \(c\).

Reconstruction of the global generator

The local modular data of Theorem [thm:backbone] carries more than the boost energy: it encodes the full global Hamiltonian. Chen, Lashkari and Leung  observe that for the interval \((-R, R)\) in a conformal vacuum the modular Hamiltonian is the local Casini–Huerta–Myers form  \[\begin{equation} K(R) \;=\; 2\pi \int_{-R}^{R} \frac{R^2 - x^2}{2R}\, T_{00}(x)\, dx, \label{eq:bb-chm} \end{equation}\] and that the global Hamiltonian \(H(R) = \int_{-R}^{R} T_{00}\,dx\) is recovered from this purely local datum by a single scale operation, \[\begin{equation} (R\,\partial_R + 1)\,K(R) \;=\; 2\pi R\, H(R). \label{eq:bb-cll} \end{equation}\] The reconstruction is exact for any \(T_{00}\): differentiating \(\eqref{eq:bb-chm}\) in \(R\) produces boundary terms carrying the weight \(\tfrac{R^2 - x^2}{2R}\big|_{x=\pm R} = 0\), which vanish because the modular weight vanishes at the entangling points, so only the weight derivative survives and combines with the \(+1\) to give \(2\pi R\, H\). Equation \(\eqref{eq:bb-cll}\) reads: the physical Hamiltonian is not extra information beyond the modular Hamiltonian of a region — it is reconstructible from it, with coefficient \(2\pi\). The \(2\pi\) recovered here is the same \(2\pi\) of \(\eqref{eq:bb-backbone}\) and of the identity this paper studies.

The gravitational target

Theorem [thm:backbone] and the reconstruction \(\eqref{eq:bb-cll}\) are matter-sector statements: \(H_{\mathrm{phys}}\) there is the field Hamiltonian on a fixed background. In the gravitational theory \(H_{\mathrm{phys}}\) is the constraint-reduced Hamiltonian, and the vacuum of the dressed diamond carries a Newtonian field with support outside the diamond. The operator identity is then a conjecture.

Equation \(\eqref{eq:bb-conjecture}\) is established at the level of expectation values and in solvable models; as a full interacting operator identity it remains a conjecture . The two correction channels it admits must be kept distinct. The \(\order{G^2}\) written in \(\eqref{eq:bb-conjecture}\) names the non-local (bi-local) residual, whose order is fixed in Sec. 6: that channel has no first-order piece. Separately, at finite depth \(h\) there is a local first-order-in-\(G\) term — the exterior leakage of the dressing, which does not vanish identically but is geometrically suppressed as \((d/h)^4\) and vanishes in the deep-source limit \(h\to\infty\); Sec. 4 isolates it and bounds it. The remainder of this paper does not prove \(\eqref{eq:bb-conjecture}\). It reduces the obstruction to a single computable functional, the boost-weighted energy of the dressing field outside the diamond, and bounds that functional.

The constraint selects the modular generator

Theorem [thm:backbone] fixes the free-field modular Hamiltonian, but a generic subregion’s modular Hamiltonian is a bilocal operator bearing no relation to any Hamiltonian. What singles out gravity is that its constraint is not an extra condition layered on the dynamics: the Hamiltonian constraint coincides with the modular generator. In a solvable two-sided model we show that imposing the constraint is imposing modular invariance, that its equilibrium solution forces the one-sided modular Hamiltonian to be \(2\pi\) times the physical Hamiltonian, and that the spatial Gauss law of electromagnetism — a constraint but not the modular generator — does none of this. This is why gravity, and not electromagnetism, points to the identity of Conjecture [conj:identity].

The constraint as the modular generator

Take two sectors, an observer/clock “L” and matter “R”, each with Hamiltonian \(H\) and energies \(E_n\). The gravitational Hamiltonian constraint — that the total super-Hamiltonian annihilates physical states  — is modelled by \[\begin{equation} \hat C\,\ket{\psi} = 0, \qquad \hat C = H_L - H_R. \label{eq:cs-constraint} \end{equation}\] The full two-sided modular Hamiltonian of the L\(|\)R split at inverse temperature \(\beta\) is \(K_{\mathrm{full}} = \beta\,(H_R - H_L)\) . The two operators therefore coincide up to the temperature factor, \[\begin{equation} \hat C \;=\; H_L - H_R \;=\; -\tfrac{1}{\beta}\,K_{\mathrm{full}}. \label{eq:cs-C-is-K} \end{equation}\] Equation \(\eqref{eq:cs-C-is-K}\) reads: the Hamiltonian constraint and the generator of modular time are the same operator. Imposing \(\hat C\ket{\psi} = 0\) is identically imposing \(K_{\mathrm{full}}\ket{\psi} = 0\), invariance under the full modular flow — the KMS equilibrium condition . The constraint is not imposed on top of the flow; it is the statement that the state is stationary under the flow.

