Introduction: two horizon-rate coincidences

Two numbers in contemporary gravitational physics sit close to the Hubble rate. Milgrom’s acceleration scale \[\begin{equation} a_0 \;\approx\; 1.2\times10^{-10}\ \mathrm{m\,s^{-2}} \;\approx\; \frac{cH_0}{2\pi}, \end{equation}\] the acceleration below which galactic rotation curves depart from Newton, is the empirical core of MOND and the scale a dark-matter halo must reproduce . The rate at which horizon-scale configurations of the cosmic wavefunction lose coherence through their own gravity, \[\begin{equation} \Gamma_{\mathrm{cosmic}}\;\approx\; \frac{g(1)\,H}{4\pi} \;\approx\; \frac{H}{10\pi} \;\approx\; 7\times10^{-20}\ \mathrm{Hz}, \end{equation}\] is derived within Quantum-Geometric Correspondence (QGC) on a de Sitter background , the cosmic-scale instance of the Diósi–Penrose gravitational decoherence rate . One belongs to galactic astronomy, the other to the foundations of quantum mechanics; this paper shows they are two readings of a single clock.

The cosmological horizon has the Gibbons–Hawking temperature  \[\begin{equation} \TdS \;=\; \frac{\hbar H}{2\pi \kB}, \end{equation}\] and any temperature defines a frequency \(\kB T/\hbar\). For the horizon, \[\begin{equation} \boxed{\;\omega_\Lambda\;\equiv\; \frac{\kB \TdS}{\hbar} \;=\; \frac{H}{2\pi}\;} \end{equation}\] the horizon clock. With no assumption beyond the two results above, \[\begin{equation} a_0 \;=\; c\,\omega_\Lambda, \qquad\qquad \Gamma_{\mathrm{cosmic}}\;=\; \tfrac12\,g(1)\,\omega_\Lambda: \end{equation}\] the MOND scale is the horizon clock carried by \(c\) (a frequency becomes an acceleration), the cosmic decoherence rate the same clock carried by a pure number. In the QGC reading, the horizon’s single thermodynamic frequency projects onto an acceleration scale (how gravity behaves) and onto a rate (how fast gravity decoheres); two coincidences with \(H_0\) are one fact about the horizon, seen twice.

The horizon clock and the surface gravity

The horizon clock as modular flow

The frequency \(\omega_\Lambda\) is the rate of the Tomita–Takesaki modular flow of the de Sitter vacuum, the Connes–Rovelli thermal time of the universe . For a KMS state at temperature \(T\), the modular automorphism group runs in a parameter \(s\) related to physical time by \(t = (\hbar/\kB T)\,s = s/\omega_\Lambda\), so the modular clock ticks at \(\kB T/\hbar\). Explicit constructions of the de Sitter modular Hamiltonian establish that, in the large-diamond limit, modular flow becomes static-patch time translation at this rate . The \(2\pi\) in \(\omega_\Lambda= H/2\pi\) is the thermal (KMS) period, the \(2\pi\) of every Unruh, Hawking, and Gibbons–Hawking temperature.

The MOND scale as horizon surface gravity

Any causal horizon with surface gravity \(\kappa\) has temperature \(T = \hbar\kappa/(2\pi c\kB)\), so its acceleration scale is \[\begin{equation} a_\star \;\equiv\; c\,\frac{\kB T}{\hbar} \;=\; \frac{\kappa}{2\pi}. \end{equation}\] The de Sitter horizon has \(\kappa_{\dS}=cH\), giving \[\begin{equation} \boxed{\;a_0 \;=\; \frac{\kappa_{\dS}}{2\pi}\;=\;\frac{cH}{2\pi}\;} \end{equation}\] the MOND acceleration as the de Sitter horizon’s surface gravity divided by \(2\pi\). A Rindler horizon of acceleration \(a\) gives \(a_\star = a/2\pi\), the inverse of the Unruh relation. The de Sitter thermal mass \(\dmdS = \kB\TdS/c^2 = \hbar H/(2\pi c^2)\) has Compton wavelength equal to the horizon circumference, \(\lambda_C(\dmdS)=\hbar/(\dmdS c)=2\pi\RH\), so \[\begin{equation} a_0 \;=\; \frac{c^2}{2\pi\RH} \;=\; \frac{c^2}{\lambda_C(\dmdS)} : \end{equation}\] the MOND scale is the de Sitter thermal mass delocalised once around the horizon.

