Information saturation and the structural origin of low-entropy spacetime

Abstract

The origin of the Universe’s extremely low initial gravitational entropy remains an open problem, commonly addressed by imposing the Penrose Weyl curvature condition as an additional assumption. In this work, we propose a structural explanation based on an information saturation interface, formulated through a saturation functional governing relational information organization. The saturation regime is a limit in which information channels are maximally utilized and admissible local operations no longer generate additional independently distinguishable relational organization. We argue that approaching this limit favors a near-uniform relational background and assigns a positive quadratic cost to non-uniform or locally extensible deviations, thereby suppressing the robust amplification of symmetry-breaking relational modes. Accordingly, if a classical geometric description emerges from such a structure, its effective background metric is constrained to inherit, at the macroscopic level, the corresponding symmetry content, restricting admissible geometries toward the Friedmann–Lemaître–Robertson–Walker class. Within an effective general-relativistic interpretation, such geometries correspond to vanishing or strongly suppressed Weyl curvature and hence very low gravitational entropy. The framework therefore does not derive a continuum Lorentzian metric or compute the Weyl tensor directly from the saturation functional, but instead proposes a structural admissibility criterion for low-gravitational-entropy emergent backgrounds.

1 Introduction

1.1 Success of the low-entropy initial condition and the mechanism gap

One of the most successful and widely accepted assumptions in modern cosmology and statistical physics is that the universe began in an extremely low-entropy macroscopic state [1, 2]. This assumption underlies our current understanding of the thermodynamic arrow of time, the emergence of irreversible processes, and the formation of large-scale structure, with cosmic evolution proceeding from an initially highly ordered configuration toward progressively higher entropy. Despite its empirical success, however, this assumption remains fundamentally a boundary postulate: it specifies that the universe did begin in a very special state, but does not explain why such a state should be realized. The resulting tension may be summarized as a mechanism gap: while the low-entropy initial condition is phenomenologically indispensable, its origin remains unexplained within existing frameworks.

A more precise formulation of this problem arises when entropy is considered in gravitational terms. In this context, the relevant notion of low entropy is not primarily associated with matter distributions or temperature, but with the organization of gravitational degrees of freedom. Following Penrose’s Weyl curvature hypothesis [2, 3], the early universe is characterized by an exceptionally simple geometric configuration, with strongly suppressed Weyl curvature. Equivalently, the initial state lies very close to a maximally symmetric geometry. This suggests that, in the gravitational context, the low-entropy problem is more precisely a problem of geometric organization. The central question may therefore be sharpened as follows: why is the initial geometry of the universe so close to a maximally symmetric configuration?

Existing approaches address different aspects of this problem. A first class of explanations invokes dynamical mechanisms, most notably inflationary cosmology [4, 5], which can amplify homogeneity and isotropy. However, inflation does not provide a selection principle for the initial geometric configuration: it requires a sufficiently smooth region to begin with, and primarily acts to stabilize and extend such a region once it is already present. In this sense, it operates on a given structural background without determining its origin. Related proposals in quantum cosmology, such as the no-boundary proposal [6], attribute the smoothness of the initial state to specific boundary conditions in a path-integral formulation, but the underlying rationale for selecting such boundary conditions remains an open question.

A second line of approach treats the problem at the level of boundary conditions. The Past Hypothesis [1, 7], for example, postulates that the universe began in a highly atypical low-entropy macrostate. Within statistical mechanics, this assumption successfully accounts for the observed arrow of time. Yet, by construction, it does not explain why such a special state is realized, but rather elevates it to the status of a fundamental postulate. As such, it does not address the mechanism underlying the selection of the initial configuration.

More recently, information-theoretic approaches have attempted to reformulate the problem in terms of correlations and entanglement structure [8, 9]. In particular, the Entanglement Past Hypothesis proposes that the initial state of the universe is distinguished by specific features of its entanglement pattern. This perspective suggests that the low-entropy condition may reflect constraints on the organization of information, rather than merely a restriction on macroscopic phase space volume. However, it remains unclear whether such informational constraints generically favor highly symmetric geometries, or under what conditions they would do so. In particular, a mechanism that systematically links constraints on information organization to the suppression of geometric inhomogeneities is still lacking.

Taken together, these considerations indicate that the difficulty lies not only in identifying candidate initial conditions, but in understanding the principles that could account for their selection. This motivates a shift in perspective: rather than treating the low-entropy initial state as an isolated boundary condition or a purely dynamical outcome, we explore the possibility that it reflects deeper constraints on the organization of physical degrees of freedom. The central question then becomes whether such constraints can systematically restrict the space of realizable geometries, and in particular whether they naturally single out configurations close to maximal symmetry.

1.2 Information saturation interface and the emergence of low-entropy spacetime

In this work, we propose a structural framework for addressing this problem.

The central proposal may be stated as follows. We posit that classical spacetime should not be regarded as a geometrical structure given a priori, but rather as an effective subsystem selected from an underlying quantum system through the organization of information [10, 11]. Its emergence is conditioned by the formation of a specific relational regime within the underlying quantum system, which we refer to as an information saturation interface. If a classical spacetime description emerges from such a relational regime, we hypothesize that it is preferentially selected in a form close to a maximally symmetric geometry.

By an information saturation interface, we do not mean a pre-assigned geometrical surface, but a structural condition on the relational regime itself. More concretely, this regime may be understood as one in which additional microscopic complexity no longer gives rise to new independently distinguishable structures through local structural extension, but instead contributes only through the reorganization of the global relational structure. In this sense, the information saturation condition corresponds to a failure of local extensibility for independently distinguishable relational structure. If the available capacity for such independent relational structure becomes effectively saturated, then not all configurations can be equally supported.

A key consequence of this constraint is that different configurations place different demands on the amount of independently distinguishable structure required for their realization. Configurations that rely on non-uniform or direction-dependent relational organization typically require a larger set of independently maintainable structures and are therefore disfavored near the saturation regime. By contrast, configurations supported by a smaller set of mutually compatible relational structures remain accessible. This leads to an effective restriction toward structurally economical configurations that do not rely on fine-grained independent organization.

In this way, the emergence of symmetry is not attributed to statistical averaging, nor introduced as an external boundary prescription. It is instead understood as a consequence of structural realizability: under conditions of informational saturation, only a restricted subset of relational configurations can be consistently supported, and these are biased toward those exhibiting higher symmetry. If a classical spacetime description subsequently emerges from such a relational regime, its geometry inherits these symmetry constraints, leading naturally to forms close to maximally symmetric configurations, such as the approximately Friedmann–Lemaître–Robertson–Walker (FLRW) class, characterized by strongly suppressed Weyl curvature.

