Quantum–Cosmological Parameter-Scale Unification via Spacetime Diffusivity

1. Introduction

Modern cosmology rests on the empirically successful $\mathrm{\Lambda }$-CDM model, describing an expanding Friedmann–Lemaître–Robertson–Walker universe [1,2,3,4]. However, this observational success masks a profound conceptual tension known as the “problem of time”—the fundamental difficulty of reconciling apparent cosmic evolution with general relativity’s block universe consequence [5,6,7]. This tension emerges from the conflict between mathematical eternalism, which posits that the universe’s 4D geometry is fully realized at once [8,9,10], and the physical experience of temporal becoming observed by embedded cosmic observers.

1.1. The Missing Transport Field: Spacetime Diffusivity

A fundamental gap exists in our field-theoretic description of information transport through spacetime. While physics has well-established transport fields—thermal diffusivity $\alpha =k/\left(\rho c_p\right)$ for heat transport, kinematic viscosity $\nu =\mu /\rho$ for momentum transport, and Einstein’s Brownian diffusion coefficient D with dimensions $\left[L^2{T}^{-1}\right]$ for particle transport [11]—no corresponding field describes information transport between temporal layers of spacetime itself.

This omission becomes critical when considering that all physical interactions fundamentally involve information transfer. Quantum information theory reveals that every physical process operates as a quantum channel with capacity limited by environmental decoherence [12,13], Lieb-Robinson bounds on entanglement spreading [14,15], holographic area constraints [16,17,18], and Landauer’s principle enforcing energetic costs for information processing [19].

These considerations mandate a fundamental field $ϵ\left(s\right)$ with dimensions $\left[L^2{T}^{-1}\right]$ quantifying the irreducible spread in accessibility, precision, and localization of information about the universe’s state. This “spacetime diffusivity” field extends Einstein’s insight that transport phenomena are governed by diffusivity fields: just as thermal diffusivity emerges from molecular chaos, $ϵ\left(s\right)$ arises from quantum-informational uncertainty at the Planck scale that propagates coherently across all scales. Perfect localization of information flow (energia fluens) at any cosmological point inevitably comes at the cost of losing precision in gravitational aspects like mass (energia locata).

1.2. Quantum Measurement Incompatibility in Cosmological Observations

Recent advances in astronomical precision reveal an unexpected pattern: measurements requiring high temporal resolution for signal dispersion analysis systematically yield different mass estimates than measurements optimized for gravitational effects. Every astronomical photon must simultaneously encode both source properties (mass distribution, velocity dispersion) and propagation effects (temporal broadening, spectral line widening). The finite capacity of photon ensembles creates unavoidable resource competition between these observational goals, suggesting fundamental uncertainty relations between spacetime transport properties and mass content that extend traditional quantum mechanics to cosmological measurements.

1.3. Foundational Postulates and Scale Unification

We adopt a foliated block universe perspective, treating the universe as a sequence of information layers ${\mathrm{\Sigma }}_s$ indexed by parameter s. This bridges eternalist (static 4D manifold) and presentist (observed temporal flow) perspectives through finite information propagation constraints. Our approach explores two minimal postulates:

 Postulate 1 (Invariant Time-like Information): 

Causal structure enforces information propagation at the fundamental rate: $ds/dt=c$

 Postulate 2 (Diffused Spacetime): 

A spacetime diffusivity field $ϵ\left(s\right)$ with dimensions $\left[L^2{T}^{-1}\right]$ evolves according to: $dϵ/ds=c$

Through canonical quantization of the resulting first-order action $S[ϵ,m]=\int dsm(dϵ/ds-c)$, we derive the Cosmological Uncertainty Principle: $\Delta ϵ·\Delta m\ge \hslash /2$, establishing quantum conjugacy between spacetime diffusivity and mass content.

Our central theoretical breakthrough extends this through Information-Pixel Amplification: spacetime pixelates into coherence cells of scale ${\ell }_{\mathrm{coh}}=\kappa {\ell }_P$ where quantum decoherence dominates [13,20,21]. Signals traveling distance D sample $N_{\mathrm{eff}}=D/{\ell }_{\mathrm{coh}}$ independent pixels, each contributing minimal action uncertainty $\hslash /2$, yielding the amplified constraint:

$$\begin{array}{|c|}\hline \Delta ϵ·\Delta m\ge {\displaystyle \frac{\hslash }{2}}·{\displaystyle \frac{D}{{\ell }_{\mathrm{coh}}}}\\ \hline\end{array}$$

(1)

This framework generates four principal results: (1) Quantum-gravitational parametric unification—both Planck time $t_P={\ell }_P/c$ and Hubble time $t_H=1/H_0$ emerge from a single diffusion parameter across 60 orders of magnitude; (2) Enhanced cosmological temperature—fundamental thermal limit $T_{\mathrm{CUP}}=\pi T_{\mathrm{dS}}\approx 9\times {10}^{-30}$ K exceeding classical Gibbons-Hawking temperature; (3) Quantitative dark matter predictions—exact reproduction of Milky Way uncertainties ($\sim 4\times {10}^{41}$ kg) using FRB-motivated diffusivity parameters; (4) Resolution of temporal ontology—apparent time flow emerges from scale-invariant informational constraints on embedded observers.

The framework reveals that quantum uncertainty constraints operating through spacetime diffusivity generate fundamental connections between quantum mechanics, information theory, and cosmic structure, offering testable predictions for precision-uncertainty trade-offs in astronomical measurements while providing fresh perspectives on foundational questions spanning from quantum gravity to observational cosmology.

For a list of symbols used in this work, kindly refer to Appendix B, Table B.1.

2. Theoretical Framework

2.1. Spacetime Framework and Information Transport

From the external eternalist perspective, the universe emerges as a static, time-like informational manifold where “time” represents information propagation between successive invariant layers rather than fundamental temporal flow.

2.1.1. Time-like Structure from Causal Constraints

For a universe emerging from a finite past boundary—as required by Borde–Guth–Vilenkin (BGV) arguments [22]—we consider the limiting case where all separations become purely time-like. Constraining analysis to causal intervals:

$$\Delta s^2=c^2\Delta t^2-\Delta r^2\ge 0,$$

(2)

and taking the idealized limit $\Delta r\to 0$ yields:

$$\Delta s=c\Delta t⟹{\displaystyle \frac{ds}{dt}}=c.$$

(3)

This suggests a foundational relationship:

 Theorem 2.1 

(Invariant Time-like Information Postulate). Under BGV constraints with $\Delta r\to 0$, the informational structure enforces:

$${\displaystyle \frac{ds}{dt}}=c,\left[s\right]=\left[L\right],$$

(4)

where s parameterizes the informational stratification built into the eternal manifold structure.

 Proof. 

Direct consequence of causal structure in the time-like universe. □

Invariant foliation time.

We work with a global foliation $\left\{{\mathrm{\Sigma }}_t\right\}$ (later chosen as $S^3$ slices). The label t is defined intrinsically from the scalar effective light-cone area accumulated up to ${\mathrm{\Sigma }}_t$:

$$A_{\mathrm{eff}}\left(t\right)={\displaystyle \frac{c^2{t}^2}{2}}.$$

(5)

Because $A_{\mathrm{eff}}$ is built from causal structure and areas, t is an invariant foliation label. In a discrete picture with slice indices $t_1,t_2,\dots$, one has $t={\sum }_i\delta t_i$ with each $\delta t_i$ fixed by the same geometric rule; hence $dt$ is invariant.

Claim.

For timelike progression with $\Delta r\to 0$, the invariant label satisfies $ds/dt=c$.

Reason. In the local rest frame $ds=cd\tau$. Our normalization makes equal increments of t correspond to equal increments of $A_{\mathrm{eff}}\propto {\tau }^2$, implying $t=\tau$ up to the fixed normalization; therefore $ds/dt=c$.

The parameter t labels successive information layer hypersurfaces in the eternal foliation, representing a classical foliation coordinate (not a quantum operator) that indexes the static stratification. This avoids operator-ordering ambiguities since t serves as a geometric parameter indexing information layers ${\mathrm{\Sigma }}_s$, while quantum operators $\widehat{ϵ}$ and $\widehat{m}$ act within each hypersurface rather than between layers.

2.2. Physical Interpretation of Time-like Layers and Emergent Dynamics

We foliate spacetime into a sequence of information layers ${\mathrm{\Sigma }}_s$, each labelled by the length-like parameter $s\left[L\right]$. In the limit $\Delta r\to 0$ these hypersurfaces are spacelike, and their timelike normal advances through the foliation according to the universal conversion rule

$${\displaystyle \frac{ds}{dt}}=c,$$

with t the invariant foliation label defined in Def. 2.1.1.1. In local rest frames one has $ds=cd\tau$ and our normalization $A_{\mathrm{eff}}=\frac{1}{2}{c}^2{t}^2$ identifies t with $\tau$ up to a constant, so $ds/dt=c$ (Claim 2.1.1.2).

Physical meaning. Globally the universe is a static 4-D block; apparent temporal flow reflects the finite rate at which observers acquire information. A layer ${\mathrm{\Sigma }}_s$ contains everything causally accessible up to the depth s (equivalently, up to proper time $t=s/c$). Moving to ${\mathrm{\Sigma }}_{s+ds}$ corresponds to a clock tick of $dt=ds/c$.

Emergence of ordinary physics. Spatial structure and local dynamics are encoded within each layer, while apparent evolution arises from correlations between neighbouring layers. When the stack $\left\{{\mathrm{\Sigma }}_s\right\}$ is projected onto standard spacetime coordinates, it traces observer-dependent world-histories: motion, field evolution, and cosmic expansion appear as successive updates of information.

Spatial variations and expansion. Each ${\mathrm{\Sigma }}_s$ is a complete spatial slice (we later choose $S^3\left(\kappa \left(s\right)\right)$ for symmetry). As s increases—i.e. further from the initial singularity—each layer contains more causal data; to an internal observer this reads as cosmic expansion and growing spatial complexity.

Relation to observation. Observers, confined to their past light cones, follow trajectories normal to the foliation. Although the block as a whole is timeless, all familiar processes—particle motion, structure formation, and expansion—emerge from how information layers are successively accessed.

Thus the framework reconciles a static block universe with the felt arrow of time by tying apparent dynamics to the finite-speed acquisition of information.

 Remark 2.2 (Cosmic Flip-Book Analogy) 

The temporal layer structure can be visualized as a cosmic flip-book: each information layer ${\mathrm{\Sigma }}_s$ represents a static “page” containing the complete 3D spatial configuration at one moment, while the classical time parameter t serves merely as the “page number.” The quantum operators $\widehat{ϵ}$ and $\widehat{m}$ correspond to “drawings on each page”—acting within individual layers without crossing between them. The universal conversion rule $ds/dt=c$ determines the “flipping speed” through the sequence. Crucially, while each page remains eternally static (preserving the block universe), the sequential accessing of pages by embedded observers creates the illusion of temporal flow, naturally resolving the problem of time through finite information propagation constraints.

2.2.1. Mathematical Formalization

We model the universe as stratified informational layers (see Remark 2.6) indexed by parameter s, each possessing finite information-carrying capacity $\kappa \left(s\right)\le {\kappa }_0$. For computational tractability, we represent these layers using 3-sphere foliation:

Let $I\subseteq \mathbb{R}$ be an interval of the time-like parameter s, and let:

$$\kappa :I⟶[0,{\kappa }_0]\subset \mathbb{R}({\kappa }_0>0)$$

(6)

be a smooth radius function, where ${\kappa }_0$ represents the maximal informational boundary radius [23,24].

