Introduction

Two of cosmology’s oldest puzzles concern quantities separated by \(120\) orders of magnitude, and both are questions about the same horizon.

Why is the dark-energy density small but nonzero? Quantum field theory estimates a vacuum energy \(\rho_{\mathrm{vac}}\sim\rho_{\mathrm{Pl}}\sim 10^{113}\) J/m\(^3\); cosmic acceleration  requires \(\rhoDE\sim 10^{-9}\) J/m\(^3\). The ratio \(\rhoDE/\rho_{\mathrm{Pl}}\sim 10^{-122}\) is the largest disagreement between theory and observation in physics . Anthropic selection , quintessence , and modified gravity  relocate rather than resolve the question of why the scale is so small.

Why was the early universe in a state of extraordinarily low entropy? The microscopic dynamics of statistical mechanics is time-symmetric, yet the macroscopic world is not. The standard resolution adds an initial-condition postulate by hand—the Past Hypothesis : at some early epoch the universe occupied a microstate of extraordinarily low coarse-grained entropy, from which the Second Law follows by ordinary statistical reasoning. Its sharp geometric form is Penrose’s Weyl Curvature Hypothesis : the Weyl tensor \(C_{\mu\nu\rho\sigma}\), the tidal gravitational-wave part of curvature, is extraordinarily close to zero at the Big Bang and diverges at black-hole singularities, an asymmetry Penrose estimated at fine-tuning \(\exp(-10^{123})\). The postulate is necessary but unmotivated: nothing in the dynamics demands it.

One budget, spent twice

The thesis of this paper is that both puzzles are statements about a single quantity, the Bekenstein–Hawking entropy of the cosmological horizon, \[\begin{equation} \SdS(t)\;=\;\frac{A_H(t)}{4\,\lP^2}\;=\;\frac{\pi c^5}{H(t)^2\,\hbar\,G},\qquad \lP=\sqrt{\hbar G/c^3}, \label{eq:S-dS-intro} \end{equation}\] the holographic capacity of a comoving observer’s causal patch at Hubble rate \(H(t)\). The budget is used twice.

First, for dark energy. Equipartition of the horizon’s thermal energy against the critical energy—equivalently, the holographic bound with the horizon as infrared cutoff—gives \[\begin{equation} \rhoDE\;=\;\alpha\,\frac{c^2 H^2}{G},\qquad \alpha = \frac{3\,\Omega_{\mathrm{DE}}}{8\pi}\approx 0.082. \label{eq:hde-intro} \end{equation}\] Extending the generalized second law to the cosmological horizon makes de Sitter space the thermodynamically consistent late-time attractor; saturation there pins the equation of state to \(w=-1\) exactly.

Second, for the arrow of time. The same budget, applied to the shrinking past horizon where \(H(t)\to\infty\) and hence \(\SdS(t)\to 0\) as \(t\to 0\), implements the Past Hypothesis as a kinematic consequence of the holographic bound \(S(t)\leq\SdS(t)\): there is no room for high entropy in the early universe. Penrose’s \(10^{123}\) becomes \(\SdS^{\infty}\) in nats—the capacity of the cosmic horizon, not a probability assigned to the Big Bang. The smallness of \(\rhoDE\) in Planck units is the reciprocal of that capacity. The Weyl asymmetry is the direction of cosmic decoherence: the Big Bang upstream of all branching, black-hole singularities downstream.

The two applications are tied by an exact identity in the de Sitter attractor of any FRW cosmology, \[\begin{equation} \rhoDE^{\infty}\;\SdS^{\infty}\;=\;\frac{3\,c^7}{8\,G^2\,\hbar}, \label{eq:central-identity-intro} \end{equation}\] a Planck-scale constant. Both sides depend on \(H_{\infty}\) alone: dark-energy density and cosmic entropy capacity are inverses of each other up to that constant. Reducing \(\rhoDE\) raises the capacity, and conversely; the two are not independent.

Scope, stated up front

The construction is a conceptual rearrangement, and its claims carry different weights, stated here and once more in full in Section 7.

The identity \(\eqref{eq:central-identity-intro}\) is a re-labeling, not a reduction: both sides are functions of \(H_{\infty}\), and the algebra occupies three lines. Versions appear in the holographic-dark-energy literature from Banks , Fischler–Susskind , and Cohen–Kaplan–Nelson  onward. The novelty here is the coupling: one horizon budget serves dark energy and the arrow of time within one framework.

This paper belongs to the Quantum-Geometric Correspondence (QGC) series. The canonical core paper  presents the three primitive axioms, the gravitational-decoherence prediction, and the holographic bound adopted here as premise; companion papers treat emergent gravity with MOND phenomenology , the laboratory decoherence floor , and cosmic gravitational decoherence on a de Sitter background . Each paper is self-contained.

Section 2 states the holographic capacity and tabulates the cosmic entropy budget. Section 3 derives the dark-energy formula, the saturation parameter, and \(w=-1\), and addresses the coincidence problem. Section 4 gives the observational status against DESI and Planck. Section 5 derives the central identity. Section 6 develops the arrow of time—the Past Hypothesis and the Weyl asymmetry—and why \(\Lambda>0\) closes the budget. Section 7 states the limits once for the whole paper and compares with alternatives; Section 8 concludes.

Conventions. We work in geometric and Planck units interchangeably, with conversions given where ambiguity may arise. The Hubble parameter \(H(t)=\dot{a}/a\) is positive throughout the cosmic history considered. The Bekenstein–Hawking entropy is \(S=A/(4\lP^2)\) with \(\lP=\sqrt{\hbar G/c^3}\). We write \(\SdS(t)\equiv\pi c^5/(H(t)^2\hbar G)\) for the de Sitter horizon entropy at instantaneous Hubble rate \(H(t)\), and \(\SdS^{\infty}\equiv\SdS(H_{\infty})\) for its asymptotic value at \(H_{\infty}=H_0\sqrt{\Omega_{\mathrm{DE}}}\).

The Horizon Budget

The central object is the cosmic entropy capacity \(\SdS(t)\): the Bekenstein–Hawking entropy of the cosmological horizon, read as the holographic bound on the observer’s causal patch. This section states the two principles the budget rests on, identifies the relevant horizons, establishes the de Sitter attractor at which the budget saturates, and tabulates the budget across cosmic history.

Two principles from black-hole physics

Both principles are imported from black-hole thermodynamics and extended to the cosmological context. Neither has been rigorously proven for cosmological horizons; each is a working assumption whose consistency the framework tests.

The bound originates in black-hole physics: black holes carry entropy proportional to horizon area, and packing more entropy into a region induces gravitational collapse . Within QGC it is a theorem of the framework, following from the first primitive axiom together with the generalized second law ; here we adopt it as premise. It underlies the holographic ansatz \(\rhoDE\propto 1/L^2\) of Section 3.

The horizon entropy takes the Bekenstein–Hawking form \(S_{\mathrm{horizon}}=A/(4\lP^2)\) . For black holes the generalized second law (GSL) is well-established by semiclassical arguments . Its extension to cosmological horizons is physically motivated but subtle: such horizons are observer-dependent, and their thermodynamic interpretation remains open. The framework proceeds on the assumption that the thermodynamic description holds, and tests it by consistency.

The two cosmological horizons

An expanding universe possesses horizons that limit causal access and carry thermodynamic properties analogous to black-hole horizons.

The Hubble horizon \(r_H=c/H\), with \(H=\dot{a}/a\), is the distance at which recession reaches the speed of light. In de Sitter space an observer perceives thermal radiation at \(T_H=\hbar H/(2\pi k_B)\) , of order \(10^{-30}\) K today.

The future event horizon \[\begin{equation} R_h(t) = a(t)\int_t^\infty\frac{c\,dt'}{a(t')} \label{eq:event-horizon} \end{equation}\] is the boundary beyond which events are never observable; it depends on the entire future evolution, not on instantaneous data. It carries temperature \(T_E=\hbar c/(2\pi k_B R_h)\), structurally the Hawking form at the event-horizon scale. In de Sitter space \(R_h=c/H\) and the two temperatures coincide, \(T_H=T_E\); this equality characterizes the late-time attractor (Section 2.3).