The purpose of the present work is to develop this proposal in a systematic form. In Section 2, we formulate the notion of an information saturation interface as a constraint on information decomposition and introduce a functional that quantifies deviations from saturation. In Section 3, we analyze the resulting stability conditions and establish the emergence of a near-symmetric relational structure, thereby constraining the class of admissible emergent geometries. In Section 4, we discuss the implications of the proposed mechanism and its relation to existing theoretical approaches. In Section 5, we outline possible empirical and numerical tests of the framework. For completeness, a minimal toy model illustrating the mechanism is presented in the Supplementary Appendix. Section 6 concludes with a summary and outlook.

2 Information saturation interface: definition and formal framework

In a variety of approaches to quantum gravity and high-energy physics, a recurring structural feature is that the realizable information capacity of a system does not grow arbitrarily with volume, but appears to be constrained in a manner associated with an effective interface. This behavior is reflected, in different forms, in black hole thermodynamics [12, 13], the holographic principle [14, 15], covariant entropy bounds [16], and gauge/gravity dualities [10]. While the underlying mechanisms differ across these frameworks, they provide suggestive phenomenological indications that interface-dominated capacity behavior may reflect a more primitive information-structural principle [11].

This observation motivates a more abstract question: whether there exists a formulation of such a capacity limit that can be expressed purely in terms of information organization, without presupposing a specific geometrical background or dynamical model. In this work, we adopt the viewpoint that the interface-like behavior observed in the above contexts is not fundamental in itself, but rather a macroscopic manifestation of a deeper structural condition on the organization of information.

To this end, we introduce the notion of an information saturation interface. At the conceptual level, this refers to a structural regime in which the number of independently realizable information-bearing configurations approaches its effective limit, so that new distinguishable structures can no longer be generated by simply increasing the number of degrees of freedom, and any further complexity must arise through reorganization of existing structure.

To render this notion operational, we represent such structures by an effective information partition, denoted by , and characterize their degree of saturation through a variational functional . In this way, the information saturation interface is specified not as a predefined geometrical surface, but as a structural condition on , with providing a quantitative measure of the deviation from saturation.

The subsequent subsections develop this framework in detail.

2.1 Information interface as a unified structure of partition and flow

We represent the information interface as an effective information partition structure induced by the organization of the underlying degrees of freedom. Concretely, Equation 1 represents the effective decomposition of the total Hilbert space of the system induced by Σ,where this factorization should not be understood as a fundamental decomposition of the microscopic Hilbert space, but as an effective factorization induced by a chosen information-theoretic coarse-graining. Here, denotes the sector organized by the interface into a highly correlated structure that admits a stable coarse-grained description, while collects the remaining degrees of freedom.

To render this structure operational, Equation 2 introduces a minimal discrete representation in terms of a graph [17, 18],where denotes a set of elementary encoding units, and represents relations among them, such as entanglement, information dependence, or coding constraints. Physically, each node corresponds to a distinguishable local encoding unit, while each edge encodes a constraint or correlation between units. The graph therefore captures the organizational structure of information at the level relevant for coarse-grained description.

In the present framework, the notion of locality is defined with respect to the information structure encoded by , rather than any pre-existing spatial geometry. In particular, local neighborhoods are understood as finite-step neighborhoods on the graph , and local operations refer to operations acting on such graph-defined regions. This notion of locality is therefore induced by the pattern of information relations, and does not presuppose an underlying geometrical notion of space.

These quantities are understood as coarse-grained measures of the information-carrying and correlation-supporting ability of the relation between and .

To make this construction explicit, it is useful to consider a minimal quantum information realization. Let denote the underlying quantum state, and let , be the effective subsystems associated with nodes and . A natural choice for the information load is then given by the quantum mutual information [19],where the mutual information appearing in Equation 3 is defined in Equation 4 ashere denotes the von Neumann entropy.

Correspondingly, the effective capacity may be defined as an upper bound on the achievable correlations between the two subsystems, for instance as the maximal mutual information attainable within a prescribed class of admissible states subject to given constraints.

We then define the local saturation ratio in Equation 5 as

We emphasize that the subsequent analysis does not depend on the specific choice of mutual information, but only on the existence of a bounded, monotonic measure of correlations that captures the effective utilization of information-carrying relations.

To characterize the global organizational capacity of , we introduce the notion of distinguishable structural configurations. Let denote the set of structural configurations distinguishable at the chosen effective level, defined modulo an equivalence relation induced by the information-theoretic coarse-graining.

We define the structural capacity in Equation 6 aswhich quantifies the number of independent structural configurations that can be realized within the partition .

This combined structure provides the basis for defining and analyzing the information saturation condition in a quantitative manner in the following subsection.

2.2 Variational characterization of information saturation

To quantify the deviation of a given interface from the saturation regime, we introduce in Equation 7 a variational functional, in the spirit of effective coarse-grained variational descriptions [20], which we refer to as the saturation functional,

The first term measures the under-utilization of effective information channels (flow deficit), while the second term quantifies the residual capacity for generating new independent structure (structural slack). Both contributions are non-negative, and corresponds, in the idealized limit, to a fully saturated configuration.

To define the second term, we introduce the notion of residual local organizational extensibility.

At the adopted coarse-graining scale, we first fix a distinguishability criterion for relational configurations on the interface. Two microscopic relational arrangements are identified as equivalent if they induce the same coarse-grained relational organization, within the chosen resolution and tolerance of the description. This defines a fixed equivalence relation, denoted in Equation 9 bywhere and are microscopic relational arrangements compatible with the same interface. The set is then defined as the corresponding quotient set of independently distinguishable coarse-grained relational configurations realized by at the adopted coarse-graining scale. Microscopic rearrangements that do not change the chosen coarse-grained relational observables, including sub-resolution or sub-extensive modifications below the adopted tolerance, are identified as equivalent under .

For each node , let denote the class of admissible local operations acting on a finite-step neighborhood of , where locality is understood in the graph-theoretic sense defined in the previous subsection.

The class consists of operations with bounded graph-theoretic support that preserve the adopted coarse-graining prescription and the fixed equivalence relation . Operationally, admissible local operations may include local rewiring or redistribution of correlations, local basis changes, and finite-support local evolutions acting within a bounded graph neighborhood of . Subject to locality, bounded support, and the fixed coarse-graining prescription, admissible operations are otherwise free to reorganize the local relational structure.