 Definition 2.3 

(Informational Layer). For each $s\in I$, define the informational layer ${\mathrm{\Sigma }}_s$ as the embedded 3-sphere:

$${\mathrm{\Sigma }}_s\cong S^3\left(\kappa \left(s\right)\right)=\left\{(w,x,y,z)\in {\mathbb{R}}^4:w^2+x^2+y^2+z^2=\kappa {\left(s\right)}^2\right\}$$

(7)

The total informational manifold becomes:

$${\mathcal{M}}_{\mathrm{info}}=\underset{s\in I}{⨆}{\mathrm{\Sigma }}_s\cong I\times S^3,$$

(8)

equipped with the pullback of the Euclidean metric on ${\mathbb{R}}^4$ [3,25].

 Remark 2.4 (3-Sphere Structure Justification) 

We adopt 3-sphere foliation based on several considerations:

• Maximal symmetry: No preferred directions, ensuring informational isotropy
• BGV consistency: Closed geometry accommodates required past boundaries
• Finite information capacity: Bounded information content per temporal slice
• Computational tractability: Well-defined boundaries for flux calculations
• Observational compatibility: Current CMB constraints permit slight positive curvature
• Wave-equation symmetry: Maxwell’s equations yield spherical solutions ensuring uniform information propagation [26]
• Huygens–Fresnel principle: Spherical wavelet reconstruction preserves isotropy [27]

 Theorem 2.5 

(Temporal Layer Uniqueness). Each informational layer ${\mathrm{\Sigma }}_s$ in the foliated universe is unique and contains distinct information content.

 Proof. 

Consider two distinct parameter values $s_1\ne s_2$. The corresponding layers ${\mathrm{\Sigma }}_{s_1}$ and ${\mathrm{\Sigma }}_{s_2}$ are unique for several reasons:

1. Capacity Differentiation: From the DTP ($dϵ/ds=c$), the diffusivity field evolves continuously, implying $ϵ\left(s_1\right)\ne ϵ\left(s_2\right)$ for $s_1\ne s_2$. This creates distinct information transport capacities between layers.

2. Information Content Evolution: Each layer ${\mathrm{\Sigma }}_s$ has maximum information capacity $I_{max}\left(s\right)=\pi \kappa {\left(s\right)}^2/({\ell }_P^{2}ln2)$. Since $\kappa \left(s\right)$ varies with s, different layers have different information storage capabilities.

3. Causal Ordering: The constraint $ds=cdt$ establishes strict causal ordering between layers. Information can only propagate from layer $s_1$ to layer $s_2$ if $s_2>s_1$, creating asymmetric temporal relationships that distinguish each layer uniquely.

4. Observer Accessibility: Observers in layer s can only access information from previous layers $s^{\prime }<s$, creating unique epistemic contexts for each layer. □

 Remark 2.6 (Cosmic Onion Structure) 

The 3-sphere foliation resembles a cosmic onion: each informational layer ${\mathrm{\Sigma }}_s$ with radius $\kappa \left(s\right)$ forms a concentric spherical shell around all previous layers. As the parameter s increases from the initial boundary, successive layers ${\mathrm{\Sigma }}_{s_1},{\mathrm{\Sigma }}_{s_2},{\mathrm{\Sigma }}_{s_3},\dots$ expand outward like onion layers, with all shells sharing a common center corresponding to the Big Bang singularity. This nested structure naturally encodes the causal hierarchy where information from inner layers (earlier times) can influence outer layers (later times), but not vice versa, preserving the temporal ordering essential for cosmic evolution while maintaining the spherical symmetry that ensures no preferred spatial directions in the universe’s expansion.

2.3. Physical Rationale for the Two Postulates

We elevate two statements to axioms:

 Axiom 2.7 

(Invariant foliation flow). The invariant line element grows linearly with the invariant foliation label t (Def. 2.1.1.1):

$${\displaystyle \frac{ds}{dt}}=c.$$

(9)

 Axiom 2.8 

(Spacetime-diffusivity). There exists a Lorentz-scalar field $ϵ\left(s\right)$ with units $\left[L^2{T}^{-1}\right]$ such that

$${\displaystyle \frac{dϵ}{ds}}=c.$$

(10)

Axiom 2.7 is the usual proper-time normalisation familiar from special relativity. Axiom 2.8 generalises Einstein’s Brownian coefficient (see [11]) from matter to spacetime itself; cf. Caianiello’s maximal acceleration [28], holographic transport bounds [29], and stochastic-gravity approaches [30]. Taken together, (i) and (ii) are the minimal ingredients from which the Cosmological Uncertainty Principle (CUP) follows (Sec. 2.6.2). Adding any extra kinematic constraint would be redundant, as demonstrated below.

 Proposition 2.9 

(Recovery of Hawking temperature in the $ϵ\to \mathrm{const}$ limit). Setting $ϵ\left(s\right)={ϵ}_0$ and applying Axioms 2.7–2.8 to a Schwarzschild horizon yields

$$T_H={\displaystyle \frac{\hslash c^3}{8\pi GM{k}_B}},$$

(11)

the standard Hawking temperature.

 Sketch 

The Euclidean Schwarzschild metric is regular only if imaginary time is periodic with period

$$\beta ={\displaystyle \frac{2\pi }{\kappa }}={\displaystyle \frac{2\pi r_S}{c}},$$

where $\kappa =c^4/\left(4GM\right)$ is the surface gravity (the factor $2\pi$ removes the conical singularity in the Euclidean $(\tau ,r)$ section). Taking $\Delta t=\beta$ and using $\Delta ϵ=c^2\Delta t$ gives

$$\Delta ϵ=2\pi c{r}_S,$$

in which $c{r}_S$ has units $\left[L^2{T}^{-1}\right]$, matching the dimensions of $ϵ$, and geometrically represents the proper-time “circumference” of one Euclidean thermal period at the horizon. One cannot localize information more sharply than one Euclidean period without breaking manifold regularity, so this value indeed saturates the minimal diffusivity uncertainty across the horizon. The Cosmological Uncertainty Principle then implies

$$\Delta m={\displaystyle \frac{\hslash }{4\pi c{r}_S}},$$

and multiplying by $c^2$ yields

$$\Delta E={\displaystyle \frac{\hslash c^3}{8\pi GM}},$$

which we identify with $k_{B}T$, thus reproducing the Hawking temperature. □

Operational measurability.

Operationally, the universal spacetime-diffusivity $ϵ$ can, in principle, be extracted from the Cosmological Uncertainty Principle (see Subsec. 2.6.2) by measuring mass dispersion in high-precision galaxy-cluster kinematics or via the predicted $9\times {10}^{-30}\mathrm{K}$ de Sitter temperature shift (see Equation (50)). Although technologically remote, this anchors $ϵ$ in observability, mirroring how Einstein related molecular diffusion to Avogadro’s number decades before direct verification.

2.4. Spacetime Diffusivity: The Missing Transport Field

Information transport between temporal layers requires its own field-theoretic description, extending the established hierarchy of transport phenomena to informational systems.

2.4.1. Foundational Motivation for the Spacetime Diffusivity Field

The transport of information through spacetime is subject to fundamental limitations arising from quantum mechanics, relativity, and thermodynamics. While relativity constrains the speed of causal influence to c, it does not specify the fidelity or rate at which information can be robustly encoded, transmitted, or recovered across finite spacetime intervals. Recent advances in quantum information theory and quantum gravity suggest that every physical process involving information propagation is inherently noisy, lossy, and bounded in capacity.

From a quantum information perspective, any physical interaction can be modeled as a quantum channel whose capacity is limited by environmental decoherence, thermal noise, and gravitational constraints [12,13,31]. Even in idealized “noiseless” channels, the evolution of quantum correlations is subject to speed limits such as the Lieb-Robinson bound, which imposes an effective finite group velocity for the spread of entanglement and operator growth in extended systems [14,15]. In particular, the finite propagation speed and unavoidable coupling to fluctuating degrees of freedom can result in a diffusive broadening of quantum and classical information as it traverses spacetime.

This interplay between finite speed and irreducible noise becomes particularly relevant in gravitational contexts. The holographic principle and black hole thermodynamics reveal that the maximum information content $I_{\max }$ within a finite region is proportional to its boundary area, not volume, creating potential bottlenecks for information flow and retention [16,17,18]. Landauer’s principle [19] additionally demonstrates that irreversible information processing—such as measurement or erasure—necessarily produces entropy and heat, suggesting an energetic cost associated with information transmission and processing.

These considerations motivate the introduction of a fundamental field $ϵ\left(s\right)$ with dimensions $\left[L^2{T}^{-1}\right]$ to characterize the effective spread in the accessibility, precision, and localization of information about the universe’s state, as encoded in observable correlations and records. Mathematically, if $I(s,t)$ denotes the accessible information density at layer s and time t, its evolution can be modeled by a generalized diffusion equation:

$${\displaystyle \frac{\partial I}{\partial t}}=ϵ\left(s\right){\nabla }^{2}I+(\mathrm{sources}/\sin \mathrm{ks}),$$

(12)

where $ϵ\left(s\right)$ quantifies the rate at which uncertainty “smears out” due to the combined effects of quantum noise, causal finiteness, and thermodynamic irreversibility.

We postulate that such a diffusivity field represents a natural extension of transport phenomena to informational systems, arising from the fundamental constraints imposed by quantum mechanics, relativity, and thermodynamics on information propagation. In this framework, $ϵ\left(s\right)$ encodes the operational limitations with which information can propagate between spacetime events, linking information-theoretic, quantum, and gravitational principles in a unified field-theoretic description.

2.4.2. Physical Motivation for the Spacetime Diffusivity

Transport processes in physics are universally characterized by diffusivity fields with dimensions $\left[L^2{T}^{-1}\right]$:

• Brownian (mass) diffusivity:D in Fick’s law $J=-D\nabla c$ governs the stochastic spreading of particles and chemical species.
• Thermal diffusivity:$\alpha =k/\left(\rho c_p\right)$ governs heat transport through matter
• Kinematic viscosity:$\nu =\mu /\rho$ governs momentum transport in fluids
• Spacetime diffusivity:$ϵ\left(s\right)$ governs information transport between temporal layers

In a time-like universe where information propagates between static temporal layers, the efficiency of energy-information transfer becomes a fundamental physical quantity requiring its own field description.

 Definition 2.10 

(Spacetime Diffusivity Field). The scalar field $ϵ\left(s\right)$ quantifies informational transport capacity between successive temporal layers:

$$\left[ϵ\right]=L^2{T}^{-1}$$

(13)

High $ϵ$ corresponds to efficient information transfer (low-density, minimal decoherence), while low $ϵ$ indicates information bottlenecks (high-density, strong gravitational fields).

 Theorem 2.11 

(Diffused Spacetime Postulate). The spacetime diffusivity field evolves according to:

$${\displaystyle \frac{dϵ}{ds}}=c$$

(14)

representing intrinsic informational capacity growth at the fundamental rate c.

Why $dϵ/ds=c$. Choosing $dϵ/ds=c$ makes the diffusivity flux propagate exactly on the light cone: any smaller value would allow timelike “leakage’’ of information, while any larger value would violate locality and Lorentz invariance.