The lock between \(a_0\) and the cosmic decoherence rate

The parameter-free ratio

The cosmic decoherence rate of QGC on de Sitter is  \[\begin{equation} \Gamma_{\mathrm{cosmic}}\;=\; \frac{g(1)\,H}{4\pi}, \qquad g(x) \;=\; 1 - \frac{2}{\pi}\Si(x), \qquad g(1) = 0.3977, \label{eq:Gcos} \end{equation}\] with \(\Si\) the sine integral and the form-factor argument evaluated at the horizon scale (Sec. 5). Writing \(H/4\pi = \tfrac12\,\omega_\Lambda\), \[\begin{equation} \Gamma_{\mathrm{cosmic}}\;=\; \tfrac12\,g(1)\,\omega_\Lambda, \end{equation}\] the same clock \(\omega_\Lambda\) as \(a_0 = c\omega_\Lambda\), multiplied by \(g(1)/2 = 0.199\). Dividing the two relations cancels every cosmological input (\(H\), \(G\), \(c\), \(\hbar\)).

A framework explaining the MOND scale and cosmic decoherence with unrelated physics has two independent knobs here; QGC has one. Measuring \(a_0\) fixes \(\Gamma_{\mathrm{cosmic}}\) and conversely.

The ratio admits three readings, all equal to \(\omega_\Lambda/\Gamma_{\mathrm{cosmic}}\):

1. The lock (acceleration vs. rate): \(a_0 = 5.03\,c\,\Gamma_{\mathrm{cosmic}}\).

2. Modular ticks to decohere: the universe decoheres in \(\omega_\Lambda/\Gamma_{\mathrm{cosmic}}= 2/g(1) = 5.03\) ticks of its own thermal-time clock. Using the closed-form Bures–Fisher trajectory of QGC pure dephasing , \(d_B^2(t) = 2-\sqrt{2(1+e^{-\Gamma t})}\), at \(\Gamma t = 2/g(1)\) the cosmic state has covered \(99.2\%\) of its geometric distance from coherent to maximally mixed.

3. Hubble units: \(\Gamma_{\mathrm{cosmic}}^{-1} = (4\pi/g(1))H^{-1} = 31.6\,H^{-1} \approx 10\pi\,H^{-1}\), i.e. decoherence after \(\sim 10\pi\) e-folds—which is approximately five modular ticks, each tick being \(2\pi\) e-folds.

The two \(\pi/2\)’s: speed limit versus form factor

Two of this framework’s results carry a factor \(2/\pi\): the Margolus–Levitin speed limit of the canonical core paper, \(\Gamma_{\mathrm{ML}} = 2E_G/(\pi\hbar)\) , and the form-factor normalisation in Eq. \(\eqref{eq:Gcos}\). They share an origin but are distinct objects, differing in exactly one way, identified below.

The spatial \(\pi/2\): an analytic, deformable constant

The QGC decoherence functional for a two-branch source is a mode integral whose integrand factorises into a space part and a time part : \[\begin{equation} \Gamma(t) \;\propto\; \int \frac{d^3k}{k^4\,\omega_k}\; \underbrace{\bigl(1-\cos \mathbf{k}\!\cdot\!\mathbf{d}\bigr)}_{\text{space}}\; \underbrace{\bigl(1-\cos \omega_k t\bigr)}_{\text{time}} . \end{equation}\] The \(3\)D angular average of the space factor is the spherical Bessel function \(1-j_0(kd) = 1-\sin(kd)/(kd)\), and its radial integral is the Dirichlet integral \[\begin{equation} \int_0^\infty j_0(kd)\,dk \;=\; \frac{\pi}{2d}, \end{equation}\] whose half-value \(\pi/2\) is the spatial \(\pi/2\). It sits inside \(E_G\): \(E_G = (2GM^2/\pi d)\!\int_0^\infty j_0 = (2GM^2/\pi d)(\pi/2) = GM^2/d\). This \(\pi/2\) is an analytic constant and it is deformable: an infrared cutoff \(k>k_H=1/\RH\) replaces it by \(\pi/2-\Si(x)\), the form factor \(g(x)\) of Sec. 5.