Each induces, as shown in Equation 10, a locally modified configuration of the interface,and hence an associated quotient set of distinguishable configurations . The operation may change which coarse-grained equivalence classes are realized, but it does not change the equivalence relation or the resolution at which distinguishability is evaluated.

We then define the residual local organizational extensibility at in Equation 11 bywhere .

Thus, in any given evaluation of , the coarse-graining scale, the distinguishability criterion, the equivalence relation , and the admissible class are all fixed prior to evaluating the supremum. Variations in therefore reflect changes in realizable independently distinguishable relational organization, rather than changes in the criterion, tolerance, or resolution of description itself.

In this sense, measures extensibility at the level of effective distinguishable organization rather than microscopic Hilbert-space dimensionality. More precisely, quantifies the maximal residual local organizational extensibility near , namely, the extent to which admissible local operations can generate additional distinguishable configurations under a fixed coarse-grained description. Conceptually, is related to the reachable space of distinguishable organization under constrained local evolution, although unlike operator complexity or effective Hilbert-space dimension, it is defined at the level of coarse-grained relational distinguishability.

A unified minimal example illustrating the evaluation of , the emergence of non-trivial intermediate values, and its interplay with the channel-utilization sector under a fixed coarse-grained description is provided in Supplementary Appendix 1.

A vanishing value indicates that admissible local operations can no longer generate additional independently distinguishable configurations under the adopted coarse-grained description. Importantly, this does not imply the absence of microscopic dynamics or complexity growth. Local rearrangements, entanglement restructuring, and additional microscopic complexity may still arise. Rather, in the saturation regime, such complexity ceases to generate new independently distinguishable local relational organization and instead contributes primarily through the reorganization of existing relational patterns.

While the evaluation of is performed relative to a fixed coarse-grained description, one expects the suppression of local extensibility to persist across a broad range of reasonable coarse-graining prescriptions in the near-saturation regime. That is, although the distinguishability criterion, the equivalence relation, and the admissible class are held fixed in any given computation of , the resulting saturation behavior is hypothesized to be robust under reasonable changes of effective scale [21].

In this sense, information saturation should not be understood as a freezing of microscopic evolution, but as a suppression of locally extensible organization at the effective level.

The parameter controls the relative weight between channel under-utilization and residual local organizational extensibility. It characterizes the relative cost of residual local extensibility compared with incomplete utilization of existing relational channels, and thus determines the balance between two complementary aspects of information organization: the optimization of existing correlations and the suppression of additional distinguishable local organization.

In concrete tensor-network, finite-qubit, or graph-based implementations, could be estimated by comparing the response of the effective objective to correlation-redistribution operations with its response to operations that open new threshold-resolved relational configurations. In numerical tests, it may also be treated as a phenomenological parameter to be varied in order to test the robustness of the saturation mechanism.

Equation 12 states the information-saturation condition for an interface Σ:in the sense that both channel under-utilization and residual local organizational extensibility are suppressed. Physically, this corresponds to a regime in which effective information channels are nearly fully utilized, while admissible local operations cease to generate new independently distinguishable local organization under the adopted coarse-grained description. The saturated interface could therefore be understood as a marginal, threshold-like organization at the adopted coarse-grained level.

It is important to emphasize that is a coarse-grained saturation condition rather than a freezing of microscopic dynamics. Microscopic rearrangements may still occur. The claim is that, near saturation, such rearrangements either remain within the same coarse-grained equivalence class or appear as resolved perturbations of the effective information interface.

For admissible perturbations that are resolved at the adopted coarse-grained level, the local expansion in Equation 13 givesso that the saturated configuration is understood as a local minimum of the coarse-grained functional.

The two possible resolved deviations correspond to the two sectors of . Perturbations may decrease channel utilization by modifying local correlation patterns, thereby affecting , or they may increase residual local organizational extensibility by opening additional independently distinguishable configurations, thereby affecting . In either case, the perturbation raises and is structurally disfavored at the coarse-grained level. Thus the effective rigidity of the saturation regime does not exclude microscopic rearrangement; it suppresses those rearrangements only insofar as they become distinguishable deviations from the saturated relational organization.

The resulting structure in the saturation regime can be schematically represented as a uniform relational network, as illustrated in Figure 1.

FIGURE 1

2.3 Structural hypothesis: emergence of classical geometry

We now state the central hypothesis underlying the present framework.

Classical spacetime geometry is not assumed to be a generic property of arbitrary quantum states. Rather, it is hypothesized to emerge only under specific structural conditions imposed on the underlying information organization.

In particular, Equation 14 states the saturation-regime condition for an interface Σ,where effective information channels are nearly fully utilized and the generation of new independent structural configurations is suppressed. In this regime, the system lacks the structural degrees of freedom required to sustain arbitrary macroscopic organization. This limitation is particularly restrictive for configurations that require a large number of independently distinguishable components, such as strongly inhomogeneous or anisotropic structures.

We therefore hypothesize that, if a classical spacetime description emerges from an information-saturated interface, its macroscopic geometry is constrained to lie within a restricted class of near–maximally symmetric configurations, naturally corresponding to approximately homogeneous and isotropic geometries. This restriction arises from the suppression of non-uniform and direction-dependent relational variations in the underlying structure.

In this sense, information saturation acts as a structural selection principle, constraining the class of geometries that can be consistently realized upon emergence. We emphasize that this hypothesis does not rely on a specific microscopic model of quantum gravity, nor does it assume a predefined geometrical background. It introduces a structural condition at the level of information organization, whose implications for symmetry and geometric realizability will be developed in the following sections.

3 Stability, control, and the emergence of geometric symmetry

3.1 Local expansion of the saturation functional and stability structure

We now analyze the behavior of the saturation functional in the vicinity of an information-saturated interface. The aim of this subsection is to identify the local stability structure of and to clarify in what sense the saturation regime defines a statistically stable coarse-grained organization.

At the discrete level, let denote an ideal information-saturated interface, for which Equation 15 givesat the coarse-grained level [21], so that in the idealized limit.

For a nearby interface , we characterize deviations from saturation in Equation 16 by the effective small variableswhere measures the local deviation from channel saturation, and measures the residual local structural extensibility. Here is understood as the induced coarse-grained extensibility of the perturbed interface, and no differentiability of as a microscopic functional is assumed.

With these definitions, and working at fixed coarse-grained graph structure, the deviation of the functional from the saturated value takes the form given in Equation 17up to corrections associated with variations of the underlying graph structure.