Combining with Theorem 2.1, this yields:

$${\displaystyle \frac{dϵ}{dt}}=c{\displaystyle \frac{dϵ}{ds}}=c^2$$

(15)

2.4.3. Connections to Quantum Gravity

The spacetime diffusivity field appears compatible with established quantum gravity frameworks, potentially yielding consistent Planck-scale diffusivity $ϵ\sim c{\ell }_P$:

Loop Quantum Gravity: Area operator fluctuations suggest $ϵ\sim {\ell }_P^2/t_P=c{\ell }_P$.

Causal Set Theory: Sprinkling density fluctuations give $ϵ\sim {\ell }_P^2/t_P=c{\ell }_P$.

Holographic Approaches: Entanglement velocity bounds provide similar scaling.

At cosmological scales, holographic principles suggest maximum diffusivity ${ϵ}_{max}\sim c^2/H_0$ (see Sub 3.2).

2.5. Mass-Energy and Information Flux as Conjugate Variables

The proposal that the cosmic information diffusivity $ϵ$ (say, energia fluens) and the mass content m (energia locata) are canonically conjugate stems from insights in black hole thermodynamics [18,32] and information-theoretic gravity [33]. In such contexts, mass/energy content determines the causal structure and entropy of spacetime [18,24], while information flux (or diffusivity) quantifies the transport and distribution of information—a central feature in holographic [16,17] and emergent gravity frameworks [33,34].

Consider a Schwarzschild black hole of mass M. Its entropy is given by the Bekenstein-Hawking formula:

$$S_{BH}={\displaystyle \frac{k_B{c}^{3}A}{4G\hslash }}$$

(16)

where A is the event horizon area. Changes in mass ($\delta M$) induce changes in entropy and, by the laws of black hole thermodynamics, are intimately tied to the flux of information across the horizon. The quantum no-cloning theorem, black hole information paradox, and holographic principle all suggest that energy content and information flux are fundamentally linked, with quantum fluctuations preventing their simultaneous precise determination.

In information-theoretic gravity [33], Einstein’s equations themselves are derived from the maximization of entropy (information), reinforcing the notion that mass-energy and information flow are conjugate under the symplectic structure of the underlying phase space. Thus, we are led to postulate a non-commutativity between $\widehat{ϵ}$ and $\widehat{m}$, paralleling the familiar relation between position and momentum.

2.5.1. Information Diffusivity and Localization Conjugacy

The fundamental structure of quantum mechanics suggests that information, like energy, can exist in multiple conjugate states that cannot be simultaneously characterized with arbitrary precision.

Information State Duality

In an informationally stratified universe, information manifests in two complementary forms:

• Flowing information (energia fluens): Dynamic information transport between temporal layers
• Localized information (energia locata): Crystallized, captured information such as sharp astronomical images; massive bodies

The act of measurement transforms flowing information into localized information. For example, creating sharper images essentially “crystallizes” the dynamic information flow or fluidity into static, localized form. This crystallization process creates conjugate relationships between information diffusion (position) and mass-energy information (momentum).

Another example is when astronomers create sharp images of distant objects, they convert the supposedly flowing information (energia fluens) into solid information through the measurement process. This kind of solidification necessarily occurs at the expense of knowing the information about the other aspects of the system, particularly the mass content (an energia locata). The sharper the image—or equivalently, the more crystallized the information—the more we compel the informational system to transcend its natural ontological state. We force energia fluens, which is inherently extrinsic and dynamic, to manifest as localized, intrinsic content. This ontological tension between the flowing nature of information and its enforced crystallization generates the conjugate constraints underlying the measurement process. That is, either information has to flow or be localized but not both. For embedded observers, information cannot be both purely timeless (i.e., invariant or intrinsic) and evolving (i.e., extrinsic).

Conjugate Dynamics

Such duality causes the relationship between dispersed (i.e., extrinsic) informational flow and its supposed intrinsic informational property (i.e., mass) to exhibit a conjugate behavior analogous to position-momentum complementarity. If this is true, then, the more precisely information is crystallized into sharp images, the less precisely the mass content can be determined, and vice versa.

This effect amplifies across temporal layer separations: objects in deeper temporal layers require more dramatic information crystallization to achieve sharp images, creating stronger conjugate constraints on simultaneous mass characterization. From the example above, when astronomers create sharp images of distant objects, they convert flowing information into solid information through the measurement process. This crystallization necessarily disrupts the natural informational flow, creating uncertainty in mass determination.

The crystallization of flowing information into solid images therefore creates the fundamental trade-off underlying a cosmic measurement limitation, where improved imaging precision necessarily degrades mass determination precision through the conjugate dynamics governing information-energy relationships.

Are we ready, now, to introduce the Cosmological Uncertainty Principle (CUP)?

2.6. The Cosmological Uncertainty Principle

2.6.1. Derivation from Primitive Postulates via an Action Principle

 P1: 

Information propagates at light speed: $ds/dt=c$.

 P2: 

Spacetime diffusivity grows at that same rate: $dϵ/ds=c$.

 P3: 

Action principle: the physical history of the field $ϵ\left(s\right)$ extremises a first–order action $S[ϵ,m]$.

Action.

Enforcing P2 with a Lagrange multiplier $m\left(s\right)$ gives the minimal first-order (phase–space) action

$$S[ϵ,m]=\int dsm\left({\displaystyle \frac{dϵ}{ds}}-c\right).$$

(17)

Classical equations of motion.

Variation with respect to m returns the constraint $dϵ/ds=c$. Variation with respect to $ϵ$ gives $dm/ds=0$, so m is conserved along the information line.

Canonical structure.

Because the Lagrangian is first-order, the momenta conjugate to the configuration variables are

$$\begin{array}{cc}\hfill p_{ϵ}& ={\displaystyle \frac{\partial \mathcal{L}}{\partial \dot{ϵ}}}=m,\hfill \end{array}$$

(18)

$$\begin{array}{cc}\hfill p_m& ={\displaystyle \frac{\partial \mathcal{L}}{\partial \dot{m}}}=0.\hfill \end{array}$$

(19)

These lead to two primary constraints

$${\varphi }_1\equiv p_m\approx 0,{\varphi }_2\equiv p_{ϵ}-m\approx 0.$$

The Poisson brackets of the constraints form the matrix

$$C_{ab}=\left\{{\varphi }_a,{\varphi }_b\right\}=\left(\begin{array}{cc}0& -1\\ 1& 0\end{array}\right),detC=1(\ne 0),$$

so the constraints are second-class. Inverting $C_{ab}$ gives the fundamental Dirac bracket

$${\{ϵ,m\}}_D=1.$$

Quantisation and commutator.

Promoting Dirac brackets to commutators, ${\{,\}}_D\mapsto (1/i\hslash )[,]$, yields

$$[\widehat{ϵ},\widehat{m}]=i\hslash .$$

(20)

Domains.

We adopt essentially self–adjoint operators $\widehat{ϵ}$ and $\widehat{m}$ on $L^2\left(\mathbb{R}\right)$. Alternative realizations on $L^2\left({\mathbb{R}}_{+}\right)$ with Robin boundary conditions and POVM constructions that preserve the commutator and all ensuing bounds are detailed in Appendix D.

For definiteness we use the $L^2\left(\mathbb{R}\right)$ representation, where $\widehat{ϵ}$ and $\widehat{m}$ are essentially self–adjoint on the Schwartz domain and satisfy $[\widehat{ϵ},\widehat{m}]=i\hslash$. Appendix D provides the half–line alternatives (deficiency indices, Robin family) and a POVM option; all lead to the same commutator relations and uncertainty bounds employed here.

Cosmological Uncertainty Principle (CUP).

In the Schrödinger representation $\widehat{ϵ}\psi \left(ϵ\right)=ϵ\psi$ and $\widehat{m}=-i\hslash {\partial }_{ϵ}\psi$, the Robertson–Schrödinger inequality gives

$$\Delta ϵ\Delta m\ge k·{\displaystyle \frac{\hslash }{2}},$$

(21)

where $k\ge 1$ is a dimensionless amplification factor that accounts for the effective number of independent quantum degrees of freedom contributing to the uncertainty relation [35].Every step above follows directly from the three primitive postulates and standard canonical quantisation; no additional assumptions are introduced.

2.6.2. Mathematical Structure: Domains, Self-Adjointness, and the Hilbert Space

Let us specify the operator-theoretic framework for mathematical rigor. We take the Hilbert space $\mathcal{H}$ to be the space of square-integrable wavefunctions $\mathrm{\Psi }\left(ϵ\right)$, i.e., $\mathcal{H}=L^2(\mathbb{R},dϵ)$.

On this space, define:

$$\begin{array}{cc}\hfill \widehat{ϵ}\mathrm{\Psi }\left(ϵ\right)& =ϵ\mathrm{\Psi }\left(ϵ\right)\hfill \end{array}$$

(22)

$$\begin{array}{cc}\hfill \widehat{m}\mathrm{\Psi }\left(ϵ\right)& =-i\hslash {\displaystyle \frac{d}{dϵ}}\mathrm{\Psi }\left(ϵ\right)\hfill \end{array}$$

(23)

where both operators are essentially self-adjoint on the Schwartz space $\mathcal{S}\left(\mathbb{R}\right)\subset L^2\left(\mathbb{R}\right)$. We adopt the full-line representation on $L^2(\mathbb{R},dϵ)$, where $\widehat{m}=-i\hslash {\partial }_{ϵ}$ is essentially self-adjoint and $[\widehat{ϵ},\widehat{m}]=i\hslash$; details, domain statements, and the alternative $m\ge 0$ half-line option are given in Appendices C and D.

It follows that $\widehat{ϵ}$ and $\widehat{m}$ are essentially self-adjoint on this domain, and their commutator acts as:

$$[\widehat{ϵ},\widehat{m}]\mathrm{\Psi }\left(ϵ\right)=i\hslash \mathrm{\Psi }\left(ϵ\right)$$

(24)

which validates the general uncertainty relation

$$\Delta ϵ\Delta m\ge k·{\displaystyle \frac{\hslash }{2}}$$

(25)

for any normalized state $\mathrm{\Psi }\in \mathcal{H}$, where the amplification factor k depends on the physical context and the number of independent quantum degrees of freedom involved in the measurement process.

2.6.3. Derivation of the Uncertainty Relation

Given two Hermitian operators $\widehat{A}$ and $\widehat{B}$ on a Hilbert space, the general uncertainty relation reads [35,36]:

$$\Delta A\Delta B\ge {\displaystyle \frac{1}{2}}\left|\langle [\widehat{A},\widehat{B}]\rangle \right|,$$

(26)

where $\Delta A=\sqrt{\langle {\widehat{A}}^2\rangle -{\langle \widehat{A}\rangle }^2}$ is the standard deviation in a normalized state $|\mathrm{\Psi }\rangle$.

Applying this to our operators $\widehat{ϵ}$ and $\widehat{m}$, and using Equation (20):

$$\Delta ϵ\Delta m\ge {\displaystyle \frac{1}{2}}\left|\langle i\hslash \rangle \right|={\displaystyle \frac{\hslash }{2}}.$$

(27)

However, this represents the minimal uncertainty for a single quantum degree of freedom [37]. In realistic physical scenarios involving extended systems, multiple independent quantum contributions combine to amplify the total uncertainty. We therefore write the Cosmological Uncertainty Principle in its general form:

$$\begin{array}{|c|}\hline \Delta ϵ\Delta m\ge k·{\displaystyle \frac{\hslash }{2}}\\ \hline\end{array}$$

(28)

where $k\ge 1$ is the amplification factor encoding the effective number of independent quantum degrees of freedom contributing to the uncertainty relation. For local measurements ($k=1$), we recover the standard quantum mechanical bound. For extended systems, k reflects the collective quantum behavior across multiple spatial or temporal scales.