The choice of the event horizon over the Hubble horizon as infrared cutoff is empirical. The Hubble cutoff predicts \(w=0\)—dark energy tracks matter, producing no acceleration—ruled out at more than \(30\sigma\) . The event-horizon cutoff yields \(w=-1\), consistent with all current data. It is chosen because it works, not because it is derived from first principles; the theoretical justification remains incomplete.

The de Sitter attractor

Any cosmology dominated by dark energy with \(w<-1/3\) asymptotically approaches de Sitter space—a consequence of the Friedmann equations, not a prediction. In de Sitter space \(R_h=c/H\), so \[\begin{equation} HR_h = c\qquad\text{(de Sitter geometric identity)}. \label{eq:deSitter-condition} \end{equation}\]

The budget across cosmic history

Applied to the de Sitter horizon of a comoving observer at instantaneous Hubble rate \(H(t)\), the holographic bound \(\eqref{eq:holo-bound}\) becomes a definite finite number. With Hubble area \(A_H=4\pi R_H^2\) and \(R_H=c/H\), \[\begin{equation} \SdS(t)\;=\;\frac{A_H(t)}{4\,\lP^2}\;=\;\frac{4\pi c^2/H(t)^2}{4\,\hbar G/c^3}\;=\;\frac{\pi c^5}{H(t)^2\,\hbar\,G}. \label{eq:S-dS} \end{equation}\] This is the capacity: the maximum entropy any subsystem in the observer’s causal patch can carry, by Axiom 1. It is observer-dependent, but in a homogeneous FRW universe all comoving observers share the same \(\SdS(t)\).

Equation \(\eqref{eq:S-dS}\) tracks the dominant energy component through \(H(t)\): \(H\propto a^{-2}\) in radiation, \(H\propto a^{-3/2}\) in matter, \(H\to H_{\infty}\) in the de Sitter attractor. The capacity grows monotonically from its smallest value at the Planck epoch to a finite asymptotic value \(\SdS^{\infty}\) set by the dark-energy density . Table 1 lists \(\SdS\) at six epochs; the growth from \(\approx\pi\) nats at the Planck time to \(\approx 3\times 10^{122}\) nats in the asymptotic future is the structural fact on which the paper hinges.

The values span \(122\) orders of magnitude. Two features recur:

• \(\SdS(t)\to 0\) as \(t\to 0\), forced by \(H\to\infty\) in any FRW cosmology with a past singularity. This is the kinematic ingredient of the Past Hypothesis (Section 6).

• \(\SdS(t)\to\SdS^{\infty}\) as \(t\to\infty\) with \(\SdS^{\infty}\) finite, provided \(\Lambda>0\). Without dark energy the asymptotic capacity is either undefined (recollapse) or infinite (Minkowski limit). The finiteness of \(\SdS^{\infty}\) is exactly the structural role of dark energy in this account.

From bound to budget: saturation

A bound is not a budget. The inequality \(\eqref{eq:holo-bound}\) states \(S(t)\leq\SdS(t)\) without requiring equality; whether the actual cosmic entropy tracks the bound or sits below it is a separate question. Assumption 1 answers it: the cosmic-horizon entropy saturates in the de Sitter attractor, so the actual entropy tracks the capacity.

The assumption is non-trivial. Within QGC it is motivated by the cosmic gravitational decoherence rate : per-mode gravitational decoherence at rate \(g(1)\,H/(4\pi)\) with \(g(1)=1-(2/\pi)\,\mathrm{Si}(1)\approx 0.398\), multiplied by the holographically-allowed number of horizon modes, gives a total cosmic entropy production rate scaling as \(H\). That cosmic rate holds within a controlled approximation assuming decoherent-histories branching, holographic finiteness of the mode count, and free/Gaussian factorization of mode contributions . The per-mode rate alone does not saturate the bound: integrated over the age of the universe (\(N_{\mathrm{efolds}}\sim 140\)), it gives only \(\sim N_{\mathrm{efolds}}/(10\pi)\sim 4\) nats per mode, far short of \(\SdS^{\infty}\sim 10^{122}\). The \(122\)-order shortfall is closed by the mode count: each Hubble volume contains \(\SdS\) holographically distinguishable horizon modes, and summing the per-mode contribution over this count multiplies by precisely \(\SdS\). The saturation is supplied by the mode count, not the per-mode rate; Assumption 1 requires the count to do the structural work. Appendix 10 works the numerics and flags the residual \(O(1)\) ambiguity.

Under Assumption 1 the budget \(\SdS(t)\) is the actual cosmic entropy at epoch \(t\), not merely an upper bound. Three readings recur, worked in the following sections: the dark-energy density (Section 3), the reciprocal cosmological-constant hierarchy (Section 5), and the Past Hypothesis with its Weyl-asymmetry corollary (Section 6).

Dark Energy from the Budget

The first use of the horizon budget is the dark-energy density. The holographic ansatz with the horizon scale as infrared cutoff fixes the \(H^2\) scaling; the de Sitter attractor of Section 2.3 fixes the coefficient’s structure, the saturation parameter, and the equation of state.

The \(H^2\) scaling and the coefficient \(\alpha\)

For a region of characteristic size \(L\) the holographic ansatz posits \(\rhoDE\propto 1/L^2\). With \(L\) the horizon scale \(c/H\), dimensional analysis gives \[\begin{equation} \rhoDE\;=\;\alpha\,\frac{c^2 H^2}{G}, \label{eq:hde-main} \end{equation}\] since \(c^2 H^2/G\) carries the dimensions of energy density. This is dimensional analysis given the ansatz, not a first-principles derivation; the coefficient \(\alpha\) encodes the unexplained physics. Equation \(\eqref{eq:hde-main}\) is the late-time de Sitter relation, where the event horizon \(R_h=c/H\) coincides with the Hubble scale; \(\alpha\) is read off at the attractor.

To determine \(\alpha\), substitute \(\eqref{eq:hde-main}\) into the Friedmann equation. In a flat universe of matter and dark energy, \[\begin{equation} H^2 = \frac{8\pi G}{3c^2}(\rho_m + \rhoDE). \end{equation}\] Substituting \(\rhoDE=\alpha c^2 H^2/G\) and solving, \[\begin{equation} H^2 = \frac{8\pi G\rho_m}{3c^2(1 - 8\pi\alpha/3)}. \end{equation}\] Positivity \(H^2>0\) requires the self-consistency bound \(\alpha<3/(8\pi)\approx 0.119\); larger \(\alpha\) would make the dark-energy contribution exceed the total. With \(\rho_{\mathrm{crit}}=3c^2 H^2/(8\pi G)\), the dark-energy fraction is \[\begin{equation} \Omega_{\mathrm{DE}} = \frac{\rhoDE}{\rho_{\mathrm{crit}}} = \frac{\alpha c^2 H^2/G}{3c^2 H^2/(8\pi G)} = \frac{8\pi\alpha}{3}, \end{equation}\] inverted to \[\begin{equation} \alpha = \frac{3\Omega_{\mathrm{DE}}}{8\pi}. \label{eq:alpha-def} \end{equation}\] The observed \(\Omega_{\mathrm{DE}}=0.689\pm 0.006\)  yields \[\begin{equation} \alpha\approx 0.082. \label{eq:alpha-value} \end{equation}\] Propagating the statistical uncertainty in \(\Omega_{\mathrm{DE}}\) alone gives a formal error at the one-percent level; this figure is statistical only and does not account for the \(H_0\)-tension systematic, so we quote \(\alpha\approx 0.082\) rather than a sharp \(\pm 0.001\).

Cosmic acceleration requires \(\Omega_{\mathrm{DE}}>1/2\), i.e. \(\alpha>3/(16\pi)\approx 0.060\). With the self-consistency bound, \[\begin{equation} 0.060\lesssim\alpha\lesssim 0.119, \label{eq:alpha-range} \end{equation}\] a range spanning a factor of two, within which \(\alpha\approx 0.082\) falls. The magnitude is not explained: the mystery of why \(H_0\ll H_P\) is equivalent to the mystery of why \(\Lambda\ll\Lambda_P\). The claim that \(\alpha\sim 0.1\) is “natural” is tautological, since \(\alpha=3\Omega_{\mathrm{DE}}/(8\pi)\) is of order \(0.1\) whenever \(\Omega_{\mathrm{DE}}=O(1)\), which holds by definition at the present epoch. The framework reparameterizes the cosmological constant problem; it does not solve it (Section 7).