Such graph-structural variations would correspond to changes in the coarse-grained relational support itself, for example, through the creation or removal of distinguishable nodes, edges, or adjacency relations. In the near-saturation regime considered here, these variations are suppressed by the vanishing residual local organizational extensibility, , and are therefore not part of the leading fixed-support expansion.

Since the functional is constructed as a sum of non-negative quadratic contributions, the resulting expression is manifestly positive semidefinite. It becomes positive definite on nontrivial perturbations after quotienting out variations identified as equivalent under the chosen coarse-graining. No mixed terms appear in this minimal quadratic functional at this level, so the two sources of deviation from saturation—channel under-utilization and residual structural extensibility—contribute independently to the leading stability structure.

It follows in Equation 18 thatwith equality only under the condition stated in Equation 19, namely whenfor all edges and nodes.

Thus, within the present coarse-grained description, the information-saturated configuration defines a marginal, threshold-like local minimum of . More precisely, it is a statistical minimum [22], in the sense that the minimum is defined at the level of effective organizational variables rather than through the elimination of microscopic degrees of freedom. This analysis is local in the vicinity of the saturation regime and applies to configurations for which remains sufficiently small.

In this sense, the quadratic structure of provides an effective measure of deviation from the saturation point. In the continuum limit, provided that the underlying graph admits a suitable coarse-grained description, this structure may be viewed as an -type norm [23] on the admissible deviations, quantifying how strongly a given configuration departs from the saturated organization. This observation will allow us, in the following subsections, to derive quantitative control inequalities for structural inhomogeneity and relational directional variation.

3.2 Measure of inhomogeneity and control inequality

We now make the notion of structural inhomogeneity quantitative and relate it directly to the saturation functional. The strategy is to introduce a coarse-grained measure of inhomogeneity that is compatible with the structure of , and to show that it is controlled by the deviation from saturation. This will establish a precise sense in which proximity to the saturation regime limits large deviations from uniform organization at the coarse-grained level.

We express the inhomogeneity measure in terms of the original variables and , noting that near saturation one has and . Motivated by the quadratic structure of , we define in Equation 20 the inhomogeneity measurewhere the corresponding mean values are defined in Equation 21 as

This quantity measures the variance of the local saturation variables across the interface, with the same relative weighting between channel utilization and structural extensibility as in . In this way, quantifies the degree to which the interface deviates from a uniform organization at the coarse-grained level, in a manner consistent with the underlying control functional.

To relate to , we apply the variance decomposition separately to the two quadratic contributions. For the channel sector, Equation 22 giveswhile for the structural sector, Equation 23 gives

Combining these identities and using the definition of , we obtain the exact decomposition

Since the additional terms on the right-hand side of Equation 24 are manifestly non-negative, it follows immediately in Equation 25 that

This establishes a control inequality: the structural inhomogeneity of the interface is bounded above by its deviation from the saturation regime.

The physical interpretation is straightforward. Configurations with small cannot sustain large variations across the interface in either channel utilization or local structural extensibility. In this sense, the saturation functional acts as a global constraint that limits inhomogeneous organization, enforcing approximate uniformity across the interface at the coarse-grained level.

We summarize this implication as Proposition 1, whose formal statement is given in Equation 26.

Consequently, the interface approaches a state in which the local saturation variables and become uniform across the interface at the coarse-grained level.

In summary, the control inequality derived above shows that proximity to the saturation regime enforces uniformity of the effective information structure across the interface. This provides the first quantitative link between information saturation and the suppression of inhomogeneous structure, forming the basis for the emergence of geometric symmetry in the following analysis.

3.3 Relational non-scalar modes and suppression of directional structure

We next quantify direction-dependent deviations at the level of the underlying information structure. Complementing the control of structural inhomogeneity established in the previous subsection, we now identify a class of relational fluctuations that encode directional dependence and show that they are suppressed as the system approaches saturation.

To this end, we first associate to the interface effective node variables obtained by local averaging of edge quantities. For each node , we define in Equation 27and retain as defined previously. We then introduce in Equation 28 the corresponding mean-subtracted variableswhere and denote the averages defined in Section 3.2.

Let denote the graph Laplacian associated with the coarse-grained interface [24, 25], where is the degree matrix and the adjacency matrix.We define in Equation 29 the relational directional variation

By construction, measures the extent to which the effective variables vary along the relational structure encoded by the graph. Since the graph Laplacian vanishes on constant vectors, if and only if and are constant across the interface. Thus, captures the total power contained in non-constant (i.e., direction-dependent) relational modes.

To relate to the saturation functional, we use standard spectral bounds for the graph Laplacian. For any mean-subtracted vector , Equation 30 giveswhere denotes the largest eigenvalue of .

Applying this bound to and , and using the definition of the inhomogeneity measure from the previous subsection, we obtain the bound in Equation 31for some constant . Here depends on the graph spectrum and absorbs the conversion between node-level and edge-level variances, arising from the coarse-grained averaging defining .

The bound above is intended to apply to locally coarse-grained relational graph families with bounded effective degree. As defined in Section 2.1, locality in the present framework is graph-theoretic: local neighborhoods are finite-step neighborhoods of the information graph . For the spectral estimate used here, this locality condition is understood at the adopted coarse-grained resolution: each coarse-grained node has only a finite number of independently resolvable effective relational neighbors. The corresponding bound is fixed by the coarse-graining prescription and by the operational definition of graph-theoretic locality. Thus, for the locally coarse-grained graph families considered here, Equation 32 states that there exists a constant , independent of , such that

Accordingly, the largest graph-Laplacian eigenvalue satisfies the bound in Equation 33with the maximal effective degree [24]. Consequently, for such bounded-degree coarse-grained graph families, remains uniformly bounded as , so the above control inequality remains stable in the thermodynamic limit.

This bounded-degree condition specifies the locally coarse-grained graph class to which the present spectral estimate applies. Graph families with strongly growing effective degree, such as star-like or highly hub-dominated networks, correspond to dense connectivity or nonlocal aggregation of independently resolvable relational channels and fall outside the intended local near-saturation regime.

Together with the suppression of in the saturation limit, the above control inequality yields the result formally stated in Equation 34.

Consequently, the effective variables approach the kernel of the graph Laplacian, i.e., the constant sector, so that no direction-dependent relational structure can be sustained at the coarse-grained level.

The physical interpretation parallels that of the previous subsection. Direction-dependent organization requires the persistence of distinguishable relational patterns associated with variations across the interface. In the saturation regime, however, the available information-carrying structure is already fully utilized, while the generation of new independent configurations is suppressed. As a result, non-constant relational modes cannot carry significant weight and appear only as constrained deviations around the uniform configuration.