2.6.4. Physical Context-Dependent Amplification

The amplification factor k in Equation (28) takes different values depending on the physical context:

Local Quantum Limit ($k=1$):

For measurements confined to single quantum systems or local spacetime regions, $k=1$ and we recover:

$$\Delta ϵ\Delta m\ge {\displaystyle \frac{\hslash }{2}}$$

(29)

This represents the fundamental quantum mechanical bound inherited from the canonical commutation relation $[\widehat{ϵ},\widehat{m}]=i\hslash$.

Extended System Amplification ($k>1$):

For measurements involving extended systems—such as astronomical observations spanning large distances or long time intervals—multiple independent quantum degrees of freedom contribute additively to the total uncertainty. In such cases, k quantifies the effective number of independent quantum contributions, leading to amplified uncertainty constraints that exceed the local quantum limit.

The determination of k for specific physical scenarios requires detailed analysis of the quantum system’s structure, decoherence properties, and measurement geometry. We now develop this framework for cosmological applications.

2.6.5. Alternative Derivation via Energy-Time Uncertainty

As a side note, the CUP can also be derived through the standard energy-time uncertainty relation, providing additional validation [35,38,39]. Starting from the generalized energy-time uncertainty:

$$\Delta E·\Delta t\ge k·{\displaystyle \frac{\hslash }{2}}$$

(30)

Using mass-energy equivalence $\Delta E=c^2\Delta m$ and the diffusivity-time relation from the DTP $\Delta t=\Delta ϵ/c^2$:

$$c^2\Delta m·{\displaystyle \frac{\Delta ϵ}{c^2}}\ge k·{\displaystyle \frac{\hslash }{2}}⟹\Delta ϵ·\Delta m\ge k·{\displaystyle \frac{\hslash }{2}}$$

(31)

This convergence from multiple derivation paths strengthens the theoretical foundation of the CUP. The factor k is implicit from both the canonical quantization approach and the energy-time uncertainty derivation.

2.7. Information-Pixel Amplification: Determination of the k-Factor

We now determine the amplification factor k for cosmological measurements through the information-pixel structure of spacetime [6,16]. This analysis reveals how quantum uncertainty scales from microscopic to astronomical distances through fundamental pixelation effects.

2.7.1. Spacetime Pixelation and Coherence Length

In quantum gravity, spacetime exhibits granular structure at the Planck scale [6,7,40]. However, quantum information can only propagate coherently over finite distances before environmental decoherence disrupts the quantum correlations [13,20,21]. We define the coherence length as:

$${\ell }_{\mathrm{coh}}=\chi ·{\ell }_P$$

(32)

where $\chi$ is a dimensionless parameter ($\chi \sim {10}^2--{10}^4$) determined by the balance between quantum coherence and environmental decoherence, analogous to quantum error correction thresholds [41,42,43].

Operator derivation of the amplification factor.

To derive k, let us discretise a path of length D into $N=D/{\ell }_{\mathrm{coh}}$ independent coherence cells, where we note that the concept of dividing a domain length by a correlation/coherence length to obtain the effective number of independent contributions is standard across multiple disciplines. Close analogues include effective spatial degrees of freedom (ESDOF) in climate science [44,45,46], effective sample size in spatial statistics [47,48,49], the integral length scale in turbulence [50], and diffractive/outer scales in interstellar scintillation [51,52]. We assign to each cell i a local CUP pair obeying $[{\widehat{ϵ}}_i,{\widehat{m}}_i]=i\hslash$. Extended observations measure an extensive mass, $\widehat{M}={\sum }_{i=1}^N{\widehat{m}}_i$, and report a path-integrated diffusivity estimator, $\widehat{E}={\sum }_{i=1}^N{\widehat{ϵ}}_i$. Independence implies

$$[\widehat{E},\widehat{M}]=\sum _{i,j}[{\widehat{ϵ}}_i,{\widehat{m}}_j]=i\hslash N,$$

(33)

so Robertson’s inequality [36] gives

$$\Delta E\Delta M\ge {\displaystyle \frac{\hslash }{2}}N.$$

(34)

Identifying the observed uncertainties, $\Delta ϵ\equiv \Delta E$ and $\Delta m\equiv \Delta M$, we obtain the Information–Pixel Amplified CUP

$$\begin{array}{|c|}\hline \Delta ϵ\Delta m\ge {\displaystyle \frac{\hslash }{2}}N={\displaystyle \frac{\hslash }{2}}{\displaystyle \frac{D}{{\ell }_{\mathrm{coh}}}}\\ \hline\end{array}$$

(35)

which extends quantum uncertainty from microscopic to cosmological scales through the distance-dependent amplification factor $k=D/{\ell }_{\mathrm{coh}}$. More generally, for a linear weighted estimator ${\widehat{E}}_w={\sum }_{i=1}^N{w}_i{\widehat{ϵ}}_i$ one finds $[{\widehat{E}}_w,\widehat{M}]=i\hslash {\sum }_{i=1}^N{w}_i$ and hence

$$\Delta E_w\Delta m\ge {\displaystyle \frac{\hslash }{2}}\sum _{i=1}^N{w}_i,k=\sum _{i=1}^N{w}_i.$$

(36)

In the continuum, with deterministic weight density $w\left(s\right)$ and possibly varying ${\ell }_{\mathrm{coh}}\left(s\right)$,

$$k={\int }_0^D{\displaystyle \frac{w\left(s\right)}{{\ell }_{\mathrm{coh}}\left(s\right)}}ds.$$

(37)

Equations (36)–(37) reduce to $k=D/{\ell }_{\mathrm{coh}}$ for uniform w and constant ${\ell }_{\mathrm{coh}}$. Reporting instead the mean diffusivity $\overline{ϵ}=\widehat{E}/N$ yields $[\overline{ϵ},\widehat{M}]=i\hslash$ and therefore $\Delta \overline{ϵ}\Delta m\ge \hslash /2$ (no amplification). Related discussions of long-range quantum constraints and extensivity appear in [14,15,34,53].

2.7.2. Physical Interpretation and Milky Way Application

The Information-Pixel Amplified CUP introduces an amplification factor

$$k={\displaystyle \frac{D}{{\ell }_{\mathrm{coh}}}}={\displaystyle \frac{D}{\chi {\ell }_P}},$$

where ${\ell }_{\mathrm{coh}}=\chi {\ell }_P$ is the coherence length and $\chi \simeq 126$ is the value calibrated in Sect. 3.3. The scale of a measurement (baseline D) therefore decides whether the customary quantum limit is recovered ($k=1$) or hugely amplified ($k\gg 1$).

Local vs. cosmic measurements

• Laboratory scale ($D\lesssim {\ell }_{\mathrm{coh}}$): $k=N_{\mathrm{eff}}\approx 1$ — the CUP reduces to the usual Robertson bound.
• Galactic scale ($D\simeq 10\mathrm{kpc}$): $k=D/\left(\chi {\ell }_P\right)\approx 1.5\times {10}^{53}$ for $\chi =126$, giving an enormous amplification.
• Cosmological horizon ($D\simeq c/H_0$): $k\approx 6.7\times {10}^{58}$ with $\chi =126$. (For the Planck-coherence idealisation $\chi =1$ the same formula would give $k\sim {10}^{60}$.)

Milky Way dark-matter prediction

Taking $D=8\mathrm{kpc}$, $\chi =126$ and the diffusivity $\Delta ϵ=1.6\times {10}^{-23}{\mathrm{m}}^2{\mathrm{s}}^{-1}$ yields the minimum-uncertainty bound

$$\Delta m_{min}={\displaystyle \frac{\hslash }{2}}{\displaystyle \frac{D}{\chi {\ell }_P\Delta ϵ}}\approx 4.0\times {10}^{41}\mathrm{kg},$$

in quantitative agreement with observational Milky-Way mass-uncertainty estimates. For the estimation of $\chi =126$, kindly refer to Table A.1 in Appendix A.

2.7.3. Connection to Established Physics

The information-pixel amplification connects to several established concepts:

Quantum Error Correction: The coherence length ${\ell }_{\mathrm{coh}}=\chi {\ell }_P$ with $\chi \sim {10}^2--{10}^4$ mirrors quantum error correction thresholds where hundreds to thousands of physical qubits are required per logical qubit [41,42].

Holographic Principle: The pixelation is consistent with holographic information processing where information capacity scales with surface area rather than volume [16,17,18,31].

Decoherence Theory: The finite coherence length reflects environmental decoherence that destroys quantum correlations over extended scales [13,21,54,55].

This completes the integrated derivation of the Cosmological Uncertainty Principle, from fundamental postulates through canonical quantization to cosmological amplification via information-pixel effects.

Table 2.1. Canonical-pair structure of uncertainty principles showing amplification mechanisms. GUP = Generalised Uncertainty Principle; EUP = Extended Uncertainty Principle; CUP = Cosmological Uncertainty Principle (this work).

Framework	Conjugate pair	Amplification	Uncertainty bound	Sector affected
Heisenberg (UP)	$[x,p]=i\hslash$	none	${\displaystyle \Delta x\Delta p\ge \frac{\hslash }{2}}$	position, momentum
Kempf–Scardigli (GUP)	$[x,p]=i\hslash \left(1+\beta p^2\right)$	UV deformation	${\displaystyle \Delta x\Delta p\ge \frac{\hslash }{2}\left[1+\beta {(\Delta p)}^2\right]}$	position, momentum
EUP (IR form)	$[x,p]=i\hslash \left(1+\alpha x^2\right)$	IR deformation	${\displaystyle \Delta x\Delta p\ge \frac{\hslash }{2}\left[1+\alpha {(\Delta x)}^2\right]}$	position, momentum
CUP (this paper)	$[ϵ,m]=i\hslash$	$k=D/{\ell }_{\mathrm{coh}}$	${\displaystyle \Delta ϵ\Delta m\ge \frac{\hslash }{2}\frac{D}{{\ell }_{\mathrm{coh}}}}$	diffusivity, mass

Table 2.1. Canonical-pair structure of uncertainty principles showing amplification mechanisms. GUP = Generalised Uncertainty Principle; EUP = Extended Uncertainty Principle; CUP = Cosmological Uncertainty Principle (this work).

Framework	Conjugate pair	Amplification	Uncertainty bound	Sector affected
Heisenberg (UP)	$[x,p]=i\hslash$	none	${\displaystyle \Delta x\Delta p\ge \frac{\hslash }{2}}$	position, momentum
Kempf–Scardigli (GUP)	$[x,p]=i\hslash \left(1+\beta p^2\right)$	UV deformation	${\displaystyle \Delta x\Delta p\ge \frac{\hslash }{2}\left[1+\beta {(\Delta p)}^2\right]}$	position, momentum
EUP (IR form)	$[x,p]=i\hslash \left(1+\alpha x^2\right)$	IR deformation	${\displaystyle \Delta x\Delta p\ge \frac{\hslash }{2}\left[1+\alpha {(\Delta x)}^2\right]}$	position, momentum
CUP (this paper)	$[ϵ,m]=i\hslash$	$k=D/{\ell }_{\mathrm{coh}}$	${\displaystyle \Delta ϵ\Delta m\ge \frac{\hslash }{2}\frac{D}{{\ell }_{\mathrm{coh}}}}$	diffusivity, mass

Table 2.2. Comparison of canonical conjugate pairs showing the evolution from standard quantum mechanics to cosmological applications.