What the constraint gives and what the temperature gives

The states annihilated by \(\eqref{eq:cs-constraint}\) are the energy-correlated thermofield-double class, \[\begin{equation} \ket{\psi} \;=\; \sum_n c_n\,\ket{n}_L\ket{n}_R, \label{eq:cs-tfd} \end{equation}\] for which the reduced matter state \(\rho_R = \sum_n |c_n|^2 \ket{n}\!\bra{n}\) is diagonal in the energy basis. Its modular Hamiltonian therefore commutes with the physical Hamiltonian and is a function of it alone: \[\begin{equation} [K_R,\, H_{\mathrm{phys}}] = 0, \qquad K_R = f(H_{\mathrm{phys}}). \label{eq:cs-function} \end{equation}\] Equation \(\eqref{eq:cs-function}\) is the structural content the constraint supplies, and it is the hard part: a generic subregion’s modular Hamiltonian is bilocal and unrelated to the dynamics, and the constraint forces it to be a function of the physical Hamiltonian.

What the constraint does not fix is that \(f\) is linear with slope \(2\pi\). The distinguished modular-invariant (KMS / Hartle–Hawking) state \(|c_n|^2 = e^{-\beta E_n}/Z\) gives \[\begin{equation} K_R \;=\; -\ln\rho_R \;=\; \beta\,H_{\mathrm{phys}}+ \mathrm{const}, \qquad \beta = 2\pi \;\Rightarrow\; K_R = 2\pi\,H_{\mathrm{phys}}. \label{eq:cs-kms} \end{equation}\] The KMS spectrum has uniform gaps equal to \(2\pi\). A non-KMS constraint-satisfying state — for instance \(|c_n|^2 \propto e^{-\beta E_n - \gamma E_n^2}\) — still satisfies the constraint and still gives \(K_R = f(H_{\mathrm{phys}})\), but now with \(f\) nonlinear. The linearity and the value \(2\pi\) are the modular KMS temperature, the Bisognano–Wichmann input of Theorem [thm:backbone], not something the constraint alone delivers.

Proposition [prop:selection] is the mechanism: the constraint delivers the structure \(K_R = f(H_{\mathrm{phys}})\), and the KMS temperature fixes \(f\) to \(2\pi H\).

Why the Gauss law does not select the identity

The contrast with electromagnetism is sharp at the operator level. The QED constraint is the spatial Gauss law \(\nabla\!\cdot\! E = \rho\), the generator of gauge transformations on a Cauchy slice . It commutes with the boost/modular generator but is not that generator: it is an internal gauge condition, not \(H_L - H_R\). Imposing it therefore does not impose modular invariance and does not tie \(\rho_R\) to the energy basis. A Gauss-satisfying physical state can entangle the observer with R’s gauge index, making \(\rho_R\) diagonal in a basis that does not commute with \(H_R\), so that \[\begin{equation} [K_R,\, H_{\mathrm{phys}}] \;\neq\; 0 \qquad \text{(Gauss law / QED)}. \label{eq:cs-qed} \end{equation}\] Under the spatial constraint the modular Hamiltonian need not even be a function of the physical Hamiltonian, let alone \(2\pi\) times it. The distinction is the operator-level content of the statement that gravity has a Hamiltonian constraint, \(H_{\mathrm{total}} = 0\), while electromagnetism has only spatial constraints:

The left column is the physical origin of the identity: gravity’s constraint is the modular generator, so equilibrium under it is \(K_R \propto H_{\mathrm{phys}}\); electromagnetism’s is not, so no such tie is forced.

The error functional and its quadrupole suppression

The identity \(K = 2\pi H_{\mathrm{phys}}\) conjectured in Sec. 2 (Conjecture [conj:identity]) equates the modular Hamiltonian of the gravitationally dressed causal diamond with the boost-frame energy of the matter it contains. This section isolates the obstruction to that identity. For a coherent (Gaussian) dressing the deviation \(K - 2\pi H_{\mathrm{phys}}\) is computed exactly and shown to reduce to a single scalar functional—the boost-weighted energy of the dressing field on the modes outside the diamond—and that functional is suppressed as \((d/h)^4\) by momentum conservation, with \(d\) the which-path separation and \(h\) the depth of the source inside the diamond.

The exact conjugation and the leakage functional

By Bisognano–Wichmann the vacuum modular Hamiltonian of the diamond is the boost generator, diagonal over the wedge modes, \[\begin{equation} K_{\mathrm{vac}}\;=\; \sum_k \Omega_k\, a_k^\dagger a_k , \label{eq:ef-kvac} \end{equation}\] with \(\Omega_k\) the modular (boost) weight of mode \(k\) . A branch carrying its Newtonian field is the field vacuum acted on by a coherent displacement, \(\lvert\Phi\rangle = D(\alpha)\lvert 0\rangle\) with \(D(\alpha) = \exp\!\big(\sum_k \alpha_k a_k^\dagger - \alpha_k^* a_k\big)\), the amplitudes \(\alpha_k\) encoding that field. The dressed modular Hamiltonian is \(D(\alpha)\,K_{\mathrm{vac}}\,D(\alpha)^\dagger\), and this conjugation is exact.