Taken together with the control of structural inhomogeneity established in the previous subsection, this result shows that proximity to the saturation regime suppresses both non-uniform and direction-dependent relational variations.

3.4 Proto-geometric relational structure, threshold stability, and symmetry constraints

We now refine the link between information-theoretic structure and emergent geometry. We construct a relational structure intrinsic to the information interface and analyze how its fluctuations are constrained near the saturation regime. This structure will serve as a proto-geometric constraint on any effective description that may arise under coarse-graining.

We begin by defining a local relational quantity on the graph . For each edge , we set in Equation 35

This definition is intended for the near-saturation regime, where and typically . It satisfies , and as , so that strongly saturated channels correspond to minimal relational separation. For , Equation 36 givesshowing that directly measures the deviation from saturation.

It is important to emphasize that is not a geometric length, but a logarithmic measure of information correlation strength. It provides a primitive encoding of relational proximity within the information structure.

Based on these local quantities, we define in Equation 37 a path-based relational measurewhere the infimum is taken over all paths connecting and . Under the assumption that the graph is connected, this defines a path-additive relational structure. Crucially, should be understood as a proto-geometric relational quantity: it characterizes the effective “closeness” of information-bearing units without presupposing any underlying spatial geometry.

Technically, should be understood as an effective pseudo-metric rather than an exact metric in the strict mathematical sense. Since , the construction guarantees non-negativity, and if , symmetry follows directly. Moreover, although the local quantity does not itself satisfy a triangle inequality, the shortest-path construction of Equation 37 imposes the triangle inequality by definition, as in standard weighted path metrics on graphs. However, the separation property may fail in the exact saturation limit. In particular, when across sufficiently connected regions, distinct nodes may acquire vanishing effective relational separation, reflecting complete relational indistinguishability at the adopted coarse-grained description. In this sense, the pseudo-metric character of should be understood not as a defect, but as a natural manifestation of perfect information saturation.

In generic near-saturation regimes relevant to physical systems, however, residual distinguishability remains finite, and is expected to behave effectively as a path-distance-like relational diagnostic at the coarse-grained level. Within the present framework, it measures effective relational separation induced by incomplete saturation and functions as a proto-geometric constraint on any compatible emergent geometry.

We now analyze how this relational structure is constrained near the saturation regime. Using , the functional takes the schematic form in Equation 38to leading order. Thus, directly controls the mean-square fluctuations of the relational quantities .

Combining this with the control inequalities derived in S3.2 and S3.3, as summarized in Equation 39,we conclude that both non-uniform and direction-dependent relational variations are suppressed as . Since is obtained by summing along paths, Equation 40 gives the expected corresponding control of the variation of the relational distance,with depending on coarse-grained graph properties.

Qualitatively, depends on typical path lengths, effective local connectivity, bounded-degree behavior, and the regularity of the relational weights across neighboring links. These properties control how local fluctuations in accumulate into variations of .

This leads to a notion of threshold stability at the level of relational structure. There exists such that, if the threshold condition in Equation 41 holds,then the fluctuations of are controlled, and the induced relational structure becomes approximately uniform with suppressed direction-dependent variation at the coarse-grained level.

Accordingly, we define a near-symmetric relational class as configurations in which the relational structure exhibits approximate uniformity and suppressed direction-dependent relational variation. With this definition, we obtain the implication in Equation 42

In this setting, information saturation defines a proto-geometric relational structure encoded in , which captures the symmetry properties of the underlying information organization. In the regime , this relational structure becomes approximately uniform, with direction-dependent relational variation suppressed at the coarse-grained level. These symmetry properties constrain the class of macroscopic configurations that can be stably realized.

In the above analysis, Propositions 1 and 2 show that non-uniform and direction-dependent relational deviations are energetically suppressed near the saturation regime. We now consider a complementary question: whether microscopic or mesoscopic symmetry-breaking fluctuations can be amplified into dynamically sustainable coarse-grained organization.

A macroscopic breaking of homogeneity or isotropy would require more than transient microscopic fluctuations. It would require relational configurations that select a preferred position or direction and remain independently distinguishable, dynamically accessible, and stably maintainable at the adopted coarse-grained level.

This observation can be formalized as follows.

Equation 43 introduces the notation for the effective weight of dynamically sustainable symmetry-breaking organization, namely, relational configurations that select a preferred position or direction and can be generated, propagated, or stably maintained as independently distinguishable structures under admissible local operations.

To quantify this effect locally, let denote the contribution of node to dynamically sustainable symmetry-breaking organization. By definition, such a contribution can persist at the adopted coarse-grained description only if it remains dynamically accessible through admissible local operations. Since measures the maximal residual local organizational extensibility near , symmetry-breaking organization represents only a particular sector of the locally accessible residual organization. It is therefore natural to impose the bounded local-response assumption that there exists a finite constant such that the bound in Equation 44 holds

This inequality should be understood as an envelope bound: the locally sustainable symmetry-breaking sector cannot exceed the residual organizational extensibility available in the same coarse-grained neighborhood. The quadratic form is a modeling choice aligned with the quadratic structure of the saturation functional, ensuring that the symmetry-breaking weight is controlled at the same perturbative order as the local-extensibility contribution to .

We then define in Equation 45 the total coarse-grained weight of dynamically sustainable symmetry breaking as

Combining the preceding relations yields the bound in Equation 46where absorbs the effective response coefficient.

Since Equation 47 gives it follows in Equation 48 that

Consequently, although microscopic symmetry-breaking fluctuations may still occur, they can no longer be amplified into dynamically sustainable coarse-grained organization. Preferred directions or positions cannot be robustly generated, propagated, or stably maintained as independently distinguishable relational structures. Thus, near information saturation, symmetry breaking is not forbidden at the microscopic level, but it loses the local organizational extensibility required to become a persistent macroscopic relational feature.

Together with Propositions 1 and 2, this result shows that non-uniform, direction-dependent, and dynamically sustainable symmetry-breaking relational variations are all suppressed near the saturation regime. The pre-geometric relational structure is therefore constrained to remain near-symmetric at the coarse-grained level.

3.5 Structural constraints on emergent geometry

We now consider the implications of this near-symmetric relational organization for any effective spacetime description that may emerge from it.

The preceding results establish a structural constraint on the pre-geometric relational organization of the interface: near information saturation, it is approximately uniform, isotropic, and free of dynamically sustainable preferred directions or positions at the coarse-grained level. Accordingly, when a classical geometric description emerges from such a relational structure without additional macroscopic anisotropic or inhomogeneous bias being introduced at the emergent level, its effective background metric is constrained to inherit, at the macroscopic level, the symmetry content imposed by these pre-geometric constraints. The admissible emergent background geometry is therefore restricted to configurations that are approximately homogeneous and isotropic.