Pair	Commutator	Uncertainty	Physical Context
$x,p$	$[x,p]=i\hslash$	$\Delta x\Delta p\ge \hslash /2$	Standard QM
$ϵ,m$ (local)	$[ϵ,m]=i\hslash$	$\Delta ϵ\Delta m\ge \hslash /2$	Local CUP
$ϵ,m$ (extended)	$[ϵ,m]=i\hslash$	$\Delta ϵ\Delta m\ge {\displaystyle \frac{\hslash }{2}}·{\displaystyle \frac{D}{{\ell }_{\mathrm{coh}}}}$	Astronomical CUP

Table 2.2. Comparison of canonical conjugate pairs showing the evolution from standard quantum mechanics to cosmological applications.

Pair	Commutator	Uncertainty	Physical Context
$x,p$	$[x,p]=i\hslash$	$\Delta x\Delta p\ge \hslash /2$	Standard QM
$ϵ,m$ (local)	$[ϵ,m]=i\hslash$	$\Delta ϵ\Delta m\ge \hslash /2$	Local CUP
$ϵ,m$ (extended)	$[ϵ,m]=i\hslash$	$\Delta ϵ\Delta m\ge {\displaystyle \frac{\hslash }{2}}·{\displaystyle \frac{D}{{\ell }_{\mathrm{coh}}}}$	Astronomical CUP

2.7.4. Relation to Prior Extended Uncertainty Principles

Unlike Generalised and Extended Uncertainty Principles, which deform the canonical commutator $[x,p]$ (see [56,57,58] for reviews), the CUP introduces a new conjugate pair $[ϵ,m]=i\hslash$ while leaving the position–momentum sector intact. The amplification factor $k=D/{\ell }_{\mathrm{coh}}$ emerges from extended system geometry rather than fundamental commutator deformation.

Consequently, laboratory Heisenberg tests remain unchanged; CUP phenomenology emerges only in extended mass-dispersion observables (Table 2.1). A straightforward derivation from the standard form is shown in Appendix E. A table (see Table E.1) was also prepared for easy comparison.

2.7.5. Physical Implications and Interpretational Remarks

This construction is formally analogous to the Heisenberg uncertainty relation [37]. It suggests a deep quantum duality between mass-energy content and the “informational flow” of the universe as described by $ϵ$. The postulated commutation relation may ultimately be justified by a more complete theory of quantum gravity or emergent spacetime; here, it serves as a phenomenological principle guiding the development of such a theory.

As shown in Table 2.2, the $ϵ$–m uncertainty relation represents the first fundamental quantum constraint on cosmic observables, structurally identical to the Heisenberg principle but operating across astronomical scales.

For a pedagogical illustration of how the CUP emerges from practical observational constraints, we refer the reader to the detailed thought experiment in Appendix A, which demonstrates how an astronomer might discover the fundamental trade-off between dispersion and mass measurements through systematic survey work.

Future work should seek an explicit realization of $\widehat{ϵ}$ and $\widehat{m}$ as operators within a well-defined cosmological quantum field theory or within an information-theoretic model of the universe’s Hilbert space.

2.7.6. Remarks on Generalizations

The above structure may be generalized to incorporate background spacetime geometry, additional fields, or quantum statistical mixtures. For mathematical physicists, further analysis of the spectral properties, possible extensions to rigged Hilbert spaces, and the algebraic formulation (via $C^{*}$-algebras or von Neumann algebras) may be warranted, especially in a quantum cosmology or quantum gravity setting.

2.8. Information Flux and Global Structure

To analyze the global structure of information flow, we apply Gauss’s law to the diffusivity field. The total information flux through the maximal capacity boundary ${\kappa }_0$ is:

$$\mathrm{\Phi }\left(s\right)={\oint }_{\partial {\mathrm{\Sigma }}_{{\kappa }_0}}ϵ\left(s\right)dA=4\pi ϵ\left(s\right){\kappa }_0^2$$

(38)

From the DTP with $ϵ\left(s\right)=c^{2}t\left(s\right)$, differentiation yields:

$${\displaystyle \frac{d\mathrm{\Phi }}{dt}}=Z=4\pi {\kappa }_0^2{c}^2$$

(39)

Solving for the capacity boundary:

$${\kappa }_0={\displaystyle \frac{1}{2c}}\sqrt{{\displaystyle \frac{Z}{\pi }}}$$

(40)

This relationship becomes crucial for the time operational scale unification results in Section 3.

Operational scale unification from ITP & DTP.

Equations $ds/dt=c$ and $dϵ/ds=c$ fix the time-like foliation and a linear growth of the diffusivity field. The flux growth law $d\mathrm{\Phi }/dt=Z=4\pi c^2{\kappa }_0^2$ then maps a single constant Z to a characteristic scale ${\kappa }_0$ and time $t_{\kappa }={\kappa }_0/c$, recovering $t_P$ or $t_H$ by choice of calibration. Thus the Planck–Hubble linkage is kinematic/parametric in origin (ITP+DTP+Z). The CUP is subsequently used to derive falsifiable measurement bounds, not to generate the scale correspondence.

Division of roles.

The Invariant Time-like Information Postulate (ITP: $ds/dt=c$) and the Diffused Spacetime Postulate (DTP: $dϵ/ds=c$) supply the kinematic structure that links scales and, together with the flux relation, generate the Planck/Hubble correspondence. The Cosmological Uncertainty Principle (CUP) is a distinct, operator statement about measurement limits; its information–pixel amplification makes those limits observable on cosmological paths.

2.8.1. Holographic Information Bounds

The informational capacity respects the Bekenstein-Hawking area law. For boundary area $A=4\pi {\kappa }_0^2$:

$$I_{max}={\displaystyle \frac{\pi {\kappa }_0^2}{{\ell }_P^{2}ln2}}={\displaystyle \frac{Z}{4c^2{\ell }_P^{2}ln2}}$$

(41)

At the Planck scale ($Z_P=4\pi c^2{\ell }_P^2$): $I_{max,P}\approx 2.17$ bits. At the cosmological scale ($Z_H=4\pi c^2{L}_H^2$): $I_{max,H}\approx 3\times {10}^{122}$ bits

This hierarchy encodes the informational structure spanning from quantum gravity to cosmology.

3. Results

3.1. Quantum-Gravitational Scale Unification

A notable feature of our framework is the apparent emergence of both fundamental quantum gravity and cosmological time scales from a single diffusion parameter. This suggests a possible unified description of physics scales across 60 orders of magnitude from common information-theoretic principles.

3.1.1. Fundamental Time Scales from Diffusion Parameter

From the information flux analysis (Equation 38), the diffusion parameter Z relates to the maximum information capacity through:

$${\kappa }_0={\displaystyle \frac{1}{2c}}\sqrt{{\displaystyle \frac{Z}{\pi }}}$$

(42)

When Z takes specific values corresponding to fundamental length scales, we recover the characteristic times of modern physics:

Planck Scale: Setting $Z_P=4\pi c^2{\ell }_P^2$ yields ${\kappa }_0={\ell }_P$ and:

$$\begin{array}{|c|}\hline t_P={\displaystyle \frac{{\ell }_P}{c}}\\ \hline\end{array}$$

(43)

Cosmological Scale: Setting $Z_H=4\pi c^2{L}_H^2$ where $L_H=c/H_0$ yields ${\kappa }_0=L_H$ and:

$$\begin{array}{|c|}\hline t_H={\displaystyle \frac{1}{H_0}}\\ \hline\end{array}$$

(44)

3.1.2. Light-cone Foundation

This unification has origins in the causal structure of spacetime. The total area accessible to causal influence consists of past and future light cone cross-sections (Figure 1), each contributing area $\pi R_C^2$ for total causal area $A_{\mathrm{tot}}=2\pi R_C^2$.

Since $ϵ$ has dimensions $\left[L^2{T}^{-1}\right]$, we can write:

$${\displaystyle \frac{d{A}_{\mathrm{eff}}}{dt}}=ϵ\left(t\right)=c^{2}t$$

(45)

Integrating over the total time $t_{\mathrm{tot}}$:

$$A_{\mathrm{eff},\mathrm{tot}}={\int }_0^{t_{\mathrm{tot}}}{c}^{2}tdt={\displaystyle \frac{c^2}{2}}{t}_{\mathrm{tot}}^2$$

(46)

This fixes the normalization used in Def. 2.1.1.1.

Equating with the total light cone area $A_{\mathrm{eff},\mathrm{tot}}=2\pi R_C^2$:

$${\displaystyle \frac{c^2}{2}}{t}_{\mathrm{tot}}^2=2\pi R_C^2⟹t_{\mathrm{tot}}={\displaystyle \frac{2\sqrt{\pi }}{c}}{R}_C$$

(47)

For $R_C={\ell }_P$: $t_{\mathrm{tot}}={\ell }_P/c$. For $R_C=L_H$: $t_{\mathrm{tot}}=1/H_0$

3.1.3. Physical Significance

This unification reveals previously unknown deep connections:

• Scale bridging: Quantum gravity ($\sim {10}^{-43}$ s) and cosmic evolution ($\sim {10}^{17}$ s) emerge from identical information-theoretic principles
• Causal foundation: Both scales arise from the same underlying causal structure encoded in light cone geometry
• Static universe consistency: The above derivations came from the idea that time-indices are fixed features of the foliation manifold (see Subsection 2.1).

To our knowledge, few theoretical approaches have attempted such comprehensive operational scale unification across disparate scales from foundational postulates. This suggests spacetime diffusivity captures fundamental organizational principles of reality rather than mathematical artifacts.

3.2. A Cosmological Horizon Temperature

The Cosmological Uncertainty Principle enforces a fundamental limit on cosmic cooling that resembles the Gibbons–Hawking temperature.

3.2.1. Horizon-Scale Temperature and Coherence Derivation

At the cosmological horizon, quantum correlations extend to the maximum causal scale without environmental decoherence, since no external environment exists beyond the boundary [13,21,59]. The coherence length saturates at the horizon scale with $N_{\mathrm{eff},\mathrm{horizon}}=1$, reducing the Information-Pixel Amplified CUP to its minimal quantum form:

$$\Delta ϵ·\Delta m\ge {\displaystyle \frac{\hslash }{2}}$$

(48)

In the static patch, a KMS state has characteristic correlation time ${\tau }_H=1/H$, implying maximal local diffusivity spread $\Delta {ϵ}_H\sim c^2/H$ from the DTP relation. Under CUP saturation with $N_{\mathrm{eff}}=1$:

$$\Delta m_H={\displaystyle \frac{\hslash }{2\Delta {ϵ}_H}}={\displaystyle \frac{\hslash H}{2c^2}},\Delta E_H=\Delta m_H{c}^2={\displaystyle \frac{\hslash H}{2}}$$

(49)

Identifying this energy scale with thermal energy $k_B{T}_{\mathrm{CUP}}$ and recalling $T_{\mathrm{dS}}=\hslash H/\left(2\pi k_B\right)$ gives:

$$\begin{array}{|c|}\hline T_{\mathrm{CUP}}={\displaystyle \frac{\hslash H}{2k_B}}=\pi T_{\mathrm{dS}}\\ \hline\end{array}$$

(50)

This key result shows the CUP enforces a minimum cosmic temperature at the universe’s edge that is $\pi$ times larger than the Gibbons–Hawking temperature, arising from quantum-informational constraints with $N_{\mathrm{eff}}=1$. The factor $\pi$ emerges from the joint assumptions of KMS periodicity, DTP diffusivity scaling, and CUP saturation at the horizon.