The saturation parameter and the equation of state

The equation of state \(w=p_{\mathrm{DE}}/\rhoDE\) governs the dark-energy dynamics. For an event-horizon cutoff, the standard derivation (Appendix 9.2) gives \[\begin{equation} w = -\frac{1}{3} - \frac{2\sqrt{\Omega_{\mathrm{DE}}}}{3\xi}, \label{eq:w-general} \end{equation}\] where \(\xi\) is the saturation parameter of the ansatz \(\rhoDE=3\xi^2 M_P^2/R_h^2\). Li’s original holographic dark-energy proposal  took \(\xi=1\), which with \(\Omega_{\mathrm{DE}}=0.689\) gives \(w_0=-0.88\), in mild tension with observations; the Hubble-scale cutoff gives the degenerate \(w=0\) .

When the de Sitter saturation condition \(\xi=\sqrt{\Omega_{\mathrm{DE}}}\) holds, holographic dark energy has the equation of state of a cosmological constant. Current observations give \(w_0=-1.03\pm 0.03\), consistent with this value.

The attractor is dynamically stable: a perturbation \(\delta\Omega_m\) about the de Sitter fixed point evolves as \(\delta\Omega_m\propto e^{\lambda N}\) with \(N=\ln a\) and \(\lambda=-3\Omega_{\mathrm{DE}}<0\), decaying over roughly half an e-fold, so initial conditions in the basin converge to the de Sitter endpoint.

The coincidence problem, ameliorated

The coincidence problem asks why the dark-energy and matter densities are comparable today, when their ratio has varied enormously over cosmic history. In \(\Lambda\)CDM, \(\Lambda\) has a fixed density while \(\rho_m\propto a^{-3}\), so \(\rho_\Lambda/\rho_m\propto a^3\) grows by a factor \(\sim 10^{30}\) from matter–radiation equality to the distant future; observing it at order unity is either explained or accepted as coincidence.

Here \(\rhoDE=\alpha c^2 H^2/G\) tracks the expansion rate rather than staying constant. During matter domination \(H^2\propto\rho_m\), so \(\rhoDE\propto\rho_m\) and \[\begin{equation} \frac{\rhoDE}{\rho_m} = \frac{\Omega_{\mathrm{DE}}}{1-\Omega_{\mathrm{DE}}} = \frac{8\pi\alpha/3}{1-8\pi\alpha/3}, \label{eq:tracking-ratio} \end{equation}\] constant during matter domination. Dark energy and matter keep a fixed proportion, removing the timing coincidence of \(\Lambda\)CDM: no time during matter domination is distinguished. The transition to dark-energy domination occurs when \(\Omega_m\sim\Omega_{\mathrm{DE}}\) by definition, an epoch set by \(\alpha\) rather than fine-tuned.

This is a rephrasing, not a solution. The timing aspect is removed, but the coincidence is relocated from cosmic time to the parameter \(\alpha\): the puzzle becomes why \(\alpha\approx 0.08\). The bounds \(\eqref{eq:alpha-range}\) within which \(\alpha\) produces an accelerating universe span a factor of two, and the naturalness argument is circular. The magnitude of \(\alpha\) encodes the same mystery as the cosmological constant—why \(H_0\sim 10^{-18}\) s\(^{-1}\) rather than \(H_P\sim 10^{44}\) s\(^{-1}\), why \(\rhoDE^{1/4}\sim 10^{-3}\) eV rather than the Planck energy. The \(10^{120}\) discrepancy reappears in disguised form; the coincidence problem is transformed, not solved.

The dark-energy formula \(\rhoDE=\alpha c^2 H^2/G\) thus follows from dimensional analysis given the holographic ansatz; the coefficient \(\alpha=0.082\) is fitted; the de Sitter condition determines \(\xi=\sqrt{\Omega_{\mathrm{DE}}}\) as a consistency requirement, yielding \(w=-1\). The framework is internally consistent but does not explain the value of \(\alpha\) or supply a microscopic mechanism for dark energy.

Observational Status

The framework makes definite statements about cosmological observables that, at present precision, do not distinguish it from \(\Lambda\)CDM. The predictions are either already known and \(\Lambda\)CDM-consistent, or in-principle testable but parametrically out of reach.

The core prediction and its falsifiers

Dark energy arises from the holographic bound on the future event horizon, giving \(\rhoDE=\alpha c^2 H^2/G\) with \(\alpha\approx 0.082\) and a de Sitter attractor with \(w=-1\). Combined Planck, baryon acoustic oscillation, and Type Ia supernova data give \(w_0=-1.03\pm 0.03\), consistent with the prediction; the DESI hints of time-varying \(w(z)\) remain at low significance and subject to systematics.

Three classes of measurement would falsify the framework.

1. Equation of state. A \(w(z)\) evolution incompatible with a constant near \(-1\) at greater than \(3\sigma\) would rule out the holographic ansatz as formulated, requiring either \(\xi\neq\sqrt{\Omega_{\mathrm{DE}}}\) against the consistency analysis or a failure of the horizon relation at some epoch. Euclid and the Vera Rubin Observatory’s LSST will constrain \(w(z)\) at the sub-percent level across redshift bins.

2. Modified gravity. The framework operates within general relativity with \(G_{\mathrm{eff}}=G_N\) exactly; a measured \(G_{\mathrm{eff}}(z)/G_N\neq 1\) at the percent level—via lensing statistics or the integrated Sachs–Wolfe effect—would rule it out. \(f(R)\) and scalar-tensor theories generically predict \(G_{\mathrm{eff}}\neq G_N\), a clean discriminant.

3. Early dark energy. The CMB constrains \(\Omega_{\mathrm{DE}}(z=1100)\lesssim 10^{-4}\), whereas the tracking value during matter domination is \(\approx 0.7\). The tracking analysis of Section 3.3 applies only during matter domination; the radiation era and earlier follow a different scaling, requiring a full solution of the Friedmann equations across all epochs, deferred to future work. Early dark energy beyond the CMB bounds would require re-examining whether the framework accommodates it.

Table 2 separates what is derived as a consistency condition, what follows from general relativity, and what is fitted to observations.

CMB consistency and late-time expansion

With \(w=-1\) from the de Sitter attractor, the framework is observationally indistinguishable from a cosmological constant at the level of the equation of state, consistent with \(w_0=-1.03\pm 0.03\).

The integrated Sachs–Wolfe (ISW) effect matches \(\Lambda\)CDM. For \(w=-1\), gravitational potentials decay at the standard rate, so at multipoles \(\ell<30\), where the ISW contributes roughly \(10\%\) of the temperature power, the framework predicts no deviation from \(\Lambda\)CDM; any difference lies within cosmic variance (\(\sigma\sim 30\%\) at \(\ell=2\)). The model with \(\alpha=0.082\) is consistent with current CMB constraints from DESI 2024 baryon acoustic oscillations and Planck. The modified-gravity parameters from the ISW–lensing bispectrum, \(\mu_0=0.05\pm 0.22\) and \(\Sigma_0=0.008\pm 0.045\), are consistent with general relativity. The primary acoustic peaks (\(30<\ell<1000\)) show no deviation from \(\Lambda\)CDM, since they probe recombination, where dark energy contributed negligibly.

The saturated attractor admits a sharper test than consistency. Any small departure of \(w_{\mathrm{eff}}(z)\) from \(-1\) at \(z\sim 0.5\)–\(1\)—the generic signature of a budget not yet fully saturated—would modify the late-time ISW effect at \(\ell<30\) by \(1\)–\(2\%\), below current Planck cosmic variance but potentially accessible to a combined CMB-S4 + large-scale-structure analysis this decade. The saturated reading, for which \(w=-1\) holds exactly, requires the absence of that signature; a detection would localize the epoch at which saturation completes and falsify the exact-\(w=-1\) statement.

The negative result: CMB non-Gaussianity

The cosmic gravitational decoherence underlying the entropy budget (Section 2.5) might have imprinted a measurable CMB non-Gaussianity. The calculation gives a negative verdict.