Lemma [lem:conjugation] makes the deviation of the dressed modular Hamiltonian from the vacuum one entirely linear in the mode displacements. The coherent-state locality theorem identifies \(D(\alpha)\,K_{\mathrm{vac}}\,D(\alpha)^\dagger\) with the true modular Hamiltonian of the diamond algebra precisely when \(D(\alpha)\) is supported inside the diamond, i.e. when \(\alpha_k = 0\) for every mode localized outside; the interior linear term is then \(2\pi H_{\mathrm{phys}}\), the boost-frame energy of the dressing field (the same \(2\pi\) recovered from local modular data in Sec. 2, Theorem [thm:backbone]). Splitting the linear term of Eq. \(\eqref{eq:ef-conjugation}\) by mode support, \[\begin{equation} K - 2\pi H_{\mathrm{phys}} \;=\; -\!\!\sum_{k\,\in\,\mathrm{out}}\!\! \Omega_k\big(\alpha_k a_k^\dagger + \alpha_k^* a_k\big) , \label{eq:ef-error} \end{equation}\] identifies the obstruction with the exterior part of the displacement alone: the interior modes reconstruct \(2\pi H_{\mathrm{phys}}\) exactly, and every remaining term lives on modes outside the diamond.

Equation \(\eqref{eq:ef-error}\) exhibits \(K - 2\pi H_{\mathrm{phys}}\) as a specific operator built from the exterior modes, and \(W_{\mathrm{ext}}[\alpha]\) is the scalar that bounds its per-mode content. The bounds of this paper are bounds on \(W_{\mathrm{ext}}[\alpha]\): it is a finite functional of the displacement, whereas \(K\) and \(H_{\mathrm{phys}}\) are individually unbounded, so a bare operator norm \(\lVert K - 2\pi H_{\mathrm{phys}}\rVert\) has no meaning on the wedge algebra without the finite-trace structure supplied in Sec. 5. The functional-level statement—that the deviation is carried entirely by \(W_{\mathrm{ext}}[\alpha]\) and vanishes when \(W_{\mathrm{ext}}[\alpha]\to 0\)—is exact. Promoting it to an operator-norm bound on the interacting algebra remains open; that promotion is what Sec. 5 and Sec. 6 address, and it is not established here.

The relative displacement and the monopole cancellation

The decoherence-relevant matrix elements are off-diagonal in the branch index, \(\langle\Phi_L\lvert\,O\,\rvert\Phi_R\rangle\) for the modular Hamiltonian and its perturbation. With \(\lvert\Phi_A\rangle = D(\alpha_A)\lvert 0\rangle\) the branch content enters only through the relative displacement.

The field that governs decoherence is therefore the difference \(\delta\Phi= \Phi_L - \Phi_R\), not each branch’s monopole. Each branch sources a Newtonian monopole \(\Phi_A(x) = -GM/\lvert x - x_A\rvert\) with \(\lvert\nabla\Phi_A\rvert^2 \propto r^{-4}\); the difference of two masses separated by \(d\) is a dipole, \[\begin{equation} \delta\Phi(x) \;=\; -GM\Big(\frac{1}{\lvert x - x_L\rvert} - \frac{1}{\lvert x - x_R\rvert}\Big) \;\sim\; \frac{GM\,d}{r^{2}} \quad (r \gg d), \label{eq:ef-dipole} \end{equation}\] with energy density \(\lvert\nabla\delta\Phi\rvert^2 \sim d^2/r^{6}\), two powers more localized than the monopole. The global \(1/r\) tail—the piece that lies outside the diamond and would break the conjugation of Lemma [lem:conjugation]—is common-mode and cancels exactly in the off-diagonal.

The cancellation is quantitative. On a boost-weighted exterior integral over a three-dimensional grid, with the source at depth \(h\) inside the diamond, the exterior fraction of the monopole weight is \(6.2\times10^{-3}\) at \(h = 32\) (in units of the separation), while that of the dipole is \(4.3\times10^{-6}\), smaller by three orders of magnitude at the same depth. Power-law fits give the monopole exterior fraction falling as \(h^{-1.6}\) and the dipole as \(h^{-3.1}\): the relative-displacement leakage is suppressed by more than one extra power of \(d/h\). The falloff exponents \(-4\) (monopole) and \(-6\) (dipole) are exact and reproduced to machine precision (Appendix 9).

Momentum conservation and the multipole ladder

The dipole difference source of Eq. \(\eqref{eq:ef-dipole}\) carries a net dipole moment, which no closed system can support: in any physical preparation the recoil is taken up by the apparatus, and the total mass distribution conserves momentum. Momentum conservation cancels the dipole moment of the difference source, and the leading surviving multipole is the quadrupole, \[\begin{equation} \Phi_{\text{quad}} \;\sim\; \frac{GM\,d^2}{r^{3}}, \qquad \lvert\nabla\Phi_{\text{quad}}\rvert^2 \;\sim\; \frac{d^4}{r^{8}}, \label{eq:ef-quad} \end{equation}\] two powers more localized than the dipole, four than the monopole. This dipole-to-quadrupole cancellation for a momentum-conserving mass distribution has been derived independently in the influence-functional treatment of gravitational decoherence by Takeda, Tomizuka and Tanaka .