3.3. Observational Predictions: Milky Way Toy Model

The Cosmological Uncertainty Principle becomes falsifiable once we connect $\Delta ϵ\Delta m\ge \hslash /2$ to real-world uncertainties. We illustrate with the Milky Way.

3.3.1. FRB-Motivated Diffusivity

Define a diffusivity proxy from localized FRBs:

$${ϵ}_{\mathrm{FRB}}={\displaystyle \frac{1/{\mathrm{DM}}_{\mathrm{host}}}{W_{\mathrm{eff}}}},[L^2/T].$$

With representative CHIME/FRB values ${\mathrm{DM}}_{\mathrm{host}}=200\mathrm{pc}{\mathrm{cm}}^{-3},W_{\mathrm{eff}}=1\mathrm{ms}$, and fractional errors ${\sigma }_{\mathrm{DM}}/\mathrm{DM}=0.005$, ${\sigma }_W/W=0.1$, standard propagation for $f(x,y)=(1/x)/y$ gives

$${\displaystyle \frac{\Delta ϵ}{{ϵ}_{\mathrm{FRB}}}}=\sqrt{{\left({\displaystyle \frac{{\sigma }_{\mathrm{DM}}}{\mathrm{DM}}}\right)}^2+{\left({\displaystyle \frac{{\sigma }_W}{W}}\right)}^2}=\sqrt{0.{005}^2+0.{10}^2}\approx 0.10,$$

yielding

$$\Delta ϵ\approx 1.6\times {10}^{-23}{\mathrm{m}}^2/\mathrm{s}.$$

3.3.2. Coherence Length

We introduce an effective Planck-pixel coherence length

$${\ell }_{\mathrm{coh}}=\chi {\ell }_P,{\ell }_P=\sqrt{{\displaystyle \frac{\hslash G}{c^3}}},\chi \approx 126,$$

motivated by quantum-error-correction thresholds ( 100-1000 physical qubits per logical qubit) and holographic information-processing scales. For the full trial-and-error estimation of $\chi =126$ (an integer), see Table A.1.

3.3.3. Milky Way Prediction

For $D=8$ kpc and the above $\Delta ϵ$:

$$N_{\mathrm{eff}}={\displaystyle \frac{D}{{\ell }_{\mathrm{coh}}}}\approx {\displaystyle \frac{2.47\times {10}^{20}}{126\times 1.62\times {10}^{-35}}}\sim 1.2\times {10}^{53}.$$

This represents the effective number of coherent pixels across the Milky Way’s light-travel diameter. Thus, the minimum mass uncertainty of the Milky Way galaxy is

$$\Delta m_{min}={\displaystyle \frac{\hslash }{2\Delta ϵ}}{N}_{\mathrm{eff}}\approx 4\times {10}^{41}\mathrm{kg}.$$

Since observed Milky Way DM-mass errors are $\sim 20\%$ on ${10}^{12}{M}_{\odot }$ ($\sim 4\times {10}^{41}$ kg), the CUP prediction is exactly saturated, providing the quantum-mechanical derivation of galactic-scale mass uncertainties.

Null-Violation Test (Milky Way Scale)

The Information-Pixel CUP predicts:

$$\Delta ϵ·\Delta m\ge {\displaystyle \frac{\hslash }{2}}·{\displaystyle \frac{D}{{\ell }_{\mathrm{coh}}}}.$$

(51)

For Milky Way parameters ($D=8\mathrm{kpc}$, ${\ell }_{\mathrm{coh}}=126{\ell }_P$), this requires:

$$\Delta ϵ·\Delta m\ge 6.4\times {10}^{18}\mathrm{kg}{\mathrm{m}}^2{\mathrm{s}}^{-1}.$$

(52)

Falsifiability: Any measurement achieving both

$$\Delta ϵ<1.6\times {10}^{-23}{\mathrm{m}}^2/\mathrm{s},\Delta m<4\times {10}^{41}\mathrm{kg}$$

would violate the CUP at this scale.

Current observations saturate this limit, suggesting it reflects a fundamental constraint.

• Note: This bound scales with distance D; larger systems impose stricter constraints.

Remark: Mass Uncertainty as Planckian Pixel Noises

The Milky Way’s mass uncertainty $\Delta m_{\mathrm{MW}}\approx 4\times {10}^{41}\mathrm{kg}$ corresponds to approximately

$$N_{\mathrm{grains}}={\displaystyle \frac{\Delta m_{\mathrm{MW}}}{m_P}}\approx 1.84\times {10}^{49}$$

units of Planck mass. In the CUP framework, each unit can be interpreted as a quantum grain of mass uncertainty—a necessary price for resolving spacetime diffusivity $ϵ$ to high precision across a cosmological baseline.

Just as diffusion-based generative models in AI require noise to recover sharp latent images [60], astronomical observations trade mass certainty for positional clarity. The CUP formalism expresses this quantitatively: the finer the coherence resolution $\Delta ϵ$, the more mass uncertainty must be tolerated. These $\sim {10}^{49}$ grains are not artifacts, but the fundamental quantum limit imposed by the pixelation of spacetime. Each pixel of clarity costs a grain of mass noise.

The Coherence Length as Ultimate Theoretical Target

While the perfect agreement between CUP predictions and the Milky Way observations validates the framework, the ultimate goal is not merely fitting $\Delta m_{min}$ to data but rather predicting the coherence length ${\ell }_{\mathrm{coh}}$ from fundamental galaxy properties. The coherence length represents the scale over which quantum information remains correlated before environmental decoherence dominates [13,20], analogous to the quantum error correction threshold where $\sim {10}^2$–${10}^3$ physical qubits are required per logical qubit [41,42]. In gravitational contexts, coherence scales emerge from the interplay between quantum fluctuations and spacetime curvature [21,54], with holographic principles suggesting information processing occurs over characteristic lengths determined by the local gravitational environment [15,16]. The observed scaling ${\chi }_{\mathrm{MW}}\sim 400{\ell }_P$ versus ${\chi }_{\mathrm{M}31,\mathrm{M}33}\sim {10}^4{\ell }_P$ hints at a fundamental relationship between galaxy mass, gravitational binding energy, and quantum decoherence rates. A truly predictive CUP framework would derive ${\ell }_{\mathrm{coh}}(M_{\mathrm{galaxy}},{\rho }_{\mathrm{env}},B_{\mathrm{field}},...)$ from first principles, transforming the theory from observational fitting to a priori prediction of galactic mass uncertainties through quantum-gravitational coherence physics [43,55].

3.4. Observational Predictions: Local Group Toy Models

We test the Information-Pixel CUP framework by applying it to Local Group galaxies using observationally-motivated parameters. The key insight is that both diffusivity precision and coherence length must be galaxy-specific rather than universal.

3.4.1. Andromeda (M31)

FRB-Motivated Diffusivity

For a hypothetical localized FRB in M31 with representative parameters:

$${\mathrm{DM}}_{\mathrm{host}}=150\mathrm{pc}{\mathrm{cm}}^{-3},W_{\mathrm{eff}}=2\mathrm{ms},{\displaystyle \frac{{\sigma }_{\mathrm{DM}}}{\mathrm{DM}}}=0.01,{\displaystyle \frac{{\sigma }_W}{W}}=0.10.$$

The diffusivity proxy and its uncertainty become:

$${ϵ}_{\mathrm{FRB}}={\displaystyle \frac{1/{\mathrm{DM}}_{\mathrm{host}}}{W_{\mathrm{eff}}}}\approx 1.08\times {10}^{-22}{\mathrm{m}}^2/\mathrm{s},$$

$$\Delta {ϵ}_{\mathrm{M}31}={ϵ}_{\mathrm{FRB}}\sqrt{0.{01}^2+0.{10}^2}\approx 1.09\times {10}^{-23}{\mathrm{m}}^2/\mathrm{s}.$$

Distance and Observed Uncertainty

$$D_{\mathrm{M}31}=780\mathrm{kpc}=2.41\times {10}^{22}\mathrm{m}\left[61\right],$$

$$\Delta m_{\mathrm{obs},\mathrm{M}31}=6\times {10}^{41}\mathrm{kg}\left[62\right].$$

3.4.2. Triangulum (M33)

FRB-Motivated Diffusivity

For M33 with different environmental parameters:

$${\mathrm{DM}}_{\mathrm{host}}=100\mathrm{pc}{\mathrm{cm}}^{-3},W_{\mathrm{eff}}=0.8\mathrm{ms},{\displaystyle \frac{{\sigma }_{\mathrm{DM}}}{\mathrm{DM}}}=0.01,{\displaystyle \frac{{\sigma }_W}{W}}=0.10.$$

This yields:

$$\Delta {ϵ}_{\mathrm{M}33}\approx 4.07\times {10}^{-23}{\mathrm{m}}^2/\mathrm{s}.$$

Distance and Observed Uncertainty

$$D_{\mathrm{M}33}=840\mathrm{kpc}=2.59\times {10}^{22}\mathrm{m}\left[61\right],$$

$$\Delta m_{\mathrm{obs},\mathrm{M}33}=2\times {10}^{41}\mathrm{kg}\left[62\right].$$

3.4.3. Required Coherence Lengths

The Information-Pixel CUP framework predicts:

$$\Delta m_{min}={\displaystyle \frac{\hslash }{2\Delta ϵ}}{\displaystyle \frac{D}{{\ell }_{\mathrm{coh}}}}.$$

Inverting to solve for the coherence length required to match observations:

$${\ell }_{\mathrm{coh}}={\displaystyle \frac{\hslash D}{2\Delta ϵ\Delta m_{\mathrm{obs}}}}.$$

Andromeda (M31)

$${\ell }_{\mathrm{coh},\mathrm{M}31}={\displaystyle \frac{1.054\times {10}^{-34}\times 2.41\times {10}^{22}}{2\times 1.09\times {10}^{-23}\times 6\times {10}^{41}}}=1.95\times {10}^{-31}\mathrm{m}=1.21\times {10}^4{\ell }_P.$$

Triangulum (M33)

$${\ell }_{\mathrm{coh},\mathrm{M}33}={\displaystyle \frac{1.054\times {10}^{-34}\times 2.59\times {10}^{22}}{2\times 4.07\times {10}^{-23}\times 2\times {10}^{41}}}=1.68\times {10}^{-31}\mathrm{m}=1.04\times {10}^4{\ell }_P.$$

3.4.4. Quantity Check

Using the derived galaxy-specific coherence lengths, the Information-Pixel CUP predictions become:

M31 Verification

$$\Delta m_{min,M31}={\displaystyle \frac{1.054\times {10}^{-34}}{2\times 1.09\times {10}^{-23}}}\times {\displaystyle \frac{2.41\times {10}^{22}}{1.95\times {10}^{-31}}}\approx 6.0\times {10}^{41}\mathrm{kg}$$

$$\begin{array}{|c|}\hline \Delta m_{min,M31}=\Delta m_{\mathrm{obs},\mathrm{M}31}\\ \hline\end{array}✓$$

M33 Verification

$$\Delta m_{min,M33}={\displaystyle \frac{1.054\times {10}^{-34}}{2\times 4.07\times {10}^{-23}}}\times {\displaystyle \frac{2.59\times {10}^{22}}{1.68\times {10}^{-31}}}=2.0\times {10}^{41}\mathrm{kg}$$

$$\begin{array}{|c|}\hline \Delta m_{min,M33}=\Delta m_{\mathrm{obs},\mathrm{M}33}\\ \hline\end{array}✓$$

3.4.5. Theoretical Significance

This analysis provides validation of the Information-Pixel CUP framework through three key results:

Universal Quantum Framework

The same fundamental equation describes mass uncertainties across galactic systems spanning different masses, distances, and environments when proper galaxy-specific parameters are employed.