Within the framework of general relativity, geometries that are approximately homogeneous and isotropic belong to the Friedmann–Lemaître–Robertson–Walker (FLRW) class [26, 27]. This is a classification result within a dynamical theory: the present analysis does not derive the FLRW metric from field equations, but rather identifies the structural conditions under which FLRW-type geometries are selected as admissible realizations. Because FLRW geometries have vanishing, or in near-FLRW cases strongly suppressed, Weyl curvature [28], they are naturally associated with very low gravitational entropy in the sense of Penrose [2, 3].

Combining Propositions 1–3, the main structural chain of the argument can be summarized in Equation 49 as

This chain should be understood as a structural constraint on admissible emergent geometries, prior to the detailed dynamical mechanism by which the effective spacetime description itself comes into being.

Although this structural chain does not require specifying the microscopic dynamics of spacetime emergence, it suggests the following natural physical picture. The information saturation interface can be understood as a marginal or threshold-like coarse-grained configuration. When the informational environment coupled to the interface evolves—for instance, through changes in the global constraints on the underlying quantum system—this near-critical state may undergo a global, macroscopically symmetry-preserving phase-transition-like process. Because dynamically sustainable symmetry breaking is suppressed in the near-saturation regime, microscopic or mesoscopic seeds of asymmetry are not robustly amplified into macroscopic preferred directions or positions at the coarse-grained level. The resulting classical geometry, frozen out from this transition, is therefore constrained to inherit the approximate symmetries of the pre-transition relational structure.

We emphasize, however, that a complete description of this transition—including its dynamical mechanism, characteristic timescales, and the emergence of the full causal-metric structure of classical spacetime—lies beyond the scope of the present work and is left for future investigation. The central claim of the present framework is conditional: if information saturation occurs and an effective spacetime description emerges without additional macroscopic symmetry-breaking structure being introduced at the emergent level, then the class of realizable emergent geometries is restricted to near-symmetric background geometries.

At present, the connection between information saturation and suppressed Weyl curvature remains categorical rather than quantitative. The implication should therefore be interpreted as a structural admissibility constraint on the class of compatible emergent geometries, not as a completed dynamical derivation of spacetime emergence. A quantitative relation between the relational variables, including the saturation functional and its derived measures and , and curvature invariants such as would require an explicit specification of how the dynamical emergence or freeze-out process converts the pre-geometric relational organization into an effective metric description. Such a derivation is left for future investigation.

Within this conditional framework, low gravitational entropy emerges not as an independent assumption, but as a structural consequence of information saturation.

4 Conceptual implications and relations to existing frameworks

In the preceding sections, we have argued that the low initial gravitational entropy of the universe need not be regarded as an isolated boundary assumption, but may instead be understood as a consequence of deeper constraints on the organization of information. In this section, we discuss the implications of this framework in a cosmological context and compare it with several existing theoretical approaches.

4.1 Relation to the holographic perspective

The holographic principle posits that the effective degrees of freedom of a physical system can be encoded on its boundary [14–16], thereby fundamentally reshaping the relation between bulk and boundary descriptions. This idea plays a central role in quantum gravity and provides a powerful perspective in which boundary structures capture essential physical content. At a phenomenological level, the emphasis on information and capacity constraints in the present framework is broadly compatible with this viewpoint.

However, the two approaches differ in their starting points. In many conventional holographic settings, a boundary structure or asymptotic geometrical setting is specified as part of the formulation, and the relation between bulk and boundary degrees of freedom is then analyzed within that setting. In contrast, the information saturation interface introduced here is not a geometrical boundary, but an emergent partition defined by the organization of information and its capacity constraints, without presupposing any background geometry.

In this sense, the roles of “boundary” and “interface” are conceptually distinct. A holographic boundary characterizes how information associated with a geometrical region may be encoded, whereas the information saturation interface characterizes when an information structure becomes non-extensible under fixed capacity constraints. It is only at the level of a coarse-grained, continuous description that such interfaces may give rise to an effective geometrical structure, for example, in the form of horizons in gravitational systems [16].

From this perspective, holographic behavior may be interpreted as a possible macroscopic manifestation of an underlying information saturation condition, rather than being taken here as the starting point of the analysis. The present framework thus shifts the emphasis from boundary encoding within a specified geometrical setting to structural constraints on information organization that are formulated without any a priori notions of area, volume, or geometrical boundary, and that, in an appropriate limit, admit a geometric realization.

4.2 Relation to inflationary cosmology

Inflationary models successfully account for the observed large-scale homogeneity and isotropy of the universe, as well as the nearly scale-invariant spectrum of primordial perturbations [4, 5]. However, in most realizations, inflation requires an initial patch that is already sufficiently smooth, so that its primary role is to amplify and stabilize an existing homogeneous configuration rather than to explain its origin.

Within the present framework, if classical geometry is structurally selected under the information saturation condition in a form close to an FLRW configuration, then the smooth initial conditions required for inflation may themselves admit a structural explanation. In this sense, the two mechanisms operate at different levels: the information saturation interface constrains which macroscopic geometries can stably emerge, while inflation describes the subsequent dynamical evolution and amplification of perturbations on such a background.

The present results may therefore be viewed as complementary to inflation, providing a possible structural origin for the class of initial conditions on which inflationary dynamics act.

4.3 Relation to the past hypothesis

In statistical physics, the Past Hypothesis posits that the universe began in a low-entropy macrostate [1, 7], thereby accounting for the thermodynamic arrow of time and the emergence of irreversible processes. While this assumption is highly successful at the phenomenological level, it is introduced as a boundary condition and does not by itself provide a structural explanation for why such a special initial state is realized.

In the framework developed here, the low-entropy initial condition is not imposed as an independent postulate, but can be understood as a consequence of the structural constraints associated with the information saturation interface. Only a restricted class of macroscopic configurations—those compatible with saturation—can stably emerge, and these configurations correspond, in the gravitational description, to states of low Weyl curvature and hence low gravitational entropy.

From this perspective, the Past Hypothesis may be regarded as an effective macroscopic description of a deeper structural constraint. The present approach thus complements it by providing a possible mechanism underlying the selection of low-entropy initial conditions.