The exterior weight follows a clean multipole ladder. With boost weight \(\lvert z\rvert \sim r\), volume element \(r^2\,dr\), and energy density \(r^{-p}\), a multipole of order \(\ell\) has field \(\sim r^{-(\ell+1)}\), density exponent \(p = 2\ell + 4\), and exterior weight \[\begin{equation} W_{\mathrm{ext}}(\ell, h) \;=\; \int_h^R r^{\,3-p}\, dr \;=\; \int_h^R r^{-(2\ell+1)}\, dr \;\propto\; h^{-2\ell} \quad (\ell \ge 1), \label{eq:ef-ladder} \end{equation}\] so the monopole (\(\ell=0\)) gives the marginal logarithm \(\ln(R/h)\), while for \(\ell \ge 1\) the infrared limit \(R\to\infty\) is finite and the ladder is exact: the dipole (\(\ell=1\)) gives \(h^{-2}\) and the quadrupole (\(\ell=2\)) gives \(h^{-4}\) (Sec. 6 takes \(R\to\infty\)).

The exponents are measured directly. On the boost-weighted exterior integral the bare dipole falls as \(h^{-2.1}\) and the bare quadrupole as \(h^{-4.0}\), and the momentum-conserving composite source falls as \(h^{-3.8}\) in the far field (\(h \gg a\)) and \(h^{-1.3}\) in the near field (\(h \ll a\)), with the crossover tracking \(a\) as it is varied (Appendix 9). For a laboratory superposition the diamond boundary is far outside any compact apparatus, so the far-field quadrupole scaling holds: a \(d = 1~\mathrm{mm}\) superposition at depth \(h = 1~\mathrm{m}\) gives \((d/h)^4 = 10^{-12}\), and the leakage functional—hence the obstruction to \(K = 2\pi H_{\mathrm{phys}}\) on the Gaussian off-diagonal sector—is quartically negligible.

Nuclearity and the split property of the diamond algebra

The leakage functional of Sec. 4 measures the boost-weighted energy of the dressing on modes outside the diamond, and bounding it requires localising a globally supported dressing operator onto a wedge-internal approximant with a finite localisation constant. The flat Rindler wedge does not supply one: its algebra is Type III\(_1\), which admits no split and hence no intermediate factor onto which the dressing could be projected. The finite diamond does supply one, and this section verifies the phase-space condition that produces it.

The curvature gap of the diamond observer Hamiltonian

The intermediate factor exists because the observer Hamiltonian of the diamond is bounded below by a strictly positive gap of geometric origin. Under the Casini–Huerta–Myers conformal map the diamond becomes the ultrastatic cylinder \(\mathbb{R}\times\mathbb{H}^3\) , on which the modular generator is the cylinder-time Hamiltonian \(q = \sum_\lambda \omega_\lambda\, a^\dagger_\lambda a_\lambda\) with one-particle energies \(\omega_\lambda = \sqrt{\lambda}\) set by the eigenvalues \(\lambda\) of the \(\mathbb{H}^3\) Laplacian (App. 8). The bottom of that spectrum is fixed exactly. In geodesic polar coordinates the \(s\)-wave radial Laplace–Beltrami operator on \(\mathbb{H}^3\) is \(-\Delta f = f'' + 2\coth(\rho)\,f'\), and the substitution \(f = g/\sinh\rho\) turns it into a constant-potential problem, \[\begin{equation} f'' + 2\coth(\rho)\,f' + \lambda f = 0 \;\Longleftrightarrow\; -g'' + g \;=\; \lambda\, g , \label{eq:ns-reduction} \end{equation}\] an exact identity with no approximation. The reduced operator \(-g'' + 1\cdot g\) on the half-line with \(g(0) = 0\) is a free particle in a constant potential \(+1\), so its spectrum is the continuum \(\lambda = 1 + k^2\), \(k\in\mathbb{R}\), and \[\begin{equation} \operatorname{spec}(-\Delta_{\mathbb{H}^3}) \;=\; \Big[\tfrac{(n-1)^2}{4},\,\infty\Big)\Big|_{n=3} \;=\; [\,1,\,\infty\,). \label{eq:ns-spectrum} \end{equation}\] Equation \(\eqref{eq:ns-spectrum}\) reads: the diamond observer Hamiltonian is bounded below by the hyperbolic curvature gap \((n-1)^2/4 = 1\), the “\(+1\)” in Eq. \(\eqref{eq:ns-reduction}\) being precisely the curvature term, with no flat-space analogue (where the floor is \(0\)). The one-particle energies therefore satisfy \(\omega_\lambda = \sqrt{\lambda}\ge 1 > 0\), and the derivation of the bound uses only this gap.

