Looped spacetime cosmology: a closed-time framework for quantum gravity and cosmology

Abstract

Looped Spacetime Cosmology (LSC) explores whether black hole interiors and the cosmological origin may be linked by a single global spacetime structure. The framework posits a compact topology in which astrophysical black hole trapped regions are globally identified with a common Big Bang hypersurface so that classical terminal curvature endpoints are replaced (at the level of induced data) by a compact global completion. As a minimal realization of nonterminal high-curvature behavior, we employ effective dynamics inspired by loop quantum cosmology (LQC), without assuming a specific ultraviolet completion. The resulting consequences are conditional: if the identification is dynamically admissible, then (i) classical singularities may be avoided in an effective description via bounded-density transitions; (ii) information carried by infalling degrees of freedom need not terminate at a singular boundary but can be globally accounted for on the identified hypersurface, contingent on an isometric mapping across the transition region; (iii) fine-grained entropy can remain globally conserved, while coarse-grained observers still experience an emergent arrow of time; and (iv) compact topology provides a setting in which infrared-regulated vacuum contributions can be consistent with a small late-time cosmological constant in representative models. LSC is formulated as a falsifiable hypothesis rather than a completed theory. Its empirical program is organized as hierarchical tests: Tier I gates probe sign-fixed curvature/topology consistency and can falsify the framework outright; Tier II tests constrain dark-sector realizations without negating the global identification if Tier I survives; and Tier III probes transition-region microphysics (e.g., model-dependent compact-object phenomenology) and is logically downstream. Near-term scrutiny is provided by spatial curvature and topology constraints, population-level compact-object spin statistics, and cross-channel consistency checks using cosmic microwave background (CMB) and large-scale structure data.

1 Introduction

Modern physics rests on two foundational frameworks: general relativity (GR), in which matter and energy curve spacetime and freely falling bodies follow geodesics of that curved geometry, and quantum field theory (QFT), which unites special relativity with quantum mechanics by describing matter and interactions as excitations of underlying fields. Each has been confirmed across vast regimes, yet their assumptions remain formally incompatible. GR treats spacetime itself as a dynamical entity, while QFT presupposes a fixed (or semiclassically prescribed) background. This tension becomes unavoidable in regimes in which strong curvature, quantum coherence, and global spacetime structure are simultaneously essential, most notably in black hole interiors and in the earliest high-curvature regime of cosmology, where no shared semiclassical description exists.

Within this context, Looped Spacetime Cosmology (LSC) is introduced as a hypothesis about the global structure of spacetime. The central idea is that astrophysical black hole trapped interior regions are globally identified with a portion of a common Big Bang hypersurface within a closed spacetime manifold. The identification is implemented as a gluing of induced data across a bounded-curvature transition region rather than as a locally traversable timelike direction. The factor, therefore, represents a global topological identification rather than periodic proper time for observers.

1.1 Framework scope and realization dependence

LSC separates the global geometric structure from the microphysical mechanism governing high-curvature evolution. The Tier I falsification burden attaches to the global identification and its associated curvature/topology consistency. The dynamics of the transition region are treated at an effective level and may admit multiple realizations.

In the minimal realization studied here, near-transition behavior is modeled using loop quantum cosmology-inspired polymer/holonomy corrections in approximately homogeneous high-curvature regimes, with semiclassical propagation in the exterior spacetime (Sections 2.2, 2.3) [1, 3]. Polymer quantization in the homogeneous sector yields a finite critical density (Section 2.3) [1, 3]. The transition region is represented through an effective matching construction, including an isometric map across the transition surface (Section 3.4).

1.2 Scope of the present construction

The analysis develops a controlled, effective realization of the black hole (BH)Big Bang (BB) identification on , formulated at the level of induced data matching across the high-curvature transition region. The framework specifies the geometric identification, introduces a bounded-curvature parametrization of the transition dynamics, and organizes the resulting observational consequences into a small number of empirical tests. Within this setup, the global spacetime identification can be analyzed independently of the specific ultraviolet completion governing the transition region.

1.3 Structure of the effective construction

The resulting framework provides a mathematically explicit realization of the BHBB identification at the level of induced data and junction conditions while leaving the detailed microphysical completion of the transition region open.

1.4 Causality and the identification

The presence of an factor in describes the global topology of spacetime rather than periodic physical time. In the realization considered here, the physical spacetime relevant to observers admits a global time function and spacelike Cauchy slices away from the bounded-curvature transition region. The BHBB identification is implemented as a gluing of induced data across that region rather than as a locally traversable timelike direction. Under these assumptions, timelike worldlines remain monotonic in proper time, and no closed timelike curves arise from the identification. Topological-censorship theorems [69–71] guarantee this in both asymptotically flat and cosmological settings. A detailed discussion of the assumptions entering this construction is provided in Supplementary Appendix S7.