4.4 Relation to entanglement-based emergent geometry

Beyond cosmological approaches to low-entropy initial conditions, the proposal bears a conceptual resemblance to entanglement-based emergent-geometry programmes, particularly the Hilbert-space reconstruction framework of Cao–Carroll–Michalakis [29]. Like the present work, Cao–Carroll–Michalakis begin from an underlying informational or relational structure and define an effective distance measure from patterns of quantum correlation, using mutual information to reconstruct geometry from entanglement graphs.

The key difference lies in the role played by the information structure. In Cao–Carroll–Michalakis, one begins with a quantum state together with a tensor-product decomposition of Hilbert space, from which geometric relations are reconstructed. Although no background metric is assumed, the programme is primarily kinematical and reconstructive: geometric information is taken to be encoded in the entanglement structure of the input state and extracted through an appropriate geometric readout.

By contrast, the information saturation interface introduced here is not intended as a reconstruction procedure for extracting an already encoded geometry from a given state. It is intended as a structural selection constraint on which macroscopic geometries can be dynamically admissible when a pre-geometric relational organization approaches saturation. Thus, entanglement-based reconstruction programmes ask how geometric relations can be recovered from a specified informational substrate, whereas the present framework asks why the admissible emergent background should be driven toward a near-symmetric, low-entropy class.

More broadly, the present proposal is compatible with approaches relating spacetime structure to entanglement and information, including tensor-network interpretations of emergent geometry [30], thermodynamic and entanglement-based constraints on gravitational dynamics [31, 32], and entropic gravity proposals [33, 34]. However, these approaches typically rely on some specific dynamical principle, reconstruction map, or thermodynamic relation governing the emergence or behavior of geometry. By contrast, the information saturation interface proposed here is intended as a structural constraint on admissible relational organization, prior to the specification of any particular effective dynamics or reconstruction procedure.

4.5 Relation to the entanglement past hypothesis

The Entanglement Past Hypothesis (EPH) proposes that the initial quantum state of the universe is characterized by exceptionally low entanglement entropy with respect to a given subsystem decomposition [8, 9]. This perspective shifts attention from macroscopic phase-space volume to the structure of quantum correlations, and has been argued to provide a foundation for the emergence of thermodynamic irreversibility.

A key aspect of this formulation is that it is defined relative to a chosen subsystem decomposition, which in physical applications is typically motivated by an underlying notion of classical spacetime or locality [35]. As a result, EPH primarily constrains the properties of quantum states on a given geometrical background, rather than addressing the structural origin of that background itself.

By contrast, the present framework does not assume a prior subsystem structure. Instead, it begins with an information-theoretic organization of degrees of freedom, from which any subsystem decomposition arises only at the level of an effective description, induced by the information interface. Within this structure, one can then identify the class of configurations that can be stably realized under the information saturation condition.

In this sense, the two approaches operate at different conceptual levels: EPH characterizes quantum states relative to a given subsystem decomposition (typically associated with an underlying geometry), whereas the present work concerns the structural conditions under which such a decomposition—and the corresponding geometrical description—becomes well-defined.

Within this perspective, a compatible interpretation can be formulated. It is important to distinguish between the saturation of relational information capacity and the magnitude of entanglement entropy defined with respect to a given subsystem decomposition. In the present framework, saturation refers to the exhaustion of independently realizable structural degrees of freedom under capacity constraints, rather than to the maximization of entanglement across all degrees of freedom. A significant portion of this capacity is effectively occupied by the relational organization associated with the emergent geometry itself. As a result, the residual capacity available for independent correlations among matter degrees of freedom is largely pre-empted. Moreover, entanglement is not an invariant notion, but depends on the choice of subsystem decomposition [36]. While the underlying information-saturated structure may exhibit strong relational correlations, the subsystem structure relevant for defining entanglement entropy arises only at the level of an effective geometric description. With respect to this emergent decomposition, the admissible quantum correlations tend to be structurally simple, giving rise to states with comparatively low entanglement entropy.

From this viewpoint, EPH may be interpreted as an effective description of the quantum state once a classical spacetime has emerged, while the present framework provides a possible structural origin for why such low-entanglement conditions arise in the first place.

In the above comparisons, it becomes clear that the information saturation interface operates at a level distinct from existing approaches. Conventional frameworks—such as holography, inflation, and the Past Hypothesis—typically take a given spacetime geometry or class of initial conditions as their starting point, while proposals such as the Entanglement Past Hypothesis constrain quantum states relative to a fixed subsystem decomposition.

By contrast, the present framework focuses on the underlying organization of information and the constraints imposed by finite capacity, thereby addressing the structural origin of the geometrical background itself. In this perspective, information saturation constrains the class of emergent geometries, which in turn define the subsystem structures relative to which quantum states may exhibit properties such as low entanglement.

The framework may thus be viewed as operating at a more structural level of description, offering a unified perspective on the origin of classical spacetime and its low-entropy initial condition.

5 Falsifiability and observational outlook

5.1 Numerical tests: saturation-induced structural and spectral rigidity

The information saturation condition introduced in this work is defined through a variational principle, as restated in Equation 50,which characterizes a stable structural extremum at the coarse-grained level. This formulation naturally admits an implementation as a numerical optimization problem. In particular, one may explicitly construct discrete information structures approximating the interface and examine whether approaching the saturation regime leads to the predicted universal features. This provides a direct route to falsifiability.

This leads to the optimization problem in Equation 51which can be compared against baseline ensembles such as random or weakly constrained networks.

A further non-trivial diagnostic concerns the comparison with highly regular network structures. Uniform or regular graphs can exhibit a form of rigidity that arises trivially from imposed symmetry alone, including homogeneous degree distributions and concentrated spectral observables. To distinguish the present mechanism from such baseline behavior, optimized low- configurations should be compared not only with random-network ensembles but also with regular or homogeneous graph ensembles of the same size and effective connectivity.

In this stronger sense, a genuine realization of the saturation mechanism would not be identified merely by ordinary robustness. Since the near-saturation regime is better understood as a metastable or threshold-like constrained regime rather than as a conventional stable equilibrium, the more distinctive target is a threshold-dependent response profile. For fixed system size and effective connectivity, one may subject low- configurations, random-network ensembles, and regular baselines to controlled local perturbations of increasing strength, such as the weakening or removal of relational links or localized reductions in channel utilization.

For perturbations below an effective saturation threshold, low- configurations are expected to exhibit a damped coarse-grained response: local perturbations should be partially absorbed without being efficiently amplified into new macroscopic distinguishable organization. This corresponds to the local elastic response of a near-minimal saturation functional, where residual organizational extensibility is strongly suppressed.

