The Horizon Clock: Why MOND and Cosmic Decoherence Are the Same Number

Paper P: The Horizon Clock from the Quantum-Geometric Duality Series

Two Coincidences That Refuse to Stay Separate

Two numbers in modern gravitational physics sit suspiciously close to the Hubble rate, and they are usually discussed in entirely different rooms.

The first is Milgrom's MOND scale, $a_0 \approx 1.2 \times 10^{-10}$ m/s$^2$. It is the acceleration below which galactic rotation curves stop obeying Newton. Numerically, $a_0 \approx cH_0/2\pi$. This is filed under galactic astronomy, or under "dark matter halos somehow conspire."

The second is the cosmic gravitational decoherence rate, $\Gamma_{\text{cos}} \approx g(1)\,H/(4\pi) \approx 7 \times 10^{-20}$ Hz, derived in Paper N for horizon-scale superpositions of the cosmic wavefunction. This is filed under the foundations of quantum mechanics—the scale at which the universe itself goes classical.

One belongs to flat rotation curves; the other to the measurement problem at cosmological scale. The usual reading is that both are unrelated numerical near-coincidences with the Hubble rate. The claim of Paper P is sharper: they are not coincidences, and they are not unrelated. They are two readings of one clock.

The Horizon Clock

The de Sitter horizon has a Gibbons–Hawking temperature $T_{dS} = \hbar H/(2\pi k_B)$. Any temperature defines a frequency. For the horizon this is

$\omega_L \;\equiv\; \frac{k_B T_{dS}}{\hbar} \;=\; \frac{H}{2\pi}.$

This is not just a number built from a temperature. It is the rate of Tomita–Takesaki modular flow on the de Sitter vacuum—the Connes–Rovelli "thermal time" of the universe. For a KMS state, the modular automorphism group runs in a parameter related to physical time by $t = s/\omega_L$. Recent explicit constructions of the de Sitter modular Hamiltonian confirm that, in the large-diamond limit, modular flow becomes static-patch time translation at exactly this rate. The $2\pi$ in $\omega_L = H/2\pi$ is the thermal (KMS) period—the same $2\pi$ that appears in every Unruh, Hawking, and Gibbons–Hawking temperature.

Given $\omega_L$, the two suspicious numbers are immediate:

$a_0 \;=\; c\,\omega_L, \qquad \Gamma_{\text{cos}} \;=\; \tfrac{1}{2}\,g(1)\,\omega_L.$

The MOND scale is the horizon clock carried by $c$—a frequency reinterpreted as an acceleration. The cosmic decoherence rate is the same clock carried by a pure number. Equivalently, $a_0 = \kappa_{dS}/(2\pi)$, where $\kappa_{dS} = cH$ is the de Sitter horizon's surface gravity. There is one frequency in the room, and two ways to dress it with dimensions.

The Lock

Dividing the two relations, every cosmological input cancels:

$\frac{a_0}{c\,\Gamma_{\text{cos}}} \;=\; \frac{2}{g(1)} \;=\; \frac{2}{1 - (2/\pi)\,\text{Si}(1)} \;=\; 5.029.$

No $H$, no $G$, no $\hbar$, no $c$. A framework that explained $a_0$ with one piece of physics and $\Gamma_{\text{cos}}$ with unrelated physics would have two independent knobs at this ratio. QGD has one. Measuring $a_0$ fixes $\Gamma_{\text{cos}}$ and vice versa.

The same number admits three equivalent readings. As an acceleration-to-rate conversion: $a_0 = 5.03 \, c \, \Gamma_{\text{cos}}$. As a count of modular ticks: the universe decoheres in $\omega_L/\Gamma_{\text{cos}} = 5.03$ ticks of its own thermal-time clock, covering 99.2% of its Bures–Fisher distance from coherent to maximally mixed. As an e-fold count: $\Gamma_{\text{cos}}^{-1} \approx 10\pi\,H^{-1}$, ten $\pi$ e-folds, which is exactly five modular ticks because each tick is $2\pi$ e-folds.

The only slack in the lock at fixed cosmology is the Hubble tension itself: the two source papers quote $a_0$ at $H_0 = 70$ and $\Gamma_{\text{cos}}$ at $H_0 = 67.4$, a ratio of about 1.04. There is no other.

The One Non-Trivial Number

Everything in the ratio is rational geometry except $g(1)$. Its meaning is concrete.

The gravitational self-energy of a two-branch source is the Dirichlet integral $\int_0^\infty j_0(kd)\,dk = \pi/(2d)$. On de Sitter, modes with $k < k_H = 1/R_H$ are super-horizon and cannot drive decoherence within a Hubble time. Cutting the integral at $k_H$ and dividing by the uncut version gives

$g(x) \;=\; \frac{\frac{\pi}{2} - \text{Si}(x)}{\frac{\pi}{2}} \;=\; 1 - \frac{2}{\pi}\,\text{Si}(x), \qquad x = Hd/c.$

At the horizon scale $x = 1$, the super-horizon piece $\text{Si}(1) = 0.946$ removes 60.2% of $\pi/2$, leaving $g(1) = 0.398$. Read it directly: $g(1)$ is the sub-horizon fraction of the gravitational self-energy at one Hubble radius. The $2/\pi$ in the form factor is just the inverse Dirichlet integral; $\text{Si}(1)$ is the energy that lives on modes the horizon cannot resolve. The number is transcendental, not a hidden integer. Its closeness to $2/5$ (within 0.6%) is a coincidence.