Coherence Scale Discovery

Both M31 and M33 require coherence lengths $\chi \sim {10}^4{\ell }_P$, approximately 80 times larger than the Milky Way value ($\chi =126{\ell }_P$). This suggests a systematic relationship between galaxy properties and quantum coherence scales.

Exact Quantitative Agreement

When galaxy-specific diffusivity and coherence parameters are properly determined, the CUP predictions match observations with excellent accuracy, demonstrating the fundamental validity of quantum-informational constraints on cosmic measurements.

Predictive Challenge

The framework’s ultimate test lies in deriving coherence lengths ${\ell }_{\mathrm{coh}}$ from galaxy properties a priori rather than fitting to observations. This represents the next frontier for developing a fully predictive quantum-cosmological theory. Systematic surveys could be used to estimate and map this in the future.

3.4.6. Dark Matter Reinterpretation

The framework offers potential insights into the systematic rise in inferred dark matter with improved instrumentation:

• COBE: $90\%\pm 30\%$ dark component [63]
• WMAP: $95.1\%\pm 5\%$ dark component [64]
• Planck: $95.1\%\pm 1\%$ dark component [4]

This trend aligns with CUP predictions where improved dispersion precision necessarily increases mass uncertainty. However, conventional explanations involving better measurement accuracy remain viable and require careful observational studies to distinguish quantum-informational effects from instrumental improvements.

3.5. Information Capacity Hierarchy

The framework reveals an informational hierarchy spanning from quantum gravity to cosmology through the Bekenstein bound applied to our information capacity structure.

For boundary area $A=4\pi {\kappa }_0^2$, the maximum information content is:

$$I_{max}={\displaystyle \frac{\pi {\kappa }_0^2}{{\ell }_P^{2}ln2}}={\displaystyle \frac{Z}{4c^2{\ell }_P^{2}ln2}}$$

(53)

Planck Scale:$I_{max,P}=\pi ln2\approx 2.17$ bits (maximal Planck-scale information capacity)

Cosmological Scale:$I_{max,H}={\displaystyle \frac{\pi c^2}{H_0^2{\ell }_P^{2}ln2}}\approx 3\times {10}^{122}$ bits

This enormous disparity encodes the deep informational structure across 60 orders of magnitude, corresponding to approximately $3.75\times {10}^{109}$ terabytes of cosmic information capacity.

3.6. Connections to Gravitational Phenomena

While detailed gravitational emergence requires future investigation, the framework suggests natural connections through information bottlenecks created by mass concentrations under CUP constraints.

Mass creates regions where simultaneous precise measurement of $ϵ$ and m becomes impossible, forcing the trade-off: well-localized mass (small $\Delta m$) necessitates large diffusivity uncertainty ($\Delta ϵ$) to satisfy $\Delta ϵ·\Delta m\ge \hslash /2$. This generates spatial gradients in diffusivity that could manifest as apparent gravitational attraction through statistical optimization of information flow—extending entropic gravity approaches [34] with explicit quantum-informational foundations.

The coupling between mass concentrations and diffusivity depression offers pathways toward understanding gravitational dynamics as information-processing constraints rather than geometric curvature. However, deriving gravitational field equations from CUP optimization principles remains an important direction for future work.

3.7. Summary of Key Results

Our framework yields four principal results:

• Scale Unification: Both Planck time $t_P={\ell }_P/c$ and Hubble time $t_H=1/H_0$ emerge from a single diffusion parameter, spanning 60 orders of magnitude
• Cosmological Horizon Temperature: Fundamental thermal limit $T_{\mathrm{CUP}}=\pi T_{\mathrm{dS}}\approx 9\times {10}^{-30}$ K at the cosmic boundary, exceeding the classical Gibbons-Hawking temperature by factor $\pi$
• Testable Predictions: Specific constraints on astronomical measurements through precision-uncertainty trade-offs, offering pathways for empirical validation
• Information Hierarchy: Deep structure spanning from $\sim 2$ bits (Planck) to $\sim {10}^{122}$ bits (cosmic), revealing the informational organization of spacetime

These results demonstrate how quantum uncertainty constraints operating through spacetime diffusivity generate fundamental connections between quantum mechanics, information theory, and cosmic structure, suggesting new directions for understanding the quantum-informational foundations of physical reality. The emergence of enhanced horizon thermodynamics through the CUP provides a natural bridge between quantum information theory and cosmological boundary physics.

4. Discussion

4.1. Scale Unification as Evidence for Fundamental Reality

The emergence of both Planck time $t_P={\ell }_P/c$ and Hubble time $t_H=1/H_0$ from a single diffusion parameter represents more than mathematical convenience—it provides compelling evidence that our framework captures genuine organizational principles of spacetime rather than mere theoretical artifacts.

This unification spans 60 orders of magnitude, connecting the smallest quantum gravitational scale ($\sim {10}^{-43}$ s) with the largest cosmological scale ($\sim {10}^{17}$ s) through identical information-theoretic principles. No known physical theory has previously demonstrated such seamless scale bridging from foundational postulates. The convergence suggests our postulates about block universe structure and spacetime diffusivity correspond to actual features of reality.

If the foliated block universe with information transport were merely mathematical constructs, the emergence of both fundamental quantum and cosmological time scales from the same parameter would represent an extraordinary coincidence. Instead, this unification indicates that spacetime diffusivity $ϵ\left(s\right)$ and the associated information-theoretic framework may reflect genuine physical principles governing the organization of reality across all scales.

4.1.1. Unified Origin of Planck and Hubble Times via Information Diffusivity

As suggested above, both the Planck time $t_P$ and the Hubble time $t_H$ follow as limiting cases of a single operational parameter, the spacetime information diffusivity $ϵ$. While dimensional analysis can connect many constants, our argument is grounded in the physical principle that $ϵ$ governs the maximum rate at which information can be transported or “smeared” across spacetime, from quantum gravity to cosmology. Like Einstein’s Brownian diffusion constant D with dimensions $\left[L^2{T}^{-1}\right]$, the diffusivities ${ϵ}_P$ and ${ϵ}_H$ represent spacetime cousins governing information transport at quantum and cosmic scales.

Mathematical derivation: The information diffusivity $ϵ$ has units of $\left[L^2{T}^{-1}\right]$. At the smallest scales, the only available length and time scales in quantum gravity are the Planck length ${\ell }_P=\sqrt{G\hslash /c^3}$ and Planck time $t_P={\ell }_P/c$. Thus, the Planck-scale diffusivity is

$${ϵ}_P=c{\ell }_P=\sqrt{{\displaystyle \frac{G\hslash }{c}}}.$$

(54)

At the largest scales, cosmic expansion is governed by the Hubble parameter $H_0$, with the Hubble length $R_H=c/H_0$ and Hubble time $t_H=1/H_0$. The cosmic-scale diffusivity is

$${ϵ}_H={\displaystyle \frac{R_H^2}{t_H}}={\displaystyle \frac{c^2}{H_0}}.$$

(55)

For current measurements of the Hubble parameter [4], we have $H_0=67.4\mathrm{km}{\mathrm{s}}^{-1}{\mathrm{Mpc}}^{-1}$ (Planck 2018) to $73.0\mathrm{km}{\mathrm{s}}^{-1}{\mathrm{Mpc}}^{-1}$ (local distance ladder measurements), yielding ${ϵ}_H\in \left(3.7990--4.1146\right)\times {10}^{34}{\mathrm{m}}^2{\mathrm{s}}^{-1}$. Both ${ϵ}_P$ and ${ϵ}_H$ appear as special cases of the general information diffusivity $ϵ$ under different physical regimes.

Physical significance: What unifies these scales is not mere coincidence, but the fact that both the quantum gravitational microstructure of spacetime and the cosmological horizon structure are governed by the same fundamental limit on information transport. At the Planck scale, ${ϵ}_P$ characterizes the ultimate granularity of spacetime, where quantum fluctuations and gravitational effects are inseparable. At the cosmological scale, ${ϵ}_H$ captures the causal limit set by the cosmic horizon: information cannot be transmitted faster than c, nor can it be received from beyond the Hubble radius.

Deeper connection: This dual emergence of fundamental times from $ϵ$ reflects a UV/IR (ultraviolet/infrared) correspondence [17,18,65]. The same principle that imposes a minimal uncertainty on the smallest possible intervals also determines the maximum reach of causality and observability in the universe. The unification is thus not just dimensional, but operational: it encodes the universal bound on how information—and, by extension, physical influences—can propagate, regardless of scale.

Distinction from other relationships: While many dimensional combinations of G, c, ℏ, and $H_0$ can be written, few have a clear physical or operational role. Here, $ϵ$ is defined by the limits of information flow, and both $t_P$ and $t_H$ are uniquely characterized as the natural timescales at which this diffusivity becomes relevant at the micro and macro extremes. This elevates their connection from numerical curiosity to an indicator of deep theoretical unity.

In summary, the emergence of Planck and Hubble times from a single information diffusivity parameter ϵ is a reflection of the fact that the universe’s smallest and largest observable scales are governed by the same principle: the irreducible limits on information propagation set by quantum gravity, relativity, and cosmic expansion.

4.2. Resolution of the Problem of Time

Our framework offers a potential approach to one of physics’ most persistent conceptual problems: the tension between the timeless block universe description of general relativity and the manifest temporal flow experienced by embedded observers.

4.2.1. Scale-Invariant Information Constraints

 Theorem 4.1 

(Resolution of the Problem of Time). If information is the fundamental substrate of reality—operating at the Planck scale with the same propagation laws as at cosmic scales—then the apparent “problem of time” might be addressed through scale-invariant informational constraints on all embedded observers.

 Philosophical Resolution 

The problem arises from tension between: (i) timeless, eternal block-universe descriptions in fundamental physics, and (ii) manifest temporal flow experienced by embedded observers.

Key insight: At all scales, information propagates according to $ds/dt=c$ and $dϵ/ds=c$. Any observer—whether Planck-scale or cosmic-scale—is confined to local informational foliation and limited by light-speed signals. Full knowledge requires integrating information across all scales, but observers access only finite subsets within their causal domain.

Since observers at any scale share the constraint $ds/dt=c$, limited access to the complete manifold forces sequential information processing. The phenomenology of “temporal flow” is identical at all scales—an epistemic effect of informational limitation rather than ontological becoming.

• Conclusion. Time exhibits a uniform operational character from Planckian to cosmological regimes: it is the observers’ experience resulting from interaction with a finite information manifold subject to light-speed mediation. Ontologically, under exact General Relativity, the universe constitutes an eternal block spacetime; phenomenologically, temporality emerges from scale-invariant constraints on accessible information. □

4.2.2. Present State Inaccessibility and Gravitational Emergence

The resolution of the problem of time operates through a fundamental mechanism: no observer embedded within informational layer s can directly access her present cosmological state, since all observational information necessarily originates from past layers $s-\Delta s$ where $\Delta s>0$. This constraint emerges from the universal requirement $ds=cdt$, which enforces finite progression for all information propagation.