This identification motivates several linked phenomenological consequences. Interior evolution proceeds through a bounded-curvature continuation rather than terminating at classical singularities. Information associated with infalling degrees of freedom can, therefore, be represented globally on the identified hypersurface . Fine-grained entropy may remain constant on the global spacetime, while coarse-grained observers perceive entropy increasing along forward-directed worldlines. Quantization of infrared modes on yields a finite finite-size vacuum contribution whose magnitude can be compared with the observed late-time cosmological constant. The compact identification also motivates correlated dark-sector phenomenology testable through structure formation, lensing, and clustering measurements.

1.5 Empirical posture and hierarchical tests

The framework is organized around hierarchical falsification gates. Tier I tests probe the global identification and curvature/topology consistency; failure at Tier I falsifies the framework. Tier II tests constrain specific dark-sector realizations but do not negate the global identification if Tier I survives. Tier III probes transition-region microphysics.

A schematic representation of the identification and bounded-curvature transition region is shown in Figure 1.

FIGURE 1

2 Theoretical framework

2.1 A guided tour of the loop

For orientation, the global identification is summarized as a four-stage ordering (see Figure 2): (i) Birth: degrees of freedom are represented on the common Big Bang hypersurface ; (ii) structure formation: gravity amplifies seeds and compact objects form; (iii) collapse: matter falls into black holes and enters a bounded-curvature transition region; and (iv) re-emergence (data-level identification): interior degrees of freedom are represented on via an identification map on induced data, not by any observer worldline looping back in time. The local arrow of time points away from the transition region on each side, so embedded observers experience forward-directed evolution and do not traverse closed causal curves (Supplementary Appendix S7).

FIGURE 2

2.2 Global closure label

Globally, denotes a closure label (loop-identification length) associated with the factor. is not introduced as an observer-measured period in proper time and is not treated as an independent observable. It functions as a bookkeeping parameter that characterizes the global identification structure in an effective description, with its value tied to the chosen high-curvature continuation scale (e.g., through as set by the polymer scale and Immirzi parameter) [1, 3–5]. Locally, timelike worldlines remain monotonic in their proper time, and the spacetime relevant to observers is taken to be globally hyperbolic away from the bounded-curvature region; the identification is imposed at the induced data level and does not generate closed timelike curves within the assumed causal structure (Supplementary Appendix S7). Box 1 below summarizes the framework’s postulates, prerequisites, and modeled structure.

Modeling stance (one-time statement). The framework is defined by the induced data identification (P1) plus the existence of a nonterminal high-curvature continuation (P2) compatible with standard low-curvature GR evolution and with observer-time monotonicity (P3). Where a full ultraviolet completion is not specified, the analysis uses controlled effective parametrizations (polymer/holonomy dynamics in approximately homogeneous sectors and a distributional junction proxy) solely to (i) render the BHBB induced data map well-posed and (ii) derive falsifiable observational discriminants.

2.3 Foundational postulates

LSC is defined by a compact set of postulates separating (i) the global identification principle from (ii) a local nonterminal high-curvature continuation required to make the identification meaningful. The identification is imposed at the level of induced data on spacelike hypersurfaces. It does not act on timelike worldlines: no observer experiences periodic proper time, and no locally traversable “time loop” is implied.

Postulate P1 (global topology and identification). Physical spacetime admits an effective compact completion with topology , in which the trapped interior region(s) of astrophysical black holes are globally identified with a unique Big Bang hypersurface . The identification is implemented as a map between induced data sets on spacelike hypersurfaces (three-metric and conjugate momentum/extrinsic curvature data, and, in a quantum description, compatible state data).

Postulate P2 (local nonterminal continuation). The would-be classical singular region is replaced by a bounded-curvature transition region such that the induced data approaching the transition remain finite and can be matched (in an effective description) to induced data on . LQC-inspired polymer/holonomy effective dynamics provide a minimal proxy for bounded-curvature behavior in approximately homogeneous sectors [1–4].

Postulate P3 (causality and observer-time monotonicity). Away from the bounded-curvature region, the spacetime relevant to embedded observers admits a global time function with spacelike Cauchy slices. The BHBB identification is a gluing of induced data across an achronal interface, not a locally traversable timelike direction; consequently, observer proper times remain monotonic, and no closed timelike curves are generated in the assumed causal structure (Supplementary Appendix S7).

This postulate specifies the causal conditions under which the induced data identification can be implemented without generating closed timelike curves for embedded observers. It should be interpreted as a consistency requirement on the effective construction, rather than as a theorem establishing the causal structure of a fully explicit global metric realization.

2.4 Quantum transition region dynamics and effective matching

In LSC, the classical singularity is replaced by a bounded-curvature transition region. An effective description supplies the minimal structure needed (i) to define the BHBB induced data identification and (ii) to connect to observational discriminants [1–4]. Concretely: (i) approximately homogeneous sectors entering the high-curvature regime are modeled by polymer/holonomy-corrected dynamics (Section 2.3); and (ii) the transfer of induced geometric data (and, in a microscopic completion, compatible state data) across is parameterized by an effective matching prescription.