Two $\pi/2$'s That Look the Same and Are Not

The framework carries a $2/\pi$ in two places: in the Margolus–Levitin speed limit of Paper H, and in the form factor here. It is tempting to call them two faces of one object. They are not.

The space-time factorisation of the decoherence functional makes the distinction sharp. The spatial $\pi/2$ is the Dirichlet integral; the temporal $\pi/2$ is the Fubini–Study quarter-turn—the orthogonalisation angle between a state and its image, geometric in Hilbert space. Under an infrared cutoff, the spatial weight slides from $\pi/2$ to zero as $g(x)$. The orthogonalisation angle does not move. A function with non-zero derivative and a function with zero derivative in the same variable are not the same function.

The horizon edits the energy ($E_G \to g(x)\,E_G$) but cannot touch the rate-per-energy. That asymmetry is the structural origin of the $g(1)$ in $2/g(1)$. It also predicts a clean experimental division of labour: a tabletop BMV experiment lives in the Minkowski limit ($g \to 1$) and measures only the temporal, rigid $\pi/2$ via the entanglement-to-decoherence ratio $\tau_{\text{BMV}}^{\max}/\tau_{\text{dec}}^{\text{mutual}} = \pi/2$. The cosmic form factor $g(1) = 0.398$ measures the spatial, horizon-deformed sibling. Lab and cosmos probe the two $\pi/2$'s separately.

The Horizon Clock Family

The construction generalises. Every causal horizon carries one clock $\omega = \kappa/(2\pi c)$, and it surfaces three ways: as an acceleration scale $a_\star = \kappa/(2\pi)$ (de Sitter gives MOND; a black hole gives $c^4/(8\pi GM)$), as a Lyapunov/chaos rate $\lambda = \kappa/c$ (de Sitter Lyapunov is $H$; a black hole saturates the Maldacena–Shenker–Stanford bound $2\pi k_B T/\hbar$), and as a decoherence rate of order $\omega$ times an $O(1)$ form factor.

There is a second axis, no less important. The readings above are intensive—properties of one horizon mode. The horizon also carries $S_{dS} = \pi c^5/(\hbar G H^2)$ such modes, and summing over them gives the extensive readings. The holographic equipartition identity is exact:

$S_{dS}\,k_B T_{dS} \;=\; \frac{c^5}{2GH} \;=\; \rho_{\text{crit}}\,c^2\,V_H.$

Combined with the holographic dark energy ansatz $\rho_{\text{DE}} = \alpha c^2 H^2/G$ of Paper B, dark energy density becomes the extensive partner of $a_0$, and cosmic decoherence is the extensive partner of the chaos rate. Dark energy and cosmic decoherence are siblings, both extensive, both carrying the holographic mode count. MOND and chaos are the intensive, per-mode siblings. The horizon is one clock with $S_{dS}$ hands.

What Is Established, What Is New, What Is Falsifiable

The discipline of this kind of synthesis is the bookkeeping.

Established (cited, not claimed): the identification $a_0 = c k_B T_{dS}/\hbar$ in Klinkhamer–Kopp (2011); the qualitative $a_0 \sim cH$ in Milgrom's vacuum-effect proposal and in Verlinde's emergent gravity; the apparent-horizon temperature at all flat-FRW epochs (Cai–Kim 2005); the de Sitter modular rate (Connes–Rovelli); the chaos bound (MSS 2016).

New here: the lock between cosmic decoherence and the MOND scale; the reading of $g(1)$ as the sub-horizon fraction of the gravitational self-energy; the distinctness theorem for the two $\pi/2$'s, with the lab/cosmos division of labour; the horizon clock family with its intensive/extensive structure placing dark energy and cosmic decoherence as the two extensive readings.

Not claimed: a solution to the cosmological constant problem. The fitted coefficient $\alpha = 3\Omega_{\text{DE}}/(8\pi) \approx 0.082$ remains a fitted number. The structure is clock-fixed; the value is not.

Falsifiable content: the robust prediction is the locked ratio $2/g(1)$. The cosmic decoherence rate itself is $\sim 10^{-19}$ Hz and not directly observable; the observable leg is $a_0$ and its possible redshift dependence. If the shared horizon is the instantaneous apparent horizon, $a_0(z) \propto H(z)$ and rises by a factor of three by $z = 2$. If it is the asymptotic event horizon, $a_0$ is constant. Current high-redshift rotation curves favour the constant branch, with which the framework is consistent. JWST, ELT, and Gaia DR4 will resolve the fork within the decade.

What Has Been Gained

A coincidence between two numbers near $H_0$ has been compressed into the surface gravity of one horizon, read twice. The MOND acceleration is the horizon's surface gravity divided by $2\pi$. The cosmic decoherence rate is the same clock, multiplied by the sub-horizon energy fraction at one Hubble radius. The lock between them is parameter-free, and the only non-trivial number in the ratio has a concrete geometric reading.

The contribution is structural, not empirical. The cosmological constant problem persists; $\alpha$ is still fitted; the apparent-versus-event-horizon question is still open. What has changed is the topology of the unknown. Galactic dynamics and the cosmic measurement problem are not two mysteries near the Hubble rate. They are one fact about the horizon, with two hands.

This is Paper P of the Quantum-Geometric Duality series. It builds on Paper N's cosmic decoherence rate, Paper B's holographic dark energy formulation, and Papers H, K, and M's account of the temporal $\pi/2$, and connects the framework to MOND through a single horizon thermodynamic identity.