The nuclearity index

The Buchholz–Wichmann nuclearity condition asks that the map \(a\mapsto e^{-\beta q}a\,\ket{\Omega}\) be nuclear, an index controlled by the partition function \(Z(\beta) = \mathrm{Tr}\,e^{-\beta q}\) . Two features of the diamond make it finite. The spectrum is gapped by Eq. \(\eqref{eq:ns-spectrum}\), so the integration starts at \(\omega = 1\); and the density of states of a three-dimensional Laplacian grows only polynomially — for \(\mathbb{H}^3\) the Plancherel density is \(\rho(\omega)\sim\omega^2\). A polynomial density damped by the modular weight integrates to a finite value, \[\begin{equation} Z(\beta) \;=\; \int_1^\infty \rho(\omega)\, e^{-\beta\omega}\, d\omega \;=\; e^{-\beta}\Big(\frac{1}{\beta} + \frac{2}{\beta^2} + \frac{2}{\beta^3}\Big) < \infty , \qquad \rho(\omega)\sim\omega^2 , \label{eq:ns-Z} \end{equation}\] the closed form holding at gap \(1\) for every \(\beta > 0\), in particular at the modular temperature \(\beta_{\mathrm{mod}}= 2\pi\) (App. 8), where \[\begin{equation} Z(2\pi) \;\approx\; 4.1\times10^{-4} . \label{eq:ns-Znum} \end{equation}\] The finiteness is not automatic. A Hagedorn density \(\rho\sim e^{\alpha\omega}\) would make the integrand \(e^{(\alpha-\beta)\omega}\) and drive \(Z(\beta)\) to diverge for \(\alpha\ge\beta\); the diamond escapes this exactly because its \(\mathbb{H}^3\)/Weyl density is polynomial and its spectrum is gapped. The polynomial degree does not matter — \(\omega^2\) and \(\omega^4\) both give a finite \(Z(2\pi)\) — so the index does not hinge on the precise Plancherel coefficient, only on the sub-exponential growth. Equation \(\eqref{eq:ns-Z}\) reads: the diamond is nuclear because the curvature gap and the polynomial mode count together make the modular weight summable.

The split property

Nuclearity of the index \(\eqref{eq:ns-Z}\) is a phase-space criterion, and the passage from it to a localising factor is a theorem of Buchholz, D’Antoni and Longo, invoked here rather than reproved.

Theorem [thm:split] is not our result; the implication nuclearity \(\Rightarrow\) split is standard operator algebra. Our contribution is the verification of its hypothesis for the gravitationally relevant object: the diamond observer Hamiltonian is gapped by Eq. \(\eqref{eq:ns-spectrum}\) and its density is polynomial, so the index \(\eqref{eq:ns-Z}\) is finite and the theorem applies. The matter-sector instance of the same argument — the \(L^2\)-nuclearity of the ball inclusion \(B(e^{-2\pi s}R)\subset B(R)\) — has been established directly by Chen, Lashkari and Leung . The Type I factor \(\mathcal{N}\) of Eq. \(\eqref{eq:ns-split}\) is the wedge-internal approximant: the globally supported dressing has a representative inside \(\mathcal{N}\) with an error equal to the split distance, and that error is finite because the index is. This is the localisation constant the leakage bound of Sec. 6 requires, absent on the Type III\(_1\) wedge and present on the diamond.

The interacting extension

The construction so far is free-field: the density \(\rho\sim\omega^2\) is the Gaussian mode count. The localisation constant remains finite once interactions are turned on, provided the level density stays sub-exponential.

The premise of Proposition [prop:interacting] — polynomial corrections to the level density below the Hagedorn/Planck threshold — is an assumption, stated as such, not a computed fact. What the proposition establishes is the conditional: given sub-Hagedorn growth, the localisation constant is finite throughout the effective-field-theory regime, which is the regime of the laboratory prediction. The explicit interacting value of the constant requires the full interacting modular data and is not computed here; it is one of the two open numbers isolated in Sec. 6.

The composition estimate and the order of the residual

The three preceding sections each fix one input to the obstruction. The constraint selection of Sec. 3 fixes the target of the identity to be \(2\pi H_{\mathrm{phys}}\). The leakage functional of Sec. 4 isolates the only deviation as the exterior boost-weighted energy of the dressing and shows it is suppressed as \((d/h)^4\) for a momentum-conserving source. The nuclearity of Sec. 5 supplies a finite localisation constant. This section composes them into a single bound on the leakage functional, and separately establishes the order at which the genuinely non-local part of the obstruction first appears.