2.4.1 Effective junction parametrization (distributional proxy)

To parameterize matching without selecting a specific completion, the standard geometric form of the Israel junction relation is used as a distributional proxy for the integrated Planck-scale stresses of the transition region [6]. Induced-metric continuity is imposed,together with the junction conditionequivalently

The tensor summarizes the integrated effect of the bounded-curvature region on the jump in extrinsic curvature required to realize the BHBB induced data identification. It is a bookkeeping object for unresolved Planck-scale dynamics. It is not interpreted as a physical low-energy thin shell or as an added material layer in the exterior spacetime. A convenient decomposition iswhere is a timelike unit vector tangent to (defined within each side’s foliation), is an effective surface energy density, is an effective tangential surface pressure, and encodes anisotropic stresses.

These junction relations are used here only to parameterize admissible induced data matching across the proxy transition region. They should not be read as a derived covariant low-curvature thin-shell model. Their role is to render the matching problem mathematically explicit at the level of induced data while leaving the underlying high-curvature microphysics unspecified.

2.4.2 Control requirement for the proxy

Because encodes unresolved microphysics, closed-form expressions for , , and are not assumed. The effective description is required to satisfy: (i) finiteness of induced data and the distributional jump quantities entering Equations 2.1–2.3; and (ii) recovery of standard GR propagation in the low-curvature exterior, so the matching does not introduce late-time tensor-sector modifications [7]. Rotation and anisotropy enter through and the jump structure; this is the technical sense in which near-extremal configurations require a more refined treatment (Section 2.5).

2.4.3 Perturbations and effective reflectivity

For linearized perturbations incident on the transition region, the effective response is summarized by a frequency-dependent reflection coefficient and a transmission coefficient ,where denotes a master perturbation variable in an exterior effective-potential problem. This parameterization is agnostic about microscopic details while enabling direct contact with observational diagnostics such as echo searches (Section 4.6) [8].

2.4.4 Minimal consistency requirements

Within this framework, the microphysical completion of the transition region is encoded in and the effective surface tensor . Meanwhile, the macroscopic bounded-curvature behavior is governed by the polymer-modified dynamics of Section 2.3.

2.5 Polymer quantization and bounded-curvature continuation

An LQC-inspired effective description of polymer/holonomy corrections is adopted in the high-curvature regime, using standard LQC results as a semiclassical proxy [1–4]. Standard holonomy replacement rules and their resulting effective Friedmann-type dynamics are used as an ansatz for approximately homogeneous sectors, chosen because they capture bounded-density/bounded-curvature behavior across a broad class of LQC quantizations [1, 3, 4].

2.5.1 Observational posture

The bounded-curvature continuation renders the BHBB induced data identification well-defined while remaining agnostic about detailed “pre-bounce” imprint mechanisms. LSC does not require enhanced primordial non-Gaussianity or other model-specific bounce signatures, so exclusions of particular phenomenological bounce scenarios based on CMB bispectrum constraints do not directly apply. The requirement is a nonsingular continuation at the induced data level compatible with the global identification on .

The essential feature of the polymer/holonomy framework is that the connection is not promoted to a well-defined operator; instead, only its holonomies along loops of minimum physical area are represented [1, 3, 4]. At the effective level, this yields characteristic modifications to the classical Hamiltonian constraint, leading to bounded energy density and curvature within the sector to which the ansatz is applied [1, 3, 4].

Operationally, the classical connection variable is replaced by its holonomy-corrected formwhere is fixed by requiring that the physical area of the elementary loop equals the minimum nonzero eigenvalue of the loop quantum gravity (LQG) area operator [1, 4, 5]. This “improved dynamics” prescription renders the resulting critical density essentially state independent [1, 4].

With this replacement, the effective Friedmann equation for a homogeneous region with energy density becomeswhere (see Figure 3) [1, 3, 4].

FIGURE 3

1, 3, 4].

In standard LQC, one finds a universal critical densitywith the Barbero–Immirzi parameter fixed by black hole entropy considerations in loop quantum gravity [5]. The numerical value is a robust feature of a wide class of polymer quantizations [1, 3, 4].

In the present work, Equation 2.7 is adopted as an effective minisuperspace ansatz for both the cosmological exterior and an approximately homogeneous sector of the black hole interior while noting that a full treatment of generic rotating interiors requires an anisotropic (Bianchi) polymer quantization [3, 4].

The bounded-curvature “bounce” in this effective description occurs when the expansion rate vanishes, , corresponding toCollapse is halted at finite density independent of the mass of the collapsing object, and the effective solution transitions smoothly from contraction to expansion within the homogeneous proxy [1, 3, 4].

For a collapsing region of proper mass , the minimum scale factor reached at the bounce isso that for stellar-mass black holes, the bounce occurs at scales many orders of magnitude above the Planck length even though the local density is Planckian [1, 3, 4].

Curvature invariants remain finite throughout the effective evolution. For example, the Ricci scalar and Kretschmann scalar are bounded above by values of orderwith analogous bounds holding for higher invariants. Curvature grows toward the Planck regime but saturates at a finite value rather than diverging [1, 3, 4].