The per-mode accumulated decoherence functional from horizon exit during inflation to last scattering, with the de Sitter form factor \(g(x)=1-(2/\pi)\,\mathrm{Si}(x)\) , is \[\begin{equation} D_k(\eta_{\mathrm{LSS}})\;=\;\frac{1}{4\pi}\int_{\ln a_*(k)}^{\ln a_{\mathrm{LSS}}} g\!\left(\frac{aH}{ck}\right) d\ln a\;\approx\;0.35\text{--}0.51\quad\text{per mode}, \label{eq:Dk-LSS} \end{equation}\] the dominant contribution coming from the deep sub-horizon period during inflation. The induced non-Gaussianity is a multiplicative suppression of the standard inflationary value, \[\begin{equation} f_{\mathrm{NL}}^{\mathrm{QGC}}\;=\;e^{-D_k(\eta_{\mathrm{LSS}})}\,f_{\mathrm{NL}}^{\mathrm{standard}},\qquad |\Delta f_{\mathrm{NL}}^{\mathrm{QGC}}|\;\sim\;5\times 10^{-3}\text{--}1.5\times 10^{-2}. \label{eq:fnl-QGC} \end{equation}\] The current Planck 2018 bound is \(f_{\mathrm{NL}}^{\mathrm{local}}=-0.9\pm 5.1\) ; the CMB-S4 target sensitivity is \(\sigma(f_{\mathrm{NL}})\sim 1\) . The prediction is \(60\)–\(200\) times below CMB-S4 sensitivity: the induced non-Gaussianity is not observable at any planned CMB experiment.

Compared to the standard squeezing-induced classicality mechanism of Kiefer, Polarski, and Starobinsky : the squeezing parameter \(r_k(\eta)=\ln(a/a_*)\) reaches \(r\sim 60\) for CMB pivot modes by the end of inflation, an \(e^{2r}\sim 10^{52}\) phase-space elongation that classicalizes inflationary perturbations without genuine decoherence. The squeezing rate \(r_{\mathrm{dot}}=H\) is \(\sim 30\) times the per-mode decoherence rate \(g(1)\,H/(4\pi)\approx 0.032\,H\): \[\begin{equation} \frac{\Gamma_{\mathrm{QGC}}}{r_{\mathrm{dot}}}\;=\;\frac{g(1)}{4\pi}\;\approx\;0.032. \label{eq:squeezing-vs-decoherence} \end{equation}\] Squeezing-induced classicality dominates the apparent quantum-to-classical transition of inflationary perturbations; the gravitational-decoherence contribution is real but observationally redundant.

The decisive tests are elsewhere

The decisive laboratory test of the underlying framework is gravitational decoherence at the microgram scale. The predicted rate \[\begin{equation} \Gamma_{\mathrm{dec}}\;=\;\frac{GM^2}{\hbar\,d}\;\equiv\;\frac{E_G}{\hbar} \label{eq:gamma-bmv} \end{equation}\] applies to two coherent mass distributions of mass \(M\) separated by distance \(d\).¹ The Bose–Marletto–Vedral experiments  test whether two masses become gravitationally entangled. For \(M\sim 10^{-9}\) kg and \(d\sim 10^{-3}\) m, \(\Gamma_{\mathrm{dec}}^{-1}\sim 10^{-9}\) s . Confirmation at the predicted rate supports the gravitational-decoherence mechanism underlying the cosmic-decoherence argument (Section 2.5); the present framework inherits its testability through that connection.

Table 3 collects the predictions with their status.

At present precision the framework predicts nearly identically to \(\Lambda\)CDM: both give \(G_{\mathrm{eff}}=G_N\) and \(w\) within current uncertainties. It remains falsifiable in principle—\(w\) significantly different from \(-1\), or \(G_{\mathrm{eff}}\neq G_N\), would rule it out—and will face stringent tests as precision improves. The value is conceptual organization, a thermodynamic language connecting dark energy to horizon physics, developed further in the identity of Section 5 and the arrow-of-time reading of Section 6.

The Dark-Energy / Horizon-Entropy Identity

The two uses of the horizon budget—dark energy (Section 3) and, in Section 6, the arrow of time—are tied by a single exact identity, the conjunction of the holographic horizon entropy \(\eqref{eq:S-dS}\) with the dark-energy formula \(\eqref{eq:hde-main}\) in the de Sitter attractor.

The central identity

In the late-time attractor, dark energy dominates the Friedmann equation: \[\begin{equation} H_{\infty}^2\;=\;\frac{8\pi G}{3c^2}\,\rhoDE^{\infty}, \label{eq:Friedmann-attractor} \end{equation}\] the standard de Sitter expansion rate set by a cosmological constant \(\Lambda\equiv 8\pi G\rhoDE^{\infty}/c^4\). Substituting \(\eqref{eq:Friedmann-attractor}\) into the horizon entropy \(\eqref{eq:S-dS}\) at \(H=H_{\infty}\), \[\begin{align} \SdS^{\infty}\;&=\;\frac{\pi c^5}{H_{\infty}^2\,\hbar\,G} \;=\;\frac{\pi c^5}{\bigl(8\pi G\,\rhoDE^{\infty}/(3c^2)\bigr)\,\hbar\,G}\\[4pt] &=\;\frac{3\,c^7}{8\,G^2\,\hbar\,\rhoDE^{\infty}}. \label{eq:S-from-rho} \end{align}\] Rearranging gives the central identity:

The derivation occupies three algebraic lines, and both sides of \(\eqref{eq:central-identity}\) are functions of the single variable \(H_{\infty}\), so the identity is a re-labeling of one quantity, not a relation between two independent ones. The substance lies not in the identity but in what it permits: coupling the dark-energy density to the arrow of time within one framework. Versions of \(\eqref{eq:central-identity}\) appear in the holographic-dark-energy literature from Banks , Fischler–Susskind , and Cohen–Kaplan–Nelson , and most explicitly in Li ; the originality claimed here is the conceptual coupling of Section 6.

The identity is a conservation law in Planck units. With the Planck density \(\rho_{\mathrm{Pl}}\equiv c^7/(\hbar G^2)\), the right-hand side of \(\eqref{eq:central-identity}\) is \((3/8)\,\rho_{\mathrm{Pl}}\), and the identity becomes \[\begin{equation} \frac{\rhoDE^{\infty}}{\rho_{\mathrm{Pl}}}\;=\;\frac{3/8}{\SdS^{\infty}}. \label{eq:rho-ratio} \end{equation}\]

Numerical check

Substituting measured values is a consistency check; today’s product and the asymptotic product differ because today is not yet at the attractor.

Today.

Using Planck 2018 values (\(H_0=67.4\) km/s/Mpc, \(\Omega_{\mathrm{DE}}=0.689\), \(\rho_{\mathrm{crit}}^{\mathrm{today}}=7.67\times 10^{-10}\) J/m\(^3\)), the observed dark-energy density is \[\rhoDE^{\mathrm{today}} = \Omega_{\mathrm{DE}}\,\rho_{\mathrm{crit}}^{\mathrm{today}}\approx 0.689\times 7.67\times 10^{-10}\ \mathrm{J/m^3}\approx 5.28\times 10^{-10}\ \mathrm{J/m^3}.\] Today’s horizon entropy is \(\SdS^{\mathrm{today}}\approx 2.27\times 10^{122}\) (Table 1), so \[\rhoDE^{\mathrm{today}}\,\SdS^{\mathrm{today}}\approx 1.20\times 10^{113}\ \mathrm{J/m^3}.\] This differs from the Planck-scale constant \(\eqref{eq:RHS-numeric}\) by a factor \(\Omega_{\mathrm{DE}}=0.689\), as expected: the identity \(\eqref{eq:central-identity}\) holds only at the attractor where \(\Omega_{\mathrm{DE}}\to 1\), while today is still in the matter–dark-energy transition.

Asymptote.

The dark-energy density does not dilute, so \(\rhoDE^{\infty}=\rhoDE^{\mathrm{today}}\). The asymptotic horizon entropy is \(\SdS^{\infty}=\SdS^{\mathrm{today}}/\Omega_{\mathrm{DE}}\approx 3.29\times 10^{122}\), and \[\rhoDE^{\infty}\,\SdS^{\infty}\approx 1.74\times 10^{113}\ \mathrm{J/m^3},\] matching the Planck-scale constant \(\eqref{eq:RHS-numeric}\) to better than \(1\%\) (saturating exactly in the limit), with residual at the level of present cosmological-parameter uncertainties. Both sides are computed from the same cosmological parameters: the identity is a structural relation, not an empirical test.