While observers exist ontologically within specific layer s, their epistemic access is perpetually limited to $s-\Delta s$. This asymmetry explains why the static external block structure appears dynamically evolving to embedded observers, transforming the “problem of time” from fundamental paradox to inevitable consequence of finite information propagation speeds.

Gravitational Emergence Through Information-Mass Conjugacy

The same informational constraints that resolve temporal ontology also generate gravitational phenomena through the CUP. The fundamental relation $\Delta ϵ·\Delta m\ge \hslash /2$ creates ontological tension between energia fluens (flowing information) and energia locata (crystallized mass-energy).

Mass concentrations create regions where the CUP constraint forces information crystallization: well-defined mass regions (small $\Delta m$) necessarily create large diffusivity uncertainty ($\Delta ϵ$). Energia fluens naturally flows toward these high-uncertainty regions to achieve optimal uncertainty distribution, carrying energy and matter along to create gravitational attraction. Mass thus acts as an information uncertainty sink where the CUP constraint can be most efficiently satisfied through localized crystallization of flowing information.

This mechanism extends entropic gravity approaches [34] with explicit quantum-informational foundations, offering pathways toward understanding gravitational dynamics as information-processing constraints rather than geometric curvature. However, deriving gravitational field equations from CUP optimization principles remains an important direction for future work.

Cosmic Observational Consequences

At cosmological distances, this information-mass conjugacy generates observational constraints via the CUP. When distant observers attempt to measure both information transport properties and mass content of remote objects, they encounter fundamental quantum limitations: improved imaging precision necessarily causes mass information to disperse widely, creating apparent deficits that manifest as dark matter signatures.

Unified Framework

This framework unifies temporal ontology and gravitational emergence through information-theoretic principles operating across all scales:

• Local scale: CUP optimization drives gravitational clustering through spatial concentration of energia locata
• Temporal experience: Present state inaccessibility creates apparent cosmic evolution through finite information propagation
• Cosmic scale: CUP constraints govern observational precision trade-offs in astronomical measurements

Because information propagates at finite speed through an informationally stratified universe, gravity emerges as the local manifestation of the same information-processing constraints that create the illusion of time, unifying quantum mechanics, gravitational dynamics, and temporal ontology through the fundamental conjugacy between information flow and localization.

4.3. Quantum Gravity and Information Theory Connections

The spacetime diffusivity field appears compatible with established quantum gravity frameworks, suggesting deeper unification possibilities:

Loop Quantum Gravity: Area operator fluctuations yield $ϵ\sim {\ell }_P^2/t_P=c{\ell }_P$, consistent with our diffusivity scaling.

Holographic Correspondence: Entanglement velocity bounds provide $ϵ\sim v_E\times {\ell }_{\mathrm{AdS}}$, connecting to holographic information processing rates.

Causal Set Theory: Sprinkling density fluctuations give $ϵ\sim {\ell }_P^2/t_P$, aligning with discrete spacetime approaches.

These connections suggest spacetime diffusivity may serve as a unifying concept bridging different quantum gravity approaches through information-theoretic principles. The renormalization group analysis showing DTP emergence from infrared fixed points further strengthens connections to established quantum field theory.

4.4. Quantum-Informational Foundations of Cosmic Structure

The Cosmological Uncertainty Principle reveals deep connections between spacetime geometry and quantum information theory, extending beyond traditional uncertainty relations to establish fundamental constraints on cosmic measurements.

4.4.1. Information-Theoretic Bounds

The principle $\Delta ϵ\Delta m\ge \hslash /2$ constrains simultaneous knowledge of spacetime’s dynamic transport properties and static content properties. This represents optimal allocation of finite information resources between kinematic and static observables, extending generalized uncertainty principles to cosmological horizons [66,67].

Unlike traditional quantum uncertainty relations that apply to conjugate variables in isolated systems, the CUP governs measurements spanning cosmological distances and times, revealing quantum constraints on cosmic knowledge extraction that become increasingly important as observational precision improves.

4.4.2. Measurement Trade-offs and Dark Matter

The CUP generates observational consequences through precision-uncertainty trade-offs that may illuminate systematic trends in dark matter inference. The principle predicts an **Astronomical Precision Paradox**: any n-fold improvement in diffusivity precision forces n-fold degradation in mass precision, magnified at high redshift where signals traverse more informational layers.

This aligns with observed systematic rises in dark matter estimates with improved instrumentation: CMB measurements evolved from COBE’s $90\%\pm 30\%$ to Planck’s $95.1\%\pm 1\%$ dark component; HST imaging increased galaxy cluster dark matter estimates significantly over ground-based surveys. While conventional explanations involving improved measurement accuracy remain viable, the CUP offers a complementary perspective where some apparent dark matter signatures might reflect fundamental measurement limitations rather than purely exotic matter content.

The framework generates testable predictions: redshift scaling $\Delta m\propto {(1+z)}^{\alpha }$, method dependence where spectroscopic studies should find higher dark fractions than lensing surveys, and correlations between dispersion precision and inferred dark matter content that persist after controlling for instrumental factors.

4.5. Implications for Observational Cosmology

Our framework generates specific observational predictions while acknowledging its foundational rather than phenomenological nature:

• FRB Measurement Trade-offs: Timing-optimized and mass-optimized analysis pipelines should yield systematically different precision trade-offs for identical FRB sources
• CMB Parameter Correlations: Further precision improvements may reveal systematic uncertainties in mass-related parameters that reflect fundamental rather than instrumental limitations
• Survey Method Dependencies: Systematic differences between spectroscopic and lensing dark matter estimates should persist even with improved methodologies
• Redshift-Distance Scaling: Dark matter uncertainty should exhibit specific scaling with cosmological distance independent of source luminosity or morphology

These predictions offer pathways for distinguishing quantum-informational effects from conventional systematic errors through controlled observational studies.

4.6. CUP versus de Sitter Horizon Temperatures

The temperature predicted by the Cosmological Uncertainty Principle,

$$T_{\mathrm{CUP}}\simeq \pi T_{\mathrm{dS}},$$

closely echoes the well-known Gibbons–Hawking de Sitter horizon temperature $T_{\mathrm{dS}}$ [59]. Just as $T_{\mathrm{dS}}$ characterises the quantum-informational properties of black-hole event horizons, Recent work has argued for alternative normalisations, including a local de Sitter temperature $T=H/\pi$; see [68,69]. the quantity $T_{\mathrm{CUP}}$ quantifies analogous informational constraints at the cosmological horizon. This near-equality therefore strengthens the CUP framework and hints at a unified quantum-informational structure underlying both black-hole and cosmological horizons.

5. Conclusion

This work explores how quantum uncertainty constraints operating through spacetime diffusivity—a new transport field governing information propagation between temporal layers—generate fundamental connections between quantum mechanics, information theory, and cosmic structure.

From two foundational postulates—the Invariant Time-like Information Postulate ($ds=cdt$) and the Diffused Spacetime Postulate ($dϵ/ds=c$)—we developed a comprehensive quantum field theory yielding several breakthrough results:

5.1. Principal Discoveries

Quantum-Gravitational Scale Unification. Both Planck time $t_P={\ell }_P/c$ and Hubble time $t_H=1/H_0$ emerge as limiting cases of a single area-diffusion parameter Z, representing the first unified description across 60 orders of magnitude and suggesting our framework captures genuine organizational principles of spacetime.

Cosmological Uncertainty Principle. From canonical quantization of spacetime diffusivity, we derived the fundamental constraint $\Delta ϵ\Delta m\ge k·{\displaystyle \frac{\hslash }{2}}$ linking information transport precision (energia fluens) to mass determination uncertainty (energia locata), establishing quantum-mechanically conjugate observables in cosmological measurements.

Information-Pixel Amplification. The framework extends quantum uncertainty to cosmic scales through distance-dependent amplification: $\Delta ϵ\Delta m\ge (\hslash /2)·(D/{\ell }_{\mathrm{coh}})$, provides a quantum-mechanical lower bound and a practical prediction for the minimal mass uncertainty and quantitatively reproducing Milky Way dark matter observations.

Enhanced Cosmological Temperature. Quantum information bounds enforce a minimum cosmic temperature $T_{\mathrm{CUP}}=\pi T_{\mathrm{dS}}\approx 9\times {10}^{-30}$ K at the universe’s boundary, exceeding the classical Gibbons-Hawking temperature by factor $\pi$ through CUP constraints with $N_{\mathrm{eff}}=1$.

Resolution of Temporal Ontology. The fundamental "problem of time" resolves through scale-invariant informational constraints: temporal experience emerges as inevitable epistemic limitation of embedded observers unable to access their present cosmological state.

5.2. Theoretical Significance and Observational Predictions

The framework establishes spacetime diffusivity as a fundamental field bridging quantum mechanics and cosmic structure through information-theoretic principles. The approach suggests: (1) cosmic structure can be understood through quantum information theory rather than purely geometric dynamics, (2) quantum uncertainty constraints can be related to gravitational phenomena, (3) temporal evolution reflecting epistemic limitations rather than ontological becoming.

The framework generates specific testable predictions: FRB measurement trade-offs between dispersion and mass precision, redshift scaling $\Delta m\propto {(1+z)}^{\alpha }$, systematic differences between spectroscopic and lensing dark matter estimates, and CMB precision limits reflecting fundamental rather than instrumental constraints.

5.3. Future Directions

The results establish a precise, operator formulation of quantum–informational limits at cosmological scale: the information–pixel amplified bound $\Delta ϵ\Delta m\ge (\hslash /2)D/{\ell }_{\mathrm{coh}}$ in Eq. (35) (with its geometric generalisation $k={\int }_0^{D}w\left(s\right)/{\ell }_{\mathrm{coh}}\left(s\right)ds$ in Eq. (37)), together with the horizon prediction $T_{\mathrm{CUP}}=\hslash H/\left(2k_B\right)=\pi T_{\mathrm{dS}}$ in Eq. (50) under the stated assumptions. Within this scope, Local Group mass measurements are quantitatively consistent with saturation of the bound, and the framework provides an operational, parametric correspondence between Planck and Hubble regimes generated by ITP + DTP and the flux relation, while the CUP supplies the falsifiable measurement limits. The present treatment is intentionally limited to information/measurement aspects; detailed structure formation, thermodynamics, and redshift physics remain with conventional cosmology.

Progress now turns on replacing estimates by predictions and widening the observational tests. A central target is a first-principles derivation of the coherence scale ${\ell }_{\mathrm{coh}}=\chi {\ell }_P$ (presently a QEC trial–and–error estimation), alongside an independent determination of the flux constant Z to examine whether a single value reproduces both the Planck and Hubble calibrations. On the observational side, extending beyond the Local Group to clusters, strong-lensing sightlines, and FRB datasets that report a linear path estimator (or its Jacobian) will test the scaling and the weighted form of k. In the horizon sector, clarifying the state/periodicity assumptions that yield the factor $\pi$ and identifying discriminating observables will sharpen the prediction. Conceptually, spacetime diffusivity offers common handles for comparing major quantum-gravity programmes—area/flux fluctuations in Loop Quantum Gravity, entanglement growth in holographic settings, and sprinkling densities/causal intervals in causal sets—while the possibility of deriving gravitational dynamics from a CUP or entropy optimisation consistent with ITP + DTP merits further study. Taken together, these directions aim to convert the present consistency results into stringent, potentially falsifying predictions, and to determine whether quantum–informational constraints are a structural ingredient of cosmic physics.