2.6 Interior–exterior matching and rotating interior geometry

An effective description of the collapsing interior, the expanding exterior, and the matching across is specified. The construction targets (i) anisotropy/rotation in the collapsing region, (ii) bounded-curvature continuation due to polymer (LQC-inspired) effects in an approximately homogeneous sector, and (iii) a well-defined data-level junction to an expanding, closed Friedmann–Lemaître–Robertson–Walker (FLRW) exterior [3, 4].

2.6.1 Rotating black hole interior as an effective Bianchi-IX model

For the interior, a homogeneous but anisotropic Bianchi-IX geometry, the most general closed homogeneous spacetime, is used as an effective proxy for a rotating black hole interior once angular momentum induces anisotropic collapse rates [9, 10]. In terms of left-invariant one-forms on , the line element iswhere and are independent scale factors describing anisotropic collapse. The resulting matching geometry is shown schematically in Figure 4.

FIGURE 4

Rotation is incorporated phenomenologically through a spin-dependent anisotropy parametrization,where is the dimensionless spin parameter, is a reference scale associated with the onset of strong curvature, controls the growth rate of anisotropy during collapse, and absorbs numerical factors. This captures the qualitative effect of centrifugal support (anisotropy growth as the interior contracts) without identifying Equation 2.12 with an exact Kerr interior.

In classical GR, Bianchi-IX dynamics can exhibit Mixmaster behavior with an oscillatory approach to a singular endpoint [9, 11]. In the present effective LSC realization, curvature growth in the approximately homogeneous sector is regulated by the polymer (holonomy) corrections of Section 2.3, which bound the effective energy density and suppress unbounded curvature growth within the proxy [3, 4].

At the effective level, the interior dynamics may be summarized schematically by a modified constraint of the formwhere includes shear contributions, is given by Equation 2.8, and denotes a spin-induced anisotropy term. The factor encodes holonomy suppression as the Planck regime is approached [3, 4].

For slowly and moderately rotating interiors, this effective description yields a bounded-curvature continuation (bounce in the homogeneous proxy) at finite scale factor , with invariants bounded as in Section 2.3. The evolution of anisotropies through the Planck regime depends on the polymerization scheme; LQC studies of anisotropic cosmologies indicate that the classical Mixmaster instability is strongly mitigated once holonomy corrections are included [3, 4].

2.6.2 Matching to the closed FLRW exterior

On the exterior side, the spacetime is modeled as a closed FLRW universe obeying the polymer-modified Friedmann Equation 2.7 [1, 3, 4]. The identification across is implemented by enforcing continuity of the induced three-metric and by allowing a controlled relation between extrinsic curvatures consistent with the global BHBB induced data identification (Section 2.1) [6].

The transition region is not interpreted as a locally traversable passage in time. Instead, functions as an achronal identification surface that transfers induced geometric data and (in a microscopic completion) compatible quantum state information from the collapsing interior to the expanding exterior while preserving monotonic proper time on each side [6]. Interior-directed timelike curves terminate at the transition region in the effective description; their induced data are globally re-encoded as part of the Big Bang initial conditions of the expanding FLRW branch.

This construction yields a single forward arrow of time locally for observers, while global closure is implemented through a data-level identification consistent with a closed topology.

2.9 Limitations: high-spin configurations and open problems

The effective construction above is most transparent for slowly and moderately rotating black holes, in which approximately homogeneous sectors provide a controlled proxy for illustrating bounded-curvature behavior [3, 4].

Consequently, robust evidence for a population of long-lived astrophysical black holes with reliably measured near-extremal spins would not directly refute the topological identification postulate. However, it would rule out the minimal continuation and matching prescription adopted here and would require a more sophisticated treatment of anisotropy and rotation (including anisotropic polymer quantization and/or numerical control) to maintain a bounded-curvature continuation in the near-extremal regime.

From an observational perspective, current gravitational wave catalogs and population analyses are broadly consistent with modest typical spins for stellar-mass black holes, while near-extremal configurations remain challenging to establish in a model-robust way [12–15]. Population inference often constrains combinations of component spins (such as the effective inspiral parameter ) rather than individual Kerr parameters [12, 15]. Electromagnetic spin estimates from accreting X-ray binaries provide complementary constraints but remain sensitive to astrophysical and radiative-transfer modeling [16–21].

3 Core results

3.1 Reader’s map (hierarchy and scope)

The results of LSC are logically hierarchical.

Prerequisite gates are local conditions required for the BHBB induced data identification to be well-defined (notably a nonterminal, bounded-curvature continuation at the induced data level).

Tier I is the framework-level falsification gate: the global identification must be consistent with a sign-fixed spatial curvature/topology in the cosmological exterior and with population-level spin/anisotropy consistency under a single identified Big Bang hypersurface.

Tier II/Tier III are downstream: model-dependent dark-sector phenomenology and transition-region microphysics (e.g., ) that are only meaningful if the prerequisite gates and Tier I survive. Accordingly, the results below are organized as (i) prerequisite gates, (ii) Tier I claims, and (iii) downstream compatibility conditions.