The cosmological constant problem reframed

The cosmological constant problem  asks why \(\rhoDE/\rho_{\mathrm{Pl}}\sim 10^{-122}\) rather than \(O(1)\) as quantum field theory naively predicts. The identity \(\eqref{eq:rho-ratio}\) inverts the question: \[\begin{equation} \frac{\rhoDE^{\infty}}{\rho_{\mathrm{Pl}}}\;=\;\frac{3/8}{\SdS^{\infty}}\;\sim\;\frac{1}{\SdS^{\infty}}. \end{equation}\] For \(\SdS^{\infty}\sim 10^{122}\), \(\rhoDE^{\infty}/\rho_{\mathrm{Pl}}\sim 10^{-122}\) automatically: the dark-energy hierarchy is the reciprocal of the cosmic entropy capacity, not a separate number requiring a separate explanation.

This does not solve the cosmological constant problem—why the cosmic horizon holds \(10^{122}\) nats and not \(10^{60}\) or \(10^{200}\) remains open—but it makes it one question. “Why is \(\rhoDE\) so small” is “why is \(\SdS^{\infty}\) so large.” The identity \(\eqref{eq:central-identity}\) links \(\rhoDE^{\infty}\) and \(\SdS^{\infty}\); it derives neither separately. The mystery survives in dimensionless form: one number, \(\alpha=3\Omega_{\mathrm{DE}}/(8\pi)\), equivalently \(1/\SdS^{\infty}\) in Planck units, fitted to observation. Whether it derives from a deeper principle is open. Section 6 shows that this same number controls the arrow of time.

The Arrow of Time from the Budget

The second use of the horizon budget is the arrow of time. The same bound \(\eqref{eq:holo-bound}\) that fixes the dark-energy density, applied to the shrinking past horizon, implements the Past Hypothesis, recasts Penrose’s \(10^{123}\) as a capacity, and orients the Weyl curvature asymmetry.

The Past Hypothesis as a corollary of the bound

The Past Hypothesis —the early universe in a microstate of extraordinarily low coarse-grained entropy, the boundary condition that selects the thermodynamic arrow from time-symmetric dynamics—is normally an external postulate. The holographic bound applied to the shrinking cosmic horizon implements it automatically. The bound reads \[\begin{equation} S(t)\;\leq\;\SdS(t)\;=\;\frac{\pi c^5}{H(t)^2\,\hbar\,G}. \label{eq:bound-cosmo} \end{equation}\] In any FRW cosmology with a past singularity, \(H(t)\to\infty\) as \(t\to 0\), so \[\begin{equation} \SdS(t)\;\longrightarrow\;0\quad\text{and hence}\quad S(t)\;\longrightarrow\;0\qquad\text{as } t\to 0. \label{eq:entropy-vanishes} \end{equation}\] Low initial entropy is not fine-tuned; it is forced. There is no room for high entropy in the early universe because there is no horizon area to support it.

Equation \(\eqref{eq:entropy-vanishes}\) is an upper bound; that the actual \(S(t)\) tracks \(\SdS(t)\) rather than sitting far below it is Assumption 1 (GSL saturation), motivated within QGC by the cosmic decoherence rate \(\Gamma_{\mathrm{total}}=g(1)\,H/(4\pi)\approx H/(10\pi)\) : the cosmic decoherence rate scales as \(H\), so entropy production keeps pace with capacity growth, making saturation generic rather than fine-tuned. The QGC Past Hypothesis is thus not merely “\(S(0)\approx 0\)” (trivial from the bound) but the stronger statement that \(S(t)\) tracks \(\SdS(t)\) throughout. The arrow of time is the direction of this growth. Against the conventional axiomatic statement—“the macrostate of the early universe had entropy far below the maximum compatible with its energy”—the framework substitutes “the cosmic entropy budget at early times was small because the holographic capacity was small,” following from Axiom 1 plus saturation. One postulate is eliminated at the cost of an axiom QGC accepts anyway.

Penrose’s \(10^{123}\) as a capacity

Penrose’s fine-tuning estimate \(\exp(-10^{123})\) for the initial state  is, in this reading, not a probability but the value of \(\SdS^{\infty}\) in nats: the capacity of the cosmic horizon. The asymptotic budget \(\SdS^{\infty}\approx 3\times 10^{122}\) (Table 1) is the maximum information any observer can ever encode in their causal patch. It is large because the universe is large; it is not a measure of fine-tuning but the size of the cosmic address book. Through the identity \(\eqref{eq:rho-ratio}\), the smallness of \(\rhoDE/\rho_{\mathrm{Pl}}\sim 10^{-122}\) is the reciprocal of this largeness: Penrose’s \(10^{123}\) and the cosmological constant hierarchy are the same number in two languages.

Why \(\Lambda>0\) closes the budget

A sharper claim is available: the QGC reading of the Past Hypothesis, as a finite, two-sided statement about the cosmic entropy budget, requires \(\Lambda>0\). This is a self-consistency condition on the chosen formulation, not a universal necessity. Three cases:

Case (i): \(\Lambda<0\) (AdS-like).

The universe recollapses. There is no asymptotic cosmic horizon and no well-defined \(\SdS^{\infty}\); the budget’s maximum is reached only in the recollapse phase, a turnaround, not a stable asymptote.

Case (ii): \(\Lambda=0\) (Minkowski asymptote).

\(H(t)\to 0\) as \(t\to\infty\), so \(\SdS(t)\to\infty\) and the bound becomes vacuous in the future. The Past Hypothesis is one-sided: bounded below in the past, unbounded above in the future.

Case (iii): \(\Lambda>0\) (de Sitter attractor; our case).

The Hubble rate asymptotes to finite \(H_{\infty}>0\), and the horizon entropy to finite \(\SdS^{\infty}\). The Past Hypothesis is two-sided: \(S(t)\) grows from \(\sim 0\) at the Planck epoch to \(\SdS^{\infty}\) in the asymptotic future. Both endpoints are finite; the arrow of time has a beginning (no room for entropy) and an asymptotic destination (capacity saturated).

Only case (iii) gives the closed, finite, two-sided budget developed here.

The Weyl asymmetry as the direction of decoherence

The Weyl tensor \(C_{\mu\nu\rho\sigma}\) vanishes at the Big Bang and diverges at black-hole singularities, two qualitatively different generic singular outcomes of gravitational dynamics . Penrose’s Weyl Curvature Hypothesis (WCH) states this as the classical initial-condition postulate \[\begin{equation} C_{\mu\nu\rho\sigma}(t\to 0)\to 0,\qquad C_{\mu\nu\rho\sigma}(\text{BH singularity})\to\infty. \label{eq:wch-classical} \end{equation}\] As a phenomenological observation this is hard to dispute—the CMB is extraordinarily isotropic, gravitational collapse generically produces BKL-like curvature divergences. As an explanation it is unsatisfying: a boundary condition on the classical field, not derived from a dynamical principle, requiring a past/future distinction with no analogue in the time-symmetric Einstein equations.

In QGC the Weyl tensor is an operator \(\hat{C}_{\mu\nu\rho\sigma}\) on the universal Hilbert space. A state with \(\langle\hat{C}\rangle=0\) but large \(\mathrm{Var}(\hat{C})\) is a superposition of non-zero values, qualitatively different from a sharp zero; the classical statement \(\eqref{eq:wch-classical}\) is incomplete. This motivates:

The first condition is the operator analogue of Penrose’s statement; the vanishing variance is the new content, forbidding the Weyl tensor from being in a superposition of non-zero values at the past boundary.

Support comes from three complementary readings of one underlying argument, not logically independent derivations. All rely on the same core step—the past conformal boundary has vanishing bounding area, \(A_{\mathrm{boundary}}\to 0\)—and differ only in emphasis. Holographic: the bound \(S\leq A/(4\lP^2)\to 0\) forces any entropy-carrying branching of the metric to vanish at the boundary. Kinematic vacuum: sharp Weyl values require environment-induced decoherence; at a past boundary no decoherence has occurred, so \(\hat{C}\) has neither a definite eigenvalue nor a definite superposition of non-zero values. No-environment plus global purity: the universe-as-a-whole is pure, so its von Neumann entropy is zero, the matter entropy is bounded by the vanishing boundary area, and the realized branching that would give nonzero \(\mathrm{Var}(\hat{C})\) goes to zero. The three are three vocabularies for one heuristic; the cumulative weight is one motivating argument, not three independent ones.