The temporal \(\pi/2\): a geometric, rigid constant

The time factor governs orthogonalisation. The two-branch overlap amplitude is \(\propto\cos(E_G t/2\hbar)\), orthogonal at the Fubini–Study quarter-turn \(\theta=\pi/2\); the Margolus–Levitin inequality \(1-\cos x \le (2/\pi)(x+\sin x)\) is tight at \(x=\pi\) (the two-level orthogonalisation point), fixing the temporal \(2/\pi\). This \(\pi/2\) is a geometric constant—the angle from a ray to an orthogonal ray—and it is rigid: an infrared cutoff rescales the rate but never the orthogonality angle.

Distinctness via opposite cutoff response

The shared value is the coincidence that the Dirichlet integral and the Hilbert-space right angle are both \(\pi/2\). The horizon edits the spatial \(\pi/2\) (turning \(E_G\) into \(g(x)E_G\)) but cannot touch the temporal \(\pi/2\) (the orthogonality angle is geometric), so the cosmic rate is \(\Gamma_{\mathrm{cosmic}}= \tfrac12 g(1)\omega_\Lambda\): the horizon deforms the energy, not the rate-per-energy. This asymmetry—one constant analytic and horizon-edited, one geometric and untouchable—is the structural origin of the \(g(1)\) in the lock \(2/g(1)\).

The physically meaningful statement of the canonical core paper  is that the Diósi–Penrose rate sits at the Margolus–Levitin scale, of order \(E_G/\hbar\), not the \(10^{35}\) below it of perturbative QFT. The exact factor \(\Gamma_{\rm DP}/\Gamma_{\rm ML}=(E_G/\hbar)/(2E_G/\pi\hbar)=\pi/2\) is a definitional restatement—the ratio of the \(1/e\) decay time to the orthogonalisation time—and carries no information about whether the two \(\pi/2\)’s are the same object. Proposition [prop:twopi] settles that they are not.

Lab and cosmos as probes of the two siblings

A BMV/QGEM experiment  entangles two masses through their mutual gravity; the two-qubit concurrence is \(C(t)=|\sin(\Phi/2)|\) with entangling phase \(\Phi=\Delta E_{\rm BMV}\,t/\hbar\), maximal at \(\Phi=\pi\). The same mutual energy \(\Delta E_{\rm BMV}\) that entangles the pair also mutually decoheres it (the entanglement–decoherence duality ), so the maximal-entanglement time and the mutual-decoherence time obey, for any geometry, \[\begin{equation} \frac{\tau_{\rm BMV}(\text{max})}{\tau_{\rm dec}^{\rm mutual}} = \frac{\pi\hbar/\Delta E_{\rm BMV}}{2\hbar/\Delta E_{\rm BMV}} = \frac{\pi}{2}, \end{equation}\] the Fubini–Study quarter-turn—identical to the Margolus–Levitin orthogonalisation time, and independent of the masses (checked over \(10^{-15}\)–\(10^{-13}\) kg) and of the source geometry. This is the temporal, rigid \(\pi/2\): the lab is Minkowski (\(x=Hd/c\approx0\), \(g\to1\)), so the spatial Dirichlet \(\pi/2\) is undeformed and only the temporal one is exposed. A lab entanglement experiment measures the temporal sibling (the quarter-turn, geometry-fixed and horizon-untouchable); the de Sitter form factor \(g(1)=1-(2/\pi)\Si(1)\) measures the spatial sibling (the Dirichlet integral, truncated by the horizon). Lab and cosmos share the temporal \(\pi/2\) (geometric, the same everywhere); the entire difference is the spatial form factor (\(g=1\) in the lab, \(0.398\) at the horizon), which is why the cosmic suppression in \(\Gamma_{\mathrm{cosmic}}=\tfrac12 g(1)\omega_\Lambda\) is carried by \(g(1)\) alone.