3.2 Prerequisite gate: bounded-curvature continuation

A prerequisite for LSC is the replacement of classical curvature singularities inside black holes and at the Big Bang by a finite, well-defined high-curvature regime at the induced data level (Postulate P2). In this manuscript, that prerequisite is implemented at the level of an effective description via the polymer-modified dynamics of Section 2.3. The bounded-curvature statement is, therefore, restricted to the minisuperspace/approximately homogeneous sectors to which the LQC-inspired ansatz is applied and is used as a controlled proxy for a non-perturbative completion rather than as a claim of a unique ultraviolet theory.

3.2.1 Relation to cosmic censorship and modern interior structure

Classical collapse is framed by Penrose’s cosmic censorship program [22], distinguishing exterior predictability (weak cosmic censorship) from the inextendibility of maximal globally hyperbolic developments (strong cosmic censorship). Mathematical and numerical results on black hole interiors, including the Dafermos program and subsequent work, show that dynamical interiors can exhibit Cauchy horizons and weak/null singularities with subtle extendibility properties rather than a universally spacelike endpoint [23, 24]. These developments emphasize that “singularity structure” is not monolithic even within classical GR and that the interior outcome can depend sensitively on perturbations and matter models.

In LSC, cosmic censorship is not used as a claim of classical GR behavior. Instead, the working assumption is that the high-curvature regime relevant to the induced data map is replaced by a bounded-curvature quantum transition region in the sector where the polymer-effective description is an adequate proxy. Recent studies probing critical behavior and singularity structure in black hole interiors [25], and analyses of strong cosmic censorship in loop quantum gravity-motivated or modified black hole settings [26, 27], provide complementary perspectives relevant to this prerequisite. The role of the present subsection is to state the continuation gate precisely (finite induced data and bounded invariants in the proxy sector) and to locate it within the modern landscape of interior structure.

3.2.2 Classical regular black hole alternatives

Singularity-free interiors can also arise in classical effective-metric constructions via modified matter sectors. Examples include the Bardeen and Hayward regular black hole models [28, 29], which replace the central singularity with a nonsingular core and typically introduce an inner (Cauchy) horizon. These models provide useful comparators: LSC does not require that only LQG-inspired dynamics can regularize interiors. Rather, LSC is organized around a global BHBB identification on , and it adopts a bounded-curvature transition region as a minimal prerequisite mechanism consistent with that identification. Recent analysis of regular black hole interiors, including work on the Hayward case [30], highlights that inner-horizon behavior and stability are key discriminators among regularization mechanisms.

For curvature control in the proxy sector, consider the Kretschmann scalar:The polymer-effective dynamics imply saturation of curvature invariants at the Planck scale, with an upper bound of schematic formwhere is the Barbero–Immirzi parameter and is the Planck length. The numerical coefficient multiplying depends on the effective scheme and matter content; the point for LSC is that saturates at a finite value controlled by the Planck scale within the proxy sector rather than diverging as in classical GR.

3.2.3 Effective energy condition violation and well-posedness as necessary condition gates

It is often convenient to rewrite the polymer-modified Friedmann evolution as a standard Einstein evolution sourced by an effective stress-energy tensor. In that rewriting, the null energy condition is violated only in an effective sense and only within a narrow Planck-regime window near , enabling and halting collapse/triggering continuation. This is treated here as a bookkeeping representation of holonomy corrections, not as exotic low-curvature matter and not as a covariant ultraviolet completion.

The Supplementary Material provides the conditional discussion of these requirements, including assumptions needed for hyperbolicity/well-posedness and anisotropic matching across the transition region. In the main text, the effective fluid rewriting is used only to state these gates transparently.

3.2.4 Rotating interiors (proxy)

For rotating black holes, the same Bianchi-IX interior proxy of Section 2.4 is used as a bookkeeping model for anisotropic collapse within the bounded-curvature prerequisite. The interior line element iswhere are SU(2) Maurer–Cartan one-forms on . The anisotropic scale factor encodes spin-induced anisotropy; one illustrative parameterization iswhere is the black hole mass and is the dimensionless spin. This is a proxy capturing the centrifugal tendency toward enhanced anisotropy during collapse; alternative parameterizations (including the family of Section 2.4) serve the same role in the present effective treatment.

Including polymer suppression and a phenomenological spin contribution, the interior evolution is written schematically aswhere includes shear contributions, is the standard holonomy suppression, and summarizes spin-induced anisotropy/centrifugal support in the approximately homogeneous sector. The function (with and finite) encodes the expectation that polymer effects also regulate the spin/anisotropy sector near the Planck regime. This is an LQC-inspired effective ansatz for the competition between centrifugal support and holonomy suppression, not the result of a complete polymer quantization of rotating interiors.

Within its domain of validity, the proxy sector summarized by Equations 3.2–3.5 implies the following: (i) curvature invariants remain finite (with bounded at the Planck scale); (ii) NEC violation arises only in an effective rewriting and only near ; and (iii) the interior undergoes a continuation/bounce at finite scale factor rather than evolving to a classical singular endpoint. The near-extremal regime, in which anisotropies may grow and rotating polymer dynamics require dedicated treatment, is treated as an open problem requiring anisotropic polymer quantization and/or numerical evolution beyond the present effective model in Section 2.5.