Conjecture [conj:qgc-wch] is symmetric in past and future conformal boundaries and so does not by itself distinguish the Big Bang from the future heat death. The Big Bang / black-hole asymmetry is not a past-versus-future boundary of the global \(|\Psi\rangle\): both are local singularities within a single cosmic aeon. The asymmetry is in the decoherence direction. Within a single aeon:

• The Big Bang is upstream of all decoherence. No semiclassical clock yet exists, no environment has measured the Weyl operator into definite branches. The Weyl operator is in its kinematic vacuum, \(\langle\hat{C}\rangle=\mathrm{Var}(\hat{C})=0\), matching \(\eqref{eq:wch-classical}\) at the Big Bang.

• The black-hole singularity is downstream. Substantial gravitational entanglement has accumulated; the Weyl operator has been measured many times into definite branches; the BKL-like chaos is the classical description of high-variance branches each with large \(\hat{C}\) eigenvalue. Both \(\langle\hat{C}\rangle\) and \(\mathrm{Var}(\hat{C})\) are large, matching \(\eqref{eq:wch-classical}\) at the singularity.

The asymmetry follows from a single fact—emergent time as the direction of cosmic decoherence—applied at both endpoints. There is no separate past–future-asymmetry postulate; the asymmetry is the arrow of time, with the mechanism cosmic gravitational decoherence at per-mode rate \(g(1)\,H/(4\pi)\) .

Scope and Comparison

Both uses of the horizon budget carry the same caveats, and this section states them once for the whole paper, then compares the construction with the main alternatives.

The limits, stated once

Additionally, the event-horizon cutoff is teleological, depending on the future evolution; we read this as the global character of solutions to Einstein’s equations, not as future causation. No microscopic mechanism is provided: the holographic ansatz is postulated, and although string theory furnishes examples of holography (AdS/CFT), no derivation of holographic dark energy from string theory exists. The framework does not explain why quantum field theory vacuum energy, \(\rho_{\mathrm{vac}}\sim 10^{113}\) J/m\(^3\), fails to gravitate down to \(\rhoDE\sim 10^{-9}\) J/m\(^3\); it takes \(\rhoDE\) as given.

Comparison with alternatives

\(\Lambda\)CDM and Li’s HDE.

\(\Lambda\)CDM treats \(\Lambda\) as a fundamental parameter, fitting the data with high precision but explaining nothing about its origin or magnitude, fine-tuned at \(10^{-120}\) relative to naive quantum field theory. The framework here replaces the unexplained \(\Lambda\) with the unexplained dimensionless \(\alpha\sim 0.08\); neither explains the magnitude, but the Past Hypothesis becomes a consequence rather than a separate postulate. Li’s HDE  posits \(\rhoDE=3\xi^2 M_P^2/L^2\) with \(\xi\) free; \(\xi=1\) gives \(w_0\approx -0.88\), in mild tension. Here \(\xi=\sqrt{\Omega_{\mathrm{DE}}}\) follows as a consistency condition of the de Sitter attractor, removing the freedom and yielding \(w=-1\) exactly. Both share the limitation that the holographic ansatz is not derived from first principles.

Quintessence and modified gravity.

Quintessence  drives acceleration with a scalar field and arbitrary potential, giving dynamical \(w(t)\); it is underconstrained, trading \(\Lambda\) for the function \(V(\phi)\) and generically predicting \(w\neq -1\). Modified gravity  alters the Einstein equations and generically gives \(G_{\mathrm{eff}}\neq G_N\); the framework here, within general relativity, predicts \(G_{\mathrm{eff}}=G_N\) exactly, so \(G_{\mathrm{eff}}(z)/G_N\neq 1\) at the percent level would falsify it and support modified gravity.

Prior holographic dark energy and the N-bound.

Prior work linked the cosmological constant to horizon entropy via essentially the algebraic identity \(\eqref{eq:central-identity}\). Banks  and Fischler–Susskind  proposed (1998–2000) that the cosmological constant is fundamentally an entropy quantity: the de Sitter horizon carries \(N=\SdS^{\infty}\) degrees of freedom, and the smallness of \(\Lambda\) in Planck units is the largeness of \(N\). The principle rests on ’t Hooft’s dimensional-reduction proposal  and Susskind’s “world as a hologram” . Cohen–Kaplan–Nelson  gave the effective-field-theory bound \(\rho_{\mathrm{vac}}\sim H^2 M_P^2\); Li  sharpened it into the dynamical \(\rhoDE\propto c^2 H^2/G\) inherited here. The closest prior synthesis is Padmanabhan’s thermodynamic-aspects-of-gravity program . The structural content of \(\eqref{eq:central-identity}\) is twenty-five years old; the novelty is the coupling of this prior-art identity to the Past Hypothesis (the shrinking past horizon making \(\SdS(t\to 0)\to 0\) automatic) and to Penrose’s WCH (the upstream/downstream-of-decoherence reading). Neither connection appears in those threads.

Bousso’s covariant entropy bound.

Bousso’s bound  is the most general holographic principle, yielding for the de Sitter horizon exactly the \(\SdS=A/(4\lP^2)\) used here. The framework specializes it to the cosmic horizon and adds the dynamical-dark-energy link \(\eqref{eq:central-identity}\); it elaborates Bousso into a dynamical statement, and does not contradict it.

Penrose’s Conformal Cyclic Cosmology.

CCC  identifies the future conformal boundary of one aeon with the past boundary of the next, so the WCH holds automatically at every junction. The framework endorses Penrose’s mass-as-clock and conformal-symmetry intuitions  and the reframing of \(10^{123}\) as geometric, but rejects cyclicity: the would-be Thermofield-Double identification between aeons fails for lack of a shared Hamiltonian (the Wheeler–DeWitt constraint is \(\hat{H}|\Psi\rangle=0\) globally), a shared inverse temperature (the spacelike past conformal boundary has no timelike Killing vector), and a canonical mode pairing. The surviving picture is block-static, not cyclic: every conformal boundary of \(|\Psi\rangle\) has the same kinematic-vacuum form, but “aeons” are not chronologically connected as Penrose proposed. CCC’s ontological mass-shedding (erebons) is replaced by operational horizon dilution: massive particles persist but are diluted past the holographic bound of any observer’s horizon.

Hartle–Hawking and anthropic explanations.

The Hartle–Hawking no-boundary proposal  fixes a specific boundary condition; the capacity argument here provides a bound for any state, so the two are consistent and complementary—a Hartle–Hawking state satisfies the capacity bound at small scale factor, both predicting \(S\to 0\) as \(A\to 0\). Anthropic resolutions  appeal to a landscape of universes; the framework offers a non-multiverse alternative in which the Past Hypothesis is forced by the bound and \(\Lambda\) is reciprocal to the capacity, without an unobservable ensemble. This refutes neither—both can be true—but is more economical.

The framework does not replace the need for a microscopic theory of quantum gravity. The holographic bound, the GSL, and the dark-energy formula are phenomenological inputs—well-motivated within QGC, none derived from a quantum-gravity Lagrangian. It organizes these inputs economically; it is not a final theory.

Conclusions

The Bekenstein–Hawking entropy of the cosmological horizon is one budget, and this paper spends it twice.

Used forward, against the critical density, the budget gives the dark-energy density \(\rhoDE=\alpha c^2 H^2/G\) with \(\alpha=3\Omega_{\mathrm{DE}}/(8\pi)\approx 0.082\); GSL saturation at the de Sitter attractor fixes \(\xi=\sqrt{\Omega_{\mathrm{DE}}}\) as a consistency requirement and pins the equation of state to \(w=-1\) exactly, indistinguishable from a cosmological constant and consistent with \(w_0=-1.03\pm 0.03\). The ratio \(\rhoDE/\rho_m\) is stable during matter domination, removing the timing aspect of the coincidence problem.

Used backward, against the shrinking past horizon, the same budget implements the Past Hypothesis: \(\SdS(t)\to 0\) as \(t\to 0\) makes low initial entropy a corollary of the holographic bound, not a postulate. Penrose’s \(10^{123}\) becomes \(\SdS^{\infty}\) in nats—the capacity of the cosmic horizon, not a probability. The Weyl curvature asymmetry is the direction of cosmic decoherence: the Big Bang upstream of all branching (kinematic vacuum, \(\langle\hat{C}\rangle=\mathrm{Var}(\hat{C})=0\)), black-hole singularities downstream (large mean and variance).