The Gaussian-sector composition estimate

Equation \(\eqref{eq:cb-bound}\) bounds the named leakage functional \(W_{\mathrm{ext}}[\alpha_{\mathrm{dressing}}]\) of Def. [def:leakage], not a bare operator norm \(\|K - 2\pi H_{\mathrm{phys}}\|\). The distinction is not cosmetic: \(K\) and \(H_{\mathrm{phys}}\) are unbounded, so a norm on their difference is not defined on the interacting algebra, and the leakage functional is the well-posed object the constraint analysis actually produces. Promoting the functional estimate \(\eqref{eq:cb-bound}\) to an operator-norm inequality on the interacting Type II algebra — the passage from a bound on the exterior weight to a bound on the operator itself — is not derived here; it is one of the two open constants named at the end of this section. What Eq. \(\eqref{eq:cb-bound}\) establishes is that the exterior leakage, the entire first-order content of the obstruction, vanishes as \((d/h)^4\) with a finite coefficient.

The deep-source limit reads the identity off the bound. As \(h\to\infty\) at fixed \(d\) the right-hand side of Eq. \(\eqref{eq:cb-bound}\) vanishes, so the identity \(K = 2\pi H_{\mathrm{phys}}\) holds exactly in the deep-source limit of the Gaussian/coherent sector — the qualifier being essential: it is the Gaussian sector in which every step, from the Bisognano–Wichmann backbone through the coherent dressing to the free-field split, is under control. At laboratory depth \(h/d\sim 10^3\) the geometric channel \(\eqref{eq:cb-ladder}\) is smaller than \(10^{-10}\), negligible, which leaves the \(\order{G^2}\) non-Gaussian channel as the sole surviving residual.

The order of the non-local residual

The obstruction has a second channel beyond the exterior leakage: a bi-local term in \(K\) coupling two points inside the diamond, which would spoil the first-order rate were it \(\order{G}\). It is not.

The graviton reading of this counting is that a one-graviton dressing is a first moment and produces no bi-local term, while a two-graviton correlation is a covariance and seeds one at \(\order{G^2}\): one power of \(G\) per moment, two moments for a bi-local term. This is a counting argument at the level of the Gaussian covariance model, not a field-theory computation; the interacting value of \(\delta K_{\mathrm{nonlocal}}\) for genuine interacting gravitons is the second of the two open numbers, and Eq. \(\eqref{eq:cb-og2}\) fixes only its order, not its coefficient.

Discussion

What is fixed and what is open

The chain of Secs. 2–6 reduces the conjecture \(K = 2\pi H_{\mathrm{phys}}+ \order{G^2}\) to two numbers. Fixed: the \(2\pi\) is the universal Bisognano–Wichmann calibration with no per-mode freedom (Theorem [thm:backbone]); the additive constant is dynamically inert (Lemma [lem:shift]); the constraint selects \(H_{\mathrm{phys}}\) as the target of the flow (Prop. [prop:selection]); the entire first-order deviation is the leakage functional, suppressed as \((d/h)^4\) by the monopole cancellation and momentum conservation (Prop. [prop:ladder]); the localisation constant is finite by nuclearity, and stays finite under sub-Hagedorn interactions (Theorem [thm:split], Prop. [prop:interacting]); the bi-local residual first appears at \(\order{G^2}\) (Prop. [prop:og2]). Open: the interacting value of the localisation constant \(C\) in Eq. \(\eqref{eq:cb-bound}\), and the interacting coefficient of the \(\order{G^2}\) residual — equivalently, the promotion of the functional estimate to an operator-norm statement on the interacting Type II algebra. These are interacting quantum-gravity quantities; no step in this paper computes them, and the identity remains a conjecture exactly as stated in Conjecture [conj:identity].

Other candidate sources of coefficient freedom in the decoherence rate built on the identity are closed elsewhere and enter no part of the present bound: the entropy-constant residues of the diamond construction — the trace-anomaly normalisation and the entangling-surface edge modes — are additive c-numbers decoupled from the flow by Lemma [lem:shift] and are treated in the companion graviton paper , and the residual window in the rate coefficient, \(C_{\mathrm{rate}}\in[2/\pi,1]\), is an observable convention pinned in the canonical-core paper . With those channels closed, the coefficient freedom of the rate and the operator content of the identity coincide: both live in the two constants above.

The experiment that measures the residual

The two open constants are not of purely formal interest: they are what the laboratory would measure. The identity at first order gives the decoherence rate \(\Gdec = \EG/\hbar\), linear in \(G\); perturbative graviton exchange with an unconstrained environment gives a rate at \(\order{G^2}\) . The expansion parameter separating them at the canonical benchmark — a microgram superposition at millimetre separation — is \(\epsilon_G = GM/(c^2 d) = 7.4\times10^{-34}\), and the ratio of the first-order to the second-order rate is \((\MP/M)^2\,(d/\lP) \approx 3\times10^{34}\): thirty-four orders of magnitude between the conjectured first-order rate (\(\tdec = \hbar d/(GM^2) = 1.6~\mathrm{ns}\) at the benchmark) and the second-order rate. A which-path experiment at picogram masses that resolves the gravitational decoherence floor reads off the order in \(G\) directly, and with it the fate of the operator equation . The spectral shape of the same decoherence channel — the modular-thermal knee at \(f = \EG/(2\pi\hbar)\) — supplies the finer discriminant between the modular mechanism and Markovian collapse models of the Diósi–Penrose class .