3.3 Tier I: sign-fixed curvature/topology consistency of the global identification

Therefore, Tier I is evaluated by combining curvature/topology inference with population-level constraints on compact-object spins and the absence of late-time tensor-sector modifications. If Tier I fails, downstream phenomenology (dark-sector realizations and transition-region microphysics) cannot be interpreted within LSC.

3.4 Tier II ingredient: infrared stabilization of vacuum energy on compact topology

A downstream ingredient of LSC is that a compact global topology can regulate the infrared (finite-size) sector of vacuum fluctuations. On the product manifold , the spectrum of a free field is discrete: for a representative massless scalar, the Laplace operator eigenvalues take the formwith degeneracy for each on . The corresponding mode frequencies are . The formal zero-point energy diverges. However, on a compact manifold, a finite renormalized finite-size contribution can be defined by analytic continuation of the associated spectral zeta function. In LSC, this serves as an illustration of how global radii enter the renormalized infrared sector; it is not a solution to the ultraviolet cosmological constant problem.

Define the zeta function of the Laplacian on as, We follow the spectral-regularization conventions of [67, 68].which converges for sufficiently large and admits analytic continuation to a meromorphic function. Using Poisson resummation in the -sum and Mellin transform techniques, one may decomposewhere is an Epstein-type zeta function encoding the zero-mode sector along and collects terms that are exponentially suppressed when (with coefficients that depend on boundary conditions and field content).

The regularized vacuum energy density is obtained from viawhere is the compactification cell volume and is an arbitrary renormalization scale. In regimes with a hierarchy of radii, one typically finds a finite-size/topological contribution with schematic dependenceup to order-one factors depending on matter content, boundary conditions, and the renormalization prescription. Equation 3.10 is used as a parametric estimate: it exhibits strong suppression of certain long-wavelength (infrared) sectors on compact manifolds. This is not treated as a Tier I claim about the observed ; it is a Tier II ingredient indicating how global structure can stabilize the infrared sector once a small renormalized cosmological constant is assumed at the matching scale.

Zeta-function methods on , therefore, provide an infrared/finite-size regulator: they show that, given a small renormalized cosmological constant at the matching scale, compact topology can prevent large additional infrared contributions from the long-wavelength sector. They do not resolve the ultraviolet cosmological constant problem, which still requires near-cancellation between the bare parameter and short-distance vacuum contributions. Within LSC, a compact topology provides a mechanism for infrared stability of a small effective vacuum energy while supplying a setting in which scaling with global radii can be analyzed. The full zeta-function regularization is given in the Supplementary Material, Section 4.

3.5 Compatibility condition: information preservation and unitarity

The black hole information problem arises because semiclassical gravitational collapse and Hawking evaporation appear to map an initial pure quantum state to an outgoing mixed state, apparently conflicting with unitary quantum evolution. Many proposed resolutions rely on detailed microscopic mechanisms or specific ultraviolet completions. The LSC framework instead imposes global consistency conditions associated with (i) a nonterminal high-curvature continuation at the induced data level (Postulate P2) and (ii) the existence of a unique identified Big Bang hypersurface within a compact spacetime (Postulate P1).

Within this global structure, interior degrees of freedom are not represented as terminating at a classical singular endpoint. Instead, collapse proceeds to a bounded-curvature transition region, after which interior data are represented on the unique identified hypersurface . This statement concerns the global representation of degrees of freedom rather than any locally traversable process. No observer crosses the transition region to recover information directly; the claim is one of global bookkeeping/representation.

Compatibility with unitary quantum evolution requires that the continuation through the high-curvature regime admits a unitary embedding of physical states. A minimal compatibility condition can be expressed through the existence of an isometric embedding mapping allowed infalling states into an appropriate near-transition Hilbert space, together with a unitary continuation operator acting on that space,with on the physical subspace. In such a completion, the continuation through the high-curvature region preserves quantum information at the level of the full Hilbert space.

Under this assumption, the fine-grained von Neumann entropy satisfieswhile any apparent entropy increase arises from coarse-graining by observers who trace over degrees of freedom not operationally accessible on a given cosmological slice.

Effective loop quantum cosmology provides an explicit example of bounded-curvature continuation in a simplified setting: polymer quantization replaces the Wheeler–DeWitt differential operator by a discrete evolution operator that extends the physical state deterministically through the high-curvature regime in a suitable internal time. In the present framework, the effective Friedmann law of Equation 2.7 represents the semiclassical limit of that continuation. The role of LSC is to combine this bounded-curvature continuation (as a prerequisite condition) with the global identification on a compact spacetime, ensuring that interior degrees of freedom remain represented on a single hypersurface rather than terminating at classical singularities.

Equations 3.11, 3.12, therefore, define the required compatibility condition: the global BHBB identification is compatible with a unitary completion if such a completion exists. The present framework does not derive a microscopic evolution law for quantum fields across the transition region, does not compute entanglement-entropy transport, and does not provide a Page-curve calculation. Instead, it isolates the structural claim relevant to LSC: a non-singular continuation together with a single identified hypersurface provides a global spacetime setting in which infalling degrees of freedom need not terminate at a singular boundary. Meanwhile, operational coarse-graining can still account for entropy production in observations.