The two uses are tied by the identity \[\begin{equation} \rhoDE^{\infty}\;\SdS^{\infty}\;=\;\frac{3\,c^7}{8\,G^2\,\hbar} = \frac{3}{8}\,\rho_{\mathrm{Pl}}, \label{eq:central-conclusions} \end{equation}\] exact in the de Sitter attractor of any FRW cosmology with \(\Lambda>0\), both sides functions of \(H_{\infty}\) alone. The smallness of \(\rhoDE\) in Planck units is the reciprocal of the largeness of \(\SdS^{\infty}\); the cosmological constant hierarchy and the cosmic entropy capacity are two faces of one number. Dark energy is the structural condition that closes the two-sided budget: \(\Lambda>0\) is required for the finite-capacity formulation developed here.

One axiom at one horizon gives dark energy and the arrow of time in parallel: dark energy makes the asymptotic capacity finite; the finite capacity implements the Past Hypothesis; the Past Hypothesis defines the arrow of time; the arrow produces the Weyl asymmetry as upstream-versus-downstream of decoherence; and the smallness of \(\rhoDE\) is the reciprocal of the capacity. One axiom, one identity, one parameter (\(\alpha\)), two vocabularies.

The contribution is conceptual, not empirical. None of the puzzles is solved in the strong sense of derivation from a deeper principle; rather, explaining \(\alpha\approx 0.082\) would simultaneously explain the smallness of the cosmological constant, the Past Hypothesis, and the Weyl asymmetry. The identity \(\eqref{eq:central-conclusions}\) is a tautology known in the holographic-dark-energy literature ; the novelty is its coupling of dark energy to the arrow of time. The limits are real and flagged throughout (Section 7.1): \(\alpha\) is fitted, GSL saturation is assumed, the operator Weyl hypothesis is heuristic, and no observable discriminates against \(\Lambda\)CDM at current precision.

The framework is a step, not a destination. The destination—a first-principles derivation of \(\alpha\) from microscopic quantum gravity—requires a theory we do not have. What we have is one organization of the puzzle landscape: the holographic bound at the cosmic horizon, coupled to the dark-energy density via the de Sitter attractor, makes dark energy and the arrow of time two readings of one horizon entropy budget. Whether that organization survives deeper scrutiny decides whether it is the right one or one among several.

Appendices

Numerical Values, the Hubble Cutoff, and the Identity Check

This appendix collects the numerical values underlying the main text—using the CODATA 2018 fundamental constants and the Planck 2018 cosmological parameters—derives the equation of state for both infrared cutoffs, and checks the central identity epoch by epoch.

Fundamental constants and cosmological parameters

The critical density and dark-energy density are derived from \(H_0\) and \(\Omega_{\mathrm{DE}}\) via \(\rho_{\mathrm{crit}}=3c^2 H_0^2/(8\pi G)\), ensuring consistency with the body-text Planck 2018 value \(H_0=67.4\) km/s/Mpc. Propagating the Planck uncertainty in \(\Omega_{\mathrm{DE}}\) alone gives \(\delta\alpha/\alpha=\delta\Omega_{\mathrm{DE}}/\Omega_{\mathrm{DE}}=0.006/0.689\approx 0.9\%\), i.e. \(\alpha=0.082\pm 0.001_{\mathrm{stat}}\); the \(H_0\) tension is a larger unmodeled systematic, so we quote \(\alpha\approx 0.082\) in the body.

Why the Hubble cutoff fails and the event horizon succeeds

The infrared cutoff determines the equation of state. With the Hubble cutoff \(r_H=c/H\), the Friedmann equation gives \[\begin{equation} H^2 = \frac{8\pi G}{3c^2}\rho_m + \frac{8\pi\alpha}{3}H^2\;\Longrightarrow\;H^2=\frac{8\pi G\rho_m}{3c^2(1-8\pi\alpha/3)}. \end{equation}\] During matter domination \(\rho_m\propto a^{-3}\), so \(H^2\propto a^{-3}\) and \(\dot{H}/H^2=-3/2\). From \(w=-1-2\dot{H}/(3H^2)\), \[\begin{equation} w_{\mathrm{intrinsic}} = -1-\frac{2\times(-3/2)}{3}=0. \end{equation}\] With the Hubble cutoff the intrinsic equation of state is \(w=0\): dark energy dilutes like matter, \(\rhoDE\propto a^{-3}\), producing no acceleration. Current data give \(w_0=-1.03\pm 0.03\), ruling out \(w=0\) at more than \(30\sigma\).

The future event horizon \(\eqref{eq:event-horizon}\) has \(dR_h/dt=HR_h-c\). With \(\rhoDE=3\xi^2 M_P^2/R_h^2\), the dark-energy fraction is \(\Omega_{\mathrm{DE}}=\xi^2/(H^2 R_h^2)\), and the continuity equation with the \(R_h\) evolution gives \[\begin{equation} w = -\frac{1}{3} - \frac{2\sqrt{\Omega_{\mathrm{DE}}}}{3\xi}. \end{equation}\] Li’s \(\xi=1\) gives \(w_0=-1/3-2\sqrt{0.689}/3\approx -0.88\) at \(\Omega_{\mathrm{DE}}=0.689\), in mild tension. Two equivalent routes fix \(\xi\). Temperature equality \(T_H=T_E\) gives \(HR_h=c\); GSL saturation gives \(dS_E/dt\to 0\Rightarrow HR_h\to c\) (Section 2.3). Both are geometric identities of de Sitter space. With \(HR_h=c\), the relation \(\Omega_{\mathrm{DE}}=\xi^2 c^2/(HR_h)^2\) gives \[\begin{equation} \xi=\sqrt{\Omega_{\mathrm{DE}}}\approx 0.83\quad\text{at }\Omega_{\mathrm{DE}}=0.689, \end{equation}\] and \(w=-1/3-2/3=-1\) exactly, indistinguishable from a cosmological constant, arising as a consistency requirement rather than imposed by hand.

The normalized Hubble parameter \(E(z)=H(z)/H_0\) separates the cases at \(z=1\): \(\Lambda\)CDM with \(\Omega_m=0.311\) gives \(E(1)=\sqrt{0.311\times 8+0.689}\approx 1.78\); Hubble-cutoff HDE with \(w=0\) gives \(E(1)=(1+z)^{3/2}\approx 2.83\), a \(60\%\) deviation ruled out; event-horizon HDE with de Sitter saturation gives \(w=-1\) and \(E(1)\approx 1.78\), identical to \(\Lambda\)CDM within observational uncertainties. The framework is degenerate with \(\Lambda\)CDM at the level of the background expansion history. In the CPL parameterization \(w(a)=w_0+w_a(1-a)\), current data give \(w_a=-0.1\pm 0.3\), consistent with no evolution.

Numerical check of the central identity

The asymptotic ratio matches unity within rounding, residuals at the level of present cosmological-parameter uncertainties. Pre-asymptotic ratios are smaller because \(\SdS(t)\) has not yet saturated at \(\SdS^{\infty}\).

Cosmological constant problem in Planck units

The standard statement is \[\begin{equation} \frac{\rhoDE^{\mathrm{obs}}}{\rho_{\mathrm{Pl}}}\;=\;\frac{5.28\times 10^{-10}\ \mathrm{J/m^3}}{4.63\times 10^{113}\ \mathrm{J/m^3}}\;\approx\;1.14\times 10^{-123}, \end{equation}\] the famous \(122\)–\(123\) orders-of-magnitude hierarchy. (The exact exponent depends on whether the comparison is to \(\rho_{\mathrm{Pl}}\) or to the QFT zero-point estimate; the difference is at most one order of magnitude.) Equivalently, \[\begin{equation} \SdS^{\infty}\;=\;\frac{3/8}{\rhoDE^{\infty}/\rho_{\mathrm{Pl}}}\;\approx\;\frac{3/8}{1.14\times 10^{-123}}\;\approx\;3.29\times 10^{122}, \end{equation}\] in nats, agreeing with the directly-computed value (Table 5) to within input precision.