Relation to nearby results

Three strands of recent work touch the same objects. The crossed-product constructions  build Type II subregion algebras by adjoining an observer with a bounded-below Hamiltonian; the diamond observer of Appendix 8, gapped by the \(\mathbb{H}^3\) curvature (Sec. 5.1), is precisely such an observer, and the finite nuclearity index of Sec. 5 is the phase-space input those constructions presuppose. Chen, Lashkari and Leung  established the matter-sector instance of the programme rigorously — the global Hamiltonian reconstructed from local modular data with the same \(2\pi\), and \(L^2\)-nuclearity for the conformal ball inclusion — which anchors the free-field end of the present chain; the gravitational, dressed case is what the bound of Sec. 6 addresses. Danielson, Satishchandran and Wald  showed that a black-hole or cosmological horizon decoheres superpositions through the soft radiation the two branches emit through the horizon, the number of entangling emissions growing linearly in the time the superposition is held open; the mechanism here is the intrinsic analogue — the observer’s own diamond horizon at the modular temperature \(\beta_{\mathrm{mod}}= 2\pi\), no external horizon required — and the momentum-conservation quadrupole structure of Sec. 4.3 matches the multipole counting of their emission argument, derived independently in the influence-functional setting by Takeda, Tomizuka and Tanaka .

Outlook

The obstruction to \(K = 2\pi H_{\mathrm{phys}}\) was a qualitative no-go: the dressing is global, the wedge does not split. It is now an estimate with a stated remainder — the leakage is quartically small in \(d/h\), the localisation constant is finite, the non-local channel is second order — and a two-constant residual that is well-posed, isolated, and shared with the observable the experiment targets. Computing either constant in interacting quantum gravity, or measuring the order in \(G\) of gravitational decoherence in the laboratory, decides the conjecture. Nothing else does.

Diamond kinematics and the observer clock

The modular temperature \(\beta_{\mathrm{mod}}= 2\pi\) and the ultrastatic \(\mathbb{R}\times\mathbb{H}^3\) frame used in Secs. 5 and 6 are fixed by the diamond’s own geometry, not imported by analogy. This appendix records the kinematic facts that fix them. The modular flow of a finite causal diamond \(D = \{\,|t| + r < R\,\}\) is generated by the conformal Killing vector  \[\begin{equation} \xi \;=\; \frac{\pi}{R}\left[\frac{R^2 - t^2 - r^2}{2}\,\partial_t \;-\; t\,r\,\partial_r\right], \label{eq:app-ckv} \end{equation}\] which reads: the diamond’s vacuum flows under a boost-like conformal symmetry that fixes the diamond and its light cone.

The observer worldline.

The diamond has isometry group \(\mathrm{SO}(3)\) of rotations about its centre. The unique timelike worldline left pointwise fixed by that group is the central geodesic \(r = 0\), \(|t| < R\) — a straight inertial line in Minkowski space, the diamond’s intrinsic analogue of the de Sitter static-patch worldline, selected by symmetry alone. It is an integral curve of the modular generator \(\eqref{eq:app-ckv}\): at \(r = 0\), \[\begin{equation} \xi^r\big|_{r=0} = -\frac{\pi}{R}\,t\,r\Big|_{r=0} = 0, \qquad \xi^t\big|_{r=0} = \frac{\pi}{2R}\big(R^2 - t^2\big), \label{eq:app-worldline} \end{equation}\] so \(\xi\) is purely timelike on the centre and the flow carries the observer along their own worldline.

Modular time and the horizon.

Integrating the flow \(dt/d\sigma = \xi^t|_{r=0} = (\pi/2R)(R^2 - t^2)\) with \(t(0) = 0\) gives \[\begin{equation} t(\sigma) \;=\; R\,\tanh\!\Big(\frac{\pi\sigma}{2}\Big), \qquad \sigma\in(-\infty,\infty)\ \Rightarrow\ t\in(-R,R), \label{eq:app-tsigma} \end{equation}\] so the observer reaches the future and past tips \(t = \pm R\) only as \(\sigma\to\pm\infty\): the diamond tips are a modular horizon at infinite modular time, the diamond crossed in finite proper time \(\int d\tau = 2R\) but in unbounded modular parameter — the hallmark of a boost-like modular flow on a finite region.

The uniform cylinder clock.

Under the Casini–Huerta–Myers Weyl rescaling \(\hat g = \Omega^{-2}\eta\) the diamond maps to the ultrastatic cylinder \(\mathbb{R}\times\mathbb{H}^3\) , on which \(\xi\) becomes a constant-norm Killing field, \(|\xi|^2_{\hat g} = -\pi^2 R^2/4\) everywhere. The \(\hat g\)-proper time along the central curve is therefore linear in the modular parameter, \[\begin{equation} \hat\tau(\sigma) \;=\; \frac{\pi R}{2}\,\sigma, \qquad \frac{d\hat\tau}{d\sigma} = \sqrt{-|\xi|^2_{\hat g}} = \frac{\pi R}{2} = \text{const}, \label{eq:app-clock} \end{equation}\] a clock ticking uniformly in modular time, exactly the uniform clock the crossed product needs; its conjugate energy is the observer Hamiltonian \(q\) whose spectrum is bounded below by the \(\mathbb{H}^3\) curvature gap of Sec. 5.1.