4 Observational predictions

This section summarizes empirical tests of LSC using the hierarchical gate structure defined in Table 1. Tier I tests target the theory-critical global structure (sign-fixed curvature/topology consistency and the high-spin control boundary of the minimal effective continuation). Tier II tests address correlated cosmological consequences in Tier I-surviving realizations (e.g., curvature–dark energy degeneracies, optional early dark energy remnants, and cross-channel distance–growth consistency). Tier III tests constrain transition-region microphysics or optional add-ons (e.g., effective nonlocal sourcing and gravitational wave (GW) echoes). Therefore, they cannot falsify the core BHBB identification by themselves.

4.1 CMB curvature and topology signatures

4.1.1 Tier I curvature sign gate

The spatial geometry assumed in LSC is slightly positively curved (closed FLRW on each cosmological branch), so the curvature parameter satisfieswhere is the curvature radius. This sign is the relevant Tier I discriminator: a robust inference (open spatial slices) would falsify the minimal LSC global geometry assumption as implemented here. The magnitude of is not fixed by topology alone and is treated as an empirical quantity to be constrained jointly with late-time parameters (Table 1).

4.1.2 Topology (matched circles) is optional, not required

Positive curvature (universal cover ) does not, by itself, determine the global spatial topology. The simply connected 3-sphere generically produces no repeated imaging. Observable cosmic topology signatures such as matched circles in the CMB arise only if the spatial manifold is a nontrivial spherical quotient , where is a discrete fixed-point-free subgroup of the isometry group of [31, 32]. LSC does not require a nontrivial spatial quotient: the core identification is BHBB on at the induced data level (Section 2). Accordingly, matched circles are treated as a model-selecting diagnostic within the class of closed models, not as a required prediction of the core framework.

4.1.3 Matched-circle geometry (if a nontrivial quotient is assumed)

In multiply connected realizations, the last-scattering surface (at co-moving distance ) can intersect its topological images so that the same physical region of the primordial plasma is observed at multiple angular positions on the sky. This motivates the classic matched-circles signature: pairs of circles whose fluctuation patterns match up to an orientation reversal and a phase shift [31]. For a broad class of detectable topologies (including the back-to-back circle approximation used in many searches), the angular radius of matched circles is related to the curvature radius and the effective holonomy scale via [32].where denotes the relevant holonomy distance between identified copies of the observer (for some homogeneous quotients, one has , but the mapping between and the injectivity radius is quotient-dependent).

A standard statistic for quantifying matched circles is the correlationwhere and are temperature (or polarization) fluctuations along two circles of angular radius corresponding to the same physical locus on the last-scattering surface.

4.1.4 Current status and conservative interpretation for LSC

4.2 Local expansion and the Hubble parameter in LSC

4.2.1 Status (Tier II consistency, not a Tier I gate)

In strictly homogeneous FLRW cosmology, the expansion is characterized by a single and . In LSC, the spacetime is globally closed but locally inhomogeneous, with BHBB tubes and a nontrivial induced data identification, so the operationally inferred “” is most cleanly treated as a probe-dependent lightcone reconstruction. This subsection provides a consistency framework for that interpretation; it is not used as a Tier I falsification gate, and it does not constitute a precision prediction for the Hubble tension.

4.2.2 Local expansion field

Let be the four-velocity of a coarse-grained cosmological fluid. The local expansion scalar,defines a local Hubble rateFor a small co-moving domain with proper volumeone may define an effective domain scale factor , for whichIt is convenient to decompose the inhomogeneous expansion into a nearly FLRW background plus a local distortion,which yieldswhere is the coarse-grained loop-averaged background rate.

4.2.3 BHBB volume map (bookkeeping representation)

Denote by the induced data mapping associated with BHBB identification. A Jacobianquantifies how the late-time co-moving volume is re-encoded into a BB patch at the induced data level. At the level of bookkeeping for initial distortions, one may writewith normalization fixed by on each slice. A quantitative evaluation requires a statistical model of the BH population and a controlled solution for in an inhomogeneous cosmology; those elements are not developed here.

4.2.4 Observed as a lightcone average

Distance–redshift measurements reconstruct an effective Hubble parameter from data along the past lightcone . Introducing a normalized window function encoding the selection of a given probe,one findsEarly-Universe probes that average over very large volumes (CMB, high- BAO) preferentially constrain , while late-time or environment-sensitive probes can inherit a small bias .

4.2.5 Loop-closure constraint (global bookkeeping, not an observable period)

Because LSC posits a single closed spacetime, the coarse-grained volume satisfies a global closure condition in the loop-label coordinate. Writing this schematically as for a loop-label interval (not an observer proper time period), one obtainsWith , this yields the global constraint(up to a backreaction from higher moments of ), so is not freely specifiable in a fully self-consistent realization.