A consequence, derived elsewhere: while the mutual channel obeys the clean \(\pi/2\), the single-particle self-decoherence is a different, much larger energy (\(E_G^{\rm self}\sim Gm^2/R\), set by the particle radius \(R\), versus \(\Delta E_{\rm BMV}\sim 2Gm^2\Delta x^2/d^3\)). Their ratio \(E_G^{\rm self}/\Delta E_{\rm BMV}\sim 0.6\,d^3/(R\Delta x^2)\sim10^{2\text{--}3}\) for realistic parameters means that, if QGC’s \(G^1\) self-decoherence is real, each superposition self-decoheres \(10^{2}\)–\(10^{3}\times\) faster than the BMV entanglement builds (and geometrically unavoidably so, since \(\tau_{\rm dec}^{\rm self}>\tau_{\rm BMV}\) would need \(R>\Delta x\)). BMV is thus, within QGC, also a \(G^1\)-vs-\(G^2\) discriminator: a positive entanglement result would constrain the very \(G^1\) self-decoherence scaling the framework rests on. Details and caveats (cutoff choice, pointer basis) are in the BMV note .

The same self-decoherence gives a near-term single-particle target (Fig. 1). For a uniform sphere the exact self-energy is \(E_G(\Delta x)=(GM^2/R)[\tfrac12 s^2-\tfrac{3}{16}s^3+\tfrac{1}{160}s^5]\) for \(s=\Delta x/R\le2\), saturating at \(1.2\,GM^2/R\) for \(\Delta x\gtrsim2R\). The decoherence time \(\tau_{G1}=\hbar/E_G\sim M^{-5/3}\) falls below a second only for \(M\gtrsim M_\star\approx7\times10^{-16}\) kg (a femtogram, \(R\sim0.4\,\mu\)m). A silica sphere of \(M=1.5\) fg in a \(\Delta x\approx1\,\mu\)m superposition held for \(1\) s gives \(\tau_{G1}=0.59\) s and visibility \(V=0.18\), versus \(V\simeq1\) for \(G^2\)—an unmistakable discriminant, requiring cryogenic extreme-high vacuum (\(\sim10^{-15}\) mbar) to suppress gas collisions below \(\Gamma_{G1}\). The first \(\gtrsim\)femtogram solid-object superposition held for \(\sim1\) s thus tests the \(G^1\) scaling directly .

\(g(1)\) as a sub-horizon energy fraction

Everything in Eq. \(\eqref{eq:lock}\) is rational geometry except \(g(1)\). The gravitational self-energy of the two-branch source is the Dirichlet integral above; on de Sitter the modes with \(k<k_H = 1/\RH\) are super-horizon and cannot drive decoherence within a Hubble time, so the surviving energy is the same integral cut at \(k_H\): \[\begin{equation} E_G^{\dS}(x) \;\propto\; \int_{k_H}^\infty \frac{\sin(kd)}{kd}\,dk \;=\; \frac{1}{d}\Bigl(\frac{\pi}{2}-\Si(x)\Bigr), \qquad x = k_H d = \frac{Hd}{c}. \end{equation}\] The form factor is the ratio of surviving to total energy, \[\begin{equation} g(x) \;=\; \frac{E_G^{\dS}(x)}{E_G^{\mathrm{Mink}}} \;=\; \frac{\tfrac{\pi}{2}-\Si(x)}{\tfrac{\pi}{2}} \;=\; 1-\frac{2}{\pi}\Si(x), \end{equation}\] so the \(2/\pi\) is the inverse Dirichlet integral, \(\Si(x)\) is the super-horizon energy removed, and \(x=1\) is forced because the horizon mode has \(d=\RH\). At \(x=1\) the super-horizon piece \(\Si(1)=0.946\) is \(60.2\%\) of \(\pi/2\), leaving