The modular temperature.

The modular flow is KMS at the normalisation-free inverse temperature \[\begin{equation} \beta_{\mathrm{mod}}\;=\; 2\pi, \label{eq:app-beta} \end{equation}\] with position-dependent Tolman temperature \(T_{\mathrm{loc}}(r) = 1/(2\pi f(r))\), \(f(r) = (R^2 - r^2)/(2R)\); at the centre \(T_{\mathrm{loc}}(0) = 1/(\pi R)\), redshifting to the uniform cylinder value \(T_{\mathrm{cyl}} = 1/(\pi R)\). Equation \(\eqref{eq:app-beta}\) is the value used throughout the paper: the observer carries a thermal clock at the universal modular temperature \(2\pi\). The associated crossed-product refinements — the trace-anomaly normalisation of the entropy constant and the entangling-surface edge modes — are not part of the present bound; they are treated elsewhere .

Exterior-weight numerics

This appendix records the grid construction and the fitted power laws behind the leakage suppression of Sec. 4, specifically the monopole–dipole cancellation of Sec. 4.2 (Lemma [lem:relative]) and the multipole ladder of Sec. 4.3 (Proposition [prop:ladder]).

The boost-weighted exterior integral

The diamond is modelled by the Rindler wedge \(z > 0\), its complement the left wedge \(z < 0\), with the entangling plane at \(z = 0\). A source of separation \(d\) is placed at depth \(h\) inside the wedge (distance \(h\) from the entangling plane). The obstruction of Definition [def:leakage] is the boost-weighted weight of the displacement field outside the wedge, so the discretized quantity is \[\begin{equation} W \;=\; \int \lvert z\rvert\,\lvert\nabla\Phi\rvert^2\, d^3x , \qquad \frac{W_{\mathrm{ext}}}{W_{\mathrm{int}}} \;=\; \frac{W(z<0)}{W(z>0)} , \label{eq:app-weight} \end{equation}\] with the boost weight \(\lvert z\rvert\) standing for the modular weight \(\Omega_k\) of Eq. \(\eqref{eq:ef-kvac}\). The field \(\Phi\) is the monopole \(-GM/r\), the dipole difference of Eq. \(\eqref{eq:ef-dipole}\), or the momentum-conserving composite of Eq. \(\eqref{eq:ef-quad}\); the integral of Eq. \(\eqref{eq:app-weight}\) is evaluated on a three-dimensional grid and the exterior fraction \(W_{\mathrm{ext}}/W_{\mathrm{int}}\) is read off. The falloff exponents are checked separately on the radial proxy \(W_{\mathrm{ext}}(p,h) = \int_h^R r^{\,3-p}\,dr\) of Eq. \(\eqref{eq:ef-ladder}\), integrated by Simpson’s rule to \(R = 10^4\).

The depth sweep and the monopole–dipole cancellation

Table 1 gives the exterior fraction of Eq. \(\eqref{eq:app-weight}\) as the source depth \(h\) is swept, for the monopole (each branch’s Newtonian field) and the dipole (the relative displacement of Lemma [lem:relative]), in units where the separation is unity.

The dipole exterior fraction is smaller than the monopole at every depth and falls by more than one extra power of \(h\); the common-mode \(1/r\) monopole tail, which carries the global support that would break the conjugation of Lemma [lem:conjugation], has cancelled in the difference.

The multipole ladder and the crossover

Table 2 gives the fitted exterior-weight exponents for the pure multipoles and for the momentum-conserving composite of Proposition [prop:ladder]. The pure-multipole values confirm the ladder \(W_{\mathrm{ext}}(\ell,h)\propto h^{-2\ell}\) of Eq. \(\eqref{eq:ef-ladder}\): the exterior exponent equals \(-2\ell\) for \(\ell = 1,2,3\) on the radial proxy, and the boost-weighted grid integral reproduces the dipole \(-2\) and quadrupole \(-4\) to within fit error.

The composite source, built as the mass dipole at depth \(h\) and the opposite apparatus dipole at depth \(h+a\), reproduces the two regimes of Proposition [prop:ladder]: the far-field exponent \(-3.8\) approaches the pure quadrupole \(-4.0\) for \(h \gg a\), the near-field exponent \(-1.3\) tracks the bare dipole for \(h \ll a\), and increasing \(a\) pushes the crossover outward. For a laboratory geometry \(h \gg a\), so the quadrupole scaling of Eq. \(\eqref{eq:ef-scaling}\) governs the leakage.

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