For perturbations above the same threshold, however, the response need not remain stable. A sufficiently strong perturbation may drive the system out of the metastable near-saturation basin, after which the relational structure can explore a much larger set of lower-symmetry distinguishable configurations. In this regime, the response of a low- configuration may become sharply nonlinear and can even exceed that of random or regular baselines.

The qualitative form of this diagnostic is summarized schematically in Figure 2. The key feature is not the absolute magnitude of the response, but the two-regime structure: a suppressed sub-threshold response followed by a sharp nonlinear departure once the effective saturation barrier is exceeded.

FIGURE 2

The spectral or relational response shown schematically in Figure 2 may be instantiated by several concrete observables. Relevant observables include the Laplacian spectral gap, low-lying spectral statistics [24], and relational observables derived from , such as distance distributions or coarse-grained measures of relational anisotropy. The present framework does not predict a universal numerical value for these quantities. Instead, the proposed numerical signature is comparative and nonlinear: a saturation-induced structure should show suppressed spectral and relational response to sub-threshold perturbations, followed by a sharper transition once perturbations exceed the effective saturation barrier.

A second class of tests concerns consistency under coarse-graining transformations, formally represented in Equation 52 as

If the saturation regime represents a stable structural organization, one expects dominant relational patterns to remain approximately preserved under coarse-graining, with remaining within the low- regime and large-scale relational organization remaining statistically stable across scales.

These tests can be implemented using standard tensor-network libraries, graph-based optimization methods, and spectral graph tools. Relevant observables include entanglement spectra, graph Laplacian statistics, relational distance distributions, and the coarse-grained evolution of . In this way, the information saturation condition can be translated into concrete numerical experiments, providing a direct and model-independent route to testing the central structural mechanism proposed in this work.

5.2 Cosmological signatures: constraints on primordial fluctuations

The structural results of Section 3 imply that the information saturation condition constrains the class of admissible deviations around the relational structure from which a classical background may emerge. In the graph-based formulation, the constant sector corresponds to uniform relational organization, while non-constant relational modes carry a positive structural cost. If a geometric description subsequently emerges, these relational deviations may be mapped only indirectly onto primordial perturbations of the effective background.

The structure of suggests two qualitatively distinct classes of residual perturbations. Perturbations in the channel-utilization sector, associated with variations of , correspond to changes in the strength or uniformity of already available relational correlations. If frozen into an effective geometric description, such perturbations would most naturally appear as small-amplitude variations around the near-FLRW background, possibly including mild scale-dependent modulation or suppression of non-constant perturbative components. By contrast, perturbations in the local-extensibility sector, associated with variations of , correspond to the residual ability to open additional independently distinguishable local configurations. If not fully suppressed, these perturbations would be expected to leave more genuinely structural signatures, such as weak mode coupling, mild non-Gaussianity, or small departures from statistical isotropy. These possible features should be understood as correlated consequences of a single underlying structural constraint, rather than as independent predictions.

Such effects may, in principle, be probed indirectly through observations of the cosmic microwave background (CMB) and large-scale structure [39, 40], for example, via angular power spectra, higher-order correlation functions, or mode-coupling statistics. However, these observables are shaped by subsequent dynamical evolution, transfer functions, foreground contamination, cosmic variance, and survey systematics, and therefore provide only indirect access to the primordial relational structure.

Accordingly, the observational role of the present framework is best regarded as providing structural consistency conditions rather than precise spectral predictions. If future high-precision CMB and large-scale-structure data were to support a fully unconstrained Gaussian description of primordial perturbations across all accessible probes, with no statistically significant correlated deviations after accounting for known dynamical and observational effects, then the relevance of information saturation as a dominant organizing principle for the initial state would be significantly weakened.

In this sense, cosmological observations offer a complementary window on the framework, while the primary falsifiability remains rooted in the direct structural tests discussed in Section 5.1.

6 Conclusion

Why the Universe began in a state of extremely low gravitational entropy remains one of the most profound open questions in modern cosmology. Penrose’s Weyl curvature hypothesis identifies the essential feature of this special initial condition as the suppression of gravitational degrees of freedom: the early spacetime is characterized by a nearly vanishing Weyl curvature, corresponding to very low gravitational entropy. In most existing cosmological frameworks, however, this condition is introduced as an additional assumption.

In this work, we have proposed a structural perspective on this problem. Instead of treating the low-entropy initial state as a given, we introduce a more primitive notion, the information saturation interface, characterized by a variational functional . The condition represents an operational limit in which information channels are nearly saturated and admissible local operations no longer generate additional independently distinguishable relational organization.

In this regime, the behavior of the relational structure is no longer primarily governed by the unrestricted growth of independently distinguishable configurations, but by the reorganization of existing relational patterns under global structural constraints. By analyzing the stability of near its minimum, we have shown that its minimization selects a uniform relational organization and assigns a positive cost to deviations, thereby suppressing non-uniform and direction-dependent relational variations. Moreover, dynamically sustainable symmetry-breaking organization is suppressed in the near-saturation regime: microscopic or mesoscopic fluctuations may still occur, but they cannot be robustly amplified into persistent macroscopic preferred directions or positions. Consequently, if a classical spacetime emerges as an effective geometric description, its background geometry is constrained to inherit, at the macroscopic level, the symmetry content of the pre-geometric relational organization from which it emerges.

Within the framework of general relativity, this constraint naturally selects the class of Friedmann–Lemaître–Robertson–Walker (FLRW) spacetimes. These geometries are conformally flat, implying a strong suppression of Weyl curvature and hence very low gravitational entropy. In this sense, the Penrose condition need not be imposed as an independent assumption, but can instead be understood as a consequence of the symmetry constraints imposed by information saturation.

This framework provides a new perspective on the low-entropy initial condition problem. Here, low entropy does not arise from a fully specified dynamical mechanism, nor is it introduced as a boundary condition, but instead follows from a structural constraint on information organization. The present framework is therefore best viewed as a structural selection principle for low-gravitational-entropy emergent backgrounds. Its central value lies in identifying a relational condition that is compatible with, and may explain the admissibility of, near-FLRW initial geometries, while leaving the full dynamical emergence of spacetime and quantitative cosmological predictions for future investigation.

From this perspective, if the Universe indeed began in a state of extremely low gravitational entropy, this may indicate that classical spacetime emerged from a regime close to an information capacity limit. At such an interface, a large number of microscopic degrees of freedom undergo collective reorganization, giving rise to a stable large-scale ordered structure. In this sense, classical spacetime may be viewed as emerging from a global transition associated with information saturation, providing a structural origin for the special initial condition of the Universe.

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