the sub-horizon fraction of the gravitational self-energy at \(d=\RH\). The merge number then decomposes transparently, \[\begin{equation} \frac{2}{g(1)} \;=\; \underbrace{\frac{4\pi}{2\pi}}_{\substack{\text{horizon area}\\/\text{KMS period}}} \Big/\; \underbrace{g(1)}_{\substack{\text{sub-horizon energy}\\\text{fraction at }\RH}} \;=\; \frac{\pi}{\tfrac{\pi}{2}-\Si(1)} \;=\; 5.029 : \end{equation}\] the universe decoheres in \(\sim 5\) modular ticks because the horizon area carries a factor \(2\) over the thermal period and only \(\sim 40\%\) of horizon-scale gravitational binding survives the super-horizon cut. The value is transcendental, not a hidden integer; the closeness \(g(1)\approx 2/5\) (to \(0.6\%\)) is a coincidence.

Redshift rigidity and the choice of horizon

The lock is redshift-rigid

Both \(a_0\) and \(\Gamma_{\mathrm{cosmic}}\) are the one clock \(\omega_\Lambda(z)\), and the decoherence form factor is pinned at \(x = H\RH/c = 1\) at every epoch (for a spatially flat universe the apparent-horizon radius is \(\RH = c/H\) exactly at all times ). Provided \(a_0\) and \(\Gamma_{\mathrm{cosmic}}\) are referred to the same horizon \(H\), every power of \(H(z)\) cancels: \[\begin{equation} \frac{a_0(z)}{c\,\Gamma_{\mathrm{cosmic}}(z)} \;=\; \frac{2}{g(1)} \quad\text{for all } z . \end{equation}\]

Which horizon sets the shared rate

The lock fixes the ratio; it does not fix which horizon sets the shared \(H\) (Fig. 2). Two readings compete:

• the instantaneous apparent-horizon rate \(H(z)\), giving an evolving \(a_0(z)=cH(z)/2\pi\) (a factor \(\sim 3\) larger at \(z=2\));

• the asymptotic event-horizon rate \(H_\Lambda = H_0\sqrt{\Omega_\Lambda}\), giving a constant \(a_0\).

The decoherence sector does not force the instantaneous branch, and the data point the other way: high-redshift rotation curves  find \(a_0\) consistent with constant over \(z=0\)–\(2\) and exclude strong evolution. The de Sitter decoherence calculation is most naturally referred to the asymptotic (event-horizon) rate \(H_\Lambda\); then both legs are constant, the lock holds trivially, and the framework is consistent with the data. Milgrom  frames the same \(a_H=cH(t)\) versus \(a_\Lambda=c\sqrt{\Lambda/3}\) choice as open. The robust prediction is the ratio \(2/g(1)\), not a particular \(a_0(z)\) law; the instantaneous branch remains a live, mildly disfavoured possibility that JWST/ELT and Gaia DR4 can settle.

The horizon clock family

Every causal horizon carries one clock \(\omega = \kappa/2\pi c\), surfacing three ways.

Two rows are textbook: the de Sitter Lyapunov exponent is exactly \(H\) , and the black-hole row is the Maldacena–Shenker–Stanford chaos bound \(\lambda\le 2\pi\kB T/\hbar\), saturated at horizons . The new content is the third row and its lock to the first: the same clock \(\omega\) that sets the chaos bound (via \(2\pi\)) and the acceleration scale (via \(c\)) also sets the decoherence rate (via an \(O(1)\) number). Black-hole chaos, the MOND scale, and cosmic decoherence are three projections of one horizon clock.

Intensive and extensive readings: dark energy as the fourth face

The readings above are intensive—properties of a single horizon mode of frequency \(\omega_\Lambda\). The horizon carries \(S_{\dS}=\pi c^5/(\hbar G H^2)\) such modes, and summing over them yields the framework’s two large-scale readings. The total horizon thermal energy is exactly \[\begin{equation} S_{\dS}\,\kB\TdS \;=\; \frac{c^5}{2GH} \;=\; \rho_{\rm crit}\,c^2\,V_H , \label{eq:equipartition} \end{equation}\] the critical energy of the Hubble volume (holographic equipartition, the de Sitter case of Padmanabhan’s relation ). Holographic dark energy  writes \(\rho_{\rm DE}=\alpha c^2H^2/G\) with \(\alpha=3\Omega_{\rm DE}/8\pi\); combining with Eq. \(\eqref{eq:equipartition}\) gives \(\rho_{\rm DE}=\Omega_{\rm DE}\,(S_{\dS}\kB\TdS)/(c^2V_H)\), so the dark-energy density is the fraction \(\Omega_{\rm DE}\) of the horizon’s thermal energy density, with the pure–de Sitter coefficient \(\alpha_\infty=3/8\pi=0.119\) fixed by the clock. The cosmic decoherence rate is likewise an extensive sum: \(\Gamma_{\mathrm{cosmic}}= S_{\dS}\times(\text{per-mode rate}) = g(1)H/4\pi\) . Thus