Consistency with \(\alpha=0.082\)

By \(\alpha=3\Omega_{\mathrm{DE}}/(8\pi)\), the today-to-asymptote entropy ratio is \[\frac{S_{\mathrm{dS}}^{\mathrm{today}}}{S_{\mathrm{dS}}^{\infty}}\;=\;\left(\frac{H_{\infty}}{H_0}\right)^2\;=\;\Omega_{\mathrm{DE}}\;=\;\frac{8\pi\alpha}{3}\;\approx\;\frac{8\pi\times 0.0822}{3}\;\approx\;0.689,\] so \(S_{\mathrm{dS}}^{\infty}/S_{\mathrm{dS}}^{\mathrm{today}}=1/\Omega_{\mathrm{DE}}=3/(8\pi\alpha)\approx 1.45\). From the numerical values, \(3.29\times 10^{122}/2.27\times 10^{122}\approx 1.45\), matching exactly. The universe today is approximately \(69\%\) of the way to its asymptotic entropy capacity—which is exactly \(\Omega_{\mathrm{DE}}\).

Cosmic Entropy History

This appendix traces the cosmic entropy capacity \(\SdS(t)\) from the Planck epoch to the de Sitter asymptote, showing how a single quantity governed by \(H(t)\) produces the entropy budget underlying both dark energy and the arrow of time.

The capacity as a function of cosmic time

The cosmic entropy capacity is \[\begin{equation} \SdS(t)\;=\;\frac{\pi c^5}{H(t)^2\,\hbar\,G}\;=\;\frac{\pi}{[H(t)\,\tP]^2}, \label{eq:S-vs-tP} \end{equation}\] the last form making the Planck-units structure manifest: \(\SdS\) is the inverse square of the dimensionless ratio \(H(t)\,\tP\). At the Planck epoch \(H\sim 1/\tP\) and \(\SdS\sim\pi\); today \(H\tP\sim 10^{-61}\) and \(\SdS\sim 10^{122}\).

The cosmic entropy capacity grows from \(\approx\pi\) nats at the Planck epoch to \(\approx 3.3\times 10^{122}\) nats in the de Sitter asymptote, \(122\) orders of magnitude over the full history of the universe.

Epoch-by-epoch evolution

Planck epoch (\(t\sim\tP\)).

\(H\sim 1/\tP\sim 1.85\times 10^{43}\) s\(^{-1}\); \(\SdS\sim\pi\) nats; horizon area \(\sim 1\,\lP^2\). Approximately one Planck area’s worth of capacity.

Inflationary epoch.

If inflation occurs with energy scale \(H_{\mathrm{inf}}\sim 10^{14}\) GeV (\(H_{\mathrm{inf}}\sim 10^{36}\) s\(^{-1}\)), the capacity is \(\SdS\sim 10^{17}\) nats, approximately constant during the de Sitter inflationary phase.

Radiation era.

After reheating \(H\propto a^{-2}\), so \(\SdS\propto a^4\). Evaluating \(\SdS=\pi c^5/(H^2\hbar G)\) at matter–radiation equality (\(z_{\mathrm{eq}}\approx 3400\)) with the full \(\Lambda\)CDM rate \(H\approx 3.40\times 10^{-13}\) s\(^{-1}\) gives \(\SdS\approx 9.3\times 10^{111}\) nats.

Matter era.

\(H\propto a^{-3/2}\), so \(\SdS\propto a^3\). At recombination (\(z=1100\), \(H\approx 5.11\times 10^{-14}\) s\(^{-1}\)), \(\SdS\approx 4.1\times 10^{113}\) nats; at \(z=1\) (\(H\approx 3.89\times 10^{-18}\) s\(^{-1}\)), \(\SdS\approx 7.1\times 10^{121}\) nats. Each value is obtained by direct evaluation of the closed-form formula at the appropriate \(H(z)\).

Dark-energy era.

Today (\(z=0\)), \(H_0\approx 2.18\times 10^{-18}\) s\(^{-1}\) gives \(\SdS^{\mathrm{today}}\approx 2.27\times 10^{122}\) nats. The asymptotic value \(H_{\infty}=H_0\sqrt{\Omega_{\mathrm{DE}}}\) gives \(\SdS^{\infty}\approx 3.29\times 10^{122}\) nats. The universe today is approximately \(69\%\) of the way (in entropy budget) to this asymptote, consistent with \(H_0^2/H_{\infty}^2=1/\Omega_{\mathrm{DE}}\approx 1.45\).

Entropy production from gravitational decoherence

The actual entropy \(S(t)\) tracks \(\SdS(t)\) under Assumption 1. The mechanism is cosmic gravitational decoherence, whose per-mode rate is  \[\begin{equation} \Gamma_{\mathrm{mode}}(t)\;=\;\frac{g(1)\,H(t)}{4\pi}\;\approx\;\frac{H(t)}{10\pi},\qquad g(1)=1-\frac{2}{\pi}\,\mathrm{Si}(1)\approx 0.398. \label{eq:Gamma-mode-app} \end{equation}\] Integrated over the age of the universe this produces only \(\sim 4\) nats per mode; the cosmic budget is reached by summing over the \(\SdS(t)\) holographically-allowed horizon modes, \[\begin{equation} \dot{S}(t)\;\sim\;\SdS(t)\,\Gamma_{\mathrm{mode}}(t)\;=\;\frac{g(1)\,\SdS(t)\,H(t)}{4\pi}, \label{eq:Sdot-app} \end{equation}\] the per-mode rate supplying the \(H\)-scaling and the mode count \(\SdS\) closing the \(\sim 122\)-order gap. At each epoch the rate at which cosmic entropy is realized scales as \(H\), tracking the rate at which the capacity grows.

The naive insertion of \(M\sim 10^{53}\) kg (the observable-universe mass) into the lab-scale formula \(\Gamma=GM^2/(\hbar d)\)  gives a super-causal rate \(\sim 10^{113}\) s\(^{-1}\). This is resolved  by (i) redoing the calculation on a de Sitter background rather than Minkowski, giving the form factor \(g(x)=1-(2/\pi)\,\mathrm{Si}(x)\) that vanishes for super-horizon mass separations; (ii) selecting the cosmic “\(\Delta m\)” as the de Sitter thermal mass \(\Delta m_{\mathrm{dS}}=\hbar H/(2\pi c^2)\); (iii) bounding mode additivity at \(O(G^2)\). The cosmic rate is then \(\Gamma_{\mathrm{total}}=g(1)\,H/(4\pi)\approx 7\times 10^{-20}\) Hz today, sub-causal.

Cross-check: the saturation gap

If saturation (Assumption 1) is generic, the total entropy produced from the Planck epoch to today should equal \(\SdS^{\mathrm{today}}\sim 10^{122}\). The naive per-mode integral is \[\begin{equation} \Delta S^{\mathrm{per-mode}}_{\mathrm{produced}}\;\sim\;\int_{\tP}^{t_0}\Gamma_{\mathrm{mode}}(t)\,dt\;\sim\;\frac{N_{\mathrm{efolds}}}{10\pi}, \end{equation}\] where \(N_{\mathrm{efolds}}\sim\ln(a_0/a_{\mathrm{Pl}})\sim 140\) (\(60\) from inflation \(+60\) subsequent \(+20\) pre-inflation if any), giving \(\Delta S\sim 4\) nats per mode, \(122\) orders short of \(\SdS^{\infty}\). The gap is closed by the mode count: each Hubble volume contains \(\SdS\) holographically distinguishable modes, so the total rate scales as \(\SdS\cdot H/(10\pi)\) and the accumulated entropy as \(\SdS\).

Status: structural, not closed.

The picture is internally consistent, but a first-principles derivation of why the per-mode rate combines with the holographic count to give precisely \(\SdS\) nats (rather than \(0.1\,\SdS\) or \(10\,\SdS\)) is not available, and the mapping between “mode-decoherence events” and “nats realized” carries unresolved \(O(1)\) ambiguities. We claim no better than order-of-magnitude agreement; the qualitative match between capacity-growth rate (\(\sim H\,\SdS\)) and entropy-production rate (\(\sim H\,\SdS\)) supports Assumption 1 but is one step short of a derivation.

Summary

The cosmic entropy capacity \(\SdS(t)\) grows monotonically from \(\sim\pi\) nats at the Planck epoch to \(\sim 3\times 10^{122}\) nats in the de Sitter asymptote, with entropy production tracking it through per-mode gravitational decoherence at rate \(g(1)\,H/(4\pi)\). Dark energy, the Past Hypothesis, the Weyl asymmetry, and the cosmological constant problem are different facets of this single arc of capacity growth.

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