Dark energy and cosmic decoherence are siblings—both extensive, both carrying the holographic mode count \(S_{\dS}\)—while MOND and chaos are the intensive, per-mode readings. The horizon is one clock with \(S_{\dS}\) hands: read per hand it gives the acceleration scale and the chaos bound; read over all hands, the dark-energy density and the cosmic decoherence rate.

Established, new, and falsifiable content

Established (cited, not claimed)

The identification \(a_0 = c\kB\TdS/\hbar = cH/2\pi\) ; \(a_0\sim cH\) as a de Sitter/vacuum scale ; the apparent-horizon temperature \(\hbar H/2\pi\kB\) valid at all flat-FRW epochs ; the de Sitter modular-flow rate \(H/2\pi\) ; the chaos bound ; the de Sitter Lyapunov exponent .

New here

(i) The lock between the cosmic decoherence rate and the MOND scale as one \(\omega_\Lambda\), Eq. \(\eqref{eq:lock}\), with its three equal readings; (ii) the identity \(a_0 = c^2/2\pi\RH\) (Compton wavelength of the de Sitter thermal mass equals the horizon circumference); (iii) the reading of \(g(1)\) as the sub-horizon fraction of the gravitational self-energy, and the resulting decomposition of \(2/g(1)\); (iv) showing the two \(\pi/2\)’s are distinct (speed limit vs. form factor) via their opposite cutoff response, and the reading of a lab BMV experiment and the cosmic form factor as measurements of the temporal and spatial siblings respectively; (v) the horizon clock family with decoherence as a reading, and its intensive/extensive structure placing dark energy (via holographic equipartition) and cosmic decoherence as the two extensive readings. No prior work we are aware of links a gravitational or cosmological decoherence rate to \(a_0\), nor reads the holographic dark-energy coefficient as the extensive face of the same clock. We do not claim to solve the cosmological constant problem: the present \(\Omega_{\rm DE}\) is unexplained, only the structure and \(\alpha_\infty=3/8\pi\) are clock-fixed.

Falsifiable content

The robust prediction is the locked ratio \(a_0/(c\Gamma_{\mathrm{cosmic}}) = 2/g(1)\): the MOND scale and the cosmic decoherence normalisation are not independent. The cosmic decoherence rate itself (\(\sim 10^{-19}\) Hz) is not directly observable; the observable leg is \(a_0\) and its possible redshift evolution. A measured \(a_0(z)\) tracking the instantaneous \(H(z)\) would favour the apparent-horizon branch; the present constraint of approximate constancy  favours the event-horizon branch, with which the framework is consistent. The framework would be stressed—though not the lock itself—if \(a_0\) were found to depend on a scale unrelated to any horizon.

Conclusion

The acceleration scale at which galaxies stop obeying Newton, and the rate at which the universe as a whole goes classical, are the same de Sitter horizon frequency \(\omega_\Lambda= \kB\TdS/\hbar = H/2\pi\), read once as an acceleration and once as a rate, and locked at \(a_0/(c\Gamma_{\mathrm{cosmic}}) = 2/g(1) = 5.03\). The one non-trivial number in that ratio is the fraction of horizon-scale gravitational binding that survives the super-horizon cut. Whether the shared horizon is the instantaneous apparent horizon or the asymptotic event horizon is an open, data-driven question; the ratio is not. In the language of Quantum-Geometric Correspondence, the cosmological horizon has a single clock, and dark-sector phenomenology and cosmic decoherence are two of its hands.

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