4.3 Curvature and topological dark energy (Tier II)

Because LSC assumes a compact spatial topology, the present-day Universe is slightly spatially closed and satisfies Equation 4.1. Joint CMB + BAO + SNe analyses, therefore, serve as a Tier I sign test and a Tier II correlated-parameter constraint once the sign is fixed. Within LSC, the same compact setting provides a natural arena for infrared/finite-size vacuum mode regularization; in an effective description, this motivates an infrared stability mechanism for a small renormalized vacuum energy once such a small value is assumed at the matching scale (Section 3.3). This does not solve the ultraviolet cosmological constant problem; it provides a controlled way to track how global radii enter the infrared sector.

Late-time acceleration: inference vs. dynamics (cross-channel consistency).

Distance–redshift data are often summarized using the deceleration parameterIn an inhomogeneous geometry with curvature/topology constraints, distance-based reconstructions of are lightcone integrals and can be biased relative to growth-based inferences if curvature and mild backreaction are not modeled consistently. Accordingly, LSC motivates a Tier II consistency discriminator: distance probes (BAO + SNe with curvature control) should be mutually consistent with growth probes, often summarized by a growth index via , once curvature/topology is treated consistently in the inference.

4.3.1 Early dark energy from the bounce remnant (optional Tier II)

In realizations in which the polymer-regulated continuation leaves a transient residual component, one may obtain an effective early dark energy (EDE)-like contribution: subdominant at very high and very low redshift, but potentially non-negligible around matter–radiation equality. A convenient phenomenological parametrization of its fractional contribution iswhere control the peak amplitude, peak redshift, and width, respectively. As an illustrative benchmark for joint CMB + LSS tests, one may considerwith chosen so that the contribution rapidly decays away from equality and with the understanding that future data may force [34, 35].

4.3.2 Optional by-product: primordial black holes from the bounce

LSC permits a subdominant primordial black hole (PBH) population as a secondary by-product of bounce-era physics in certain realizations. Enhanced small-scale curvature perturbations near the high-density transition could produce overdense patches that collapse into PBHs shortly after the bounce. A dimensional estimate yieldswhere the numerical value is indicative and carries order-of-magnitude uncertainty from the collapse threshold and the detailed perturbation spectrum. The associated abundance is parameterized asso that future closure of the relevant mass windows constrains this realization without falsifying Tier I.

4.4 Optional effective image-mass contribution (phenomenological Tier III)

The core LSC claim is the global induced data identification of black hole interiors with a common Big Bang hypersurface. Separately, one may ask whether that compact identification could, in some realizations, induce a small additional contribution to the effective weak-field sourcing inferred by embedded observers. No first-principles microphysical derivation is assumed here; instead, the effect is parameterized in a minimal, falsifiable way and treated as an optional Tier III add-on that can be constrained (or excluded) by data without affecting the topological identification itself.

4.4.1 Phenomenological parametrization

In the scalar, weak-field regime, write the potential in co-moving coordinates aswith an augmented sourceHere, denotes the clustered matter component(s), is a dimensionless amplitude (expected if present), and is a normalized kernel encoding an effective nonlocality associated with the global identification,The scale is a phenomenological co-moving length at which the modification turns on.

A convenient one-parameter choice is a Yukawa-type kernelwhich yields the Fourier-space relationThe signature is a scale-dependent enhancement of the effective lensing/clustering source that turns on for and saturates to on sufficiently large scales.

4.4.2 Observational discriminator: cross-channel consistency

Weak-lensing convergence directly constrains the augmented source. In the Born approximation,so the image-mass term modifies lensing through in Equation 4.25. The empirical requirement is cross-channel consistency: any nonzero preferred by cluster lensing must also be consistent with galaxy–galaxy lensing and large-scale structure clustering constraints. If joint analyses enforce (or exclude any common parameter pair), this optional add-on is ruled out while leaving Tier I intact.

4.5 Dark matter and dark energy (realization-dependent, Tier II)

In standard CDM, roughly a quarter of the present energy density is in non-baryonic dark matter, and approximately two-thirds in a nearly constant dark energy component. In LSC, correlated phenomenology can arise from (i) particle relics and PBHs produced near the transition region in certain realizations and (ii) the compact setting, which fixes the sign of curvature and provides an infrared stability mechanism for a small, renormalized vacuum energy. This subsection states minimal, explicitly optional parameterizations sufficient for the prediction matrix and for comparison to data.

4.5.1 Sterile neutrino and PBH components (illustrative benchmark)

As an organizational benchmark for how a bounce-era relic could interface with late-time phenomenology, include a single sterile neutrino species characterized byand contributingIn decay-through-mixing scenarios, this benchmark corresponds to a line energyreferenced here only to define a representative keV-scale parameter point. There is no confirmed detection, and strong null results have been reported [36, 37]. Joint X-ray and structure formation constraints, therefore, treat this benchmark as a constrained (often excluded) illustrative realization rather than evidentiary support.

4.6 Gravitational wave echoes (Tier III microphysics diagnostic)

Gravitational wave echoes are realization-dependent and model-sensitive: they can arise only if the LSC transition-region microphysics induces a partially reflective effective response. All Tier I global structure tests (curvature sign/topology consistency and the high-spin population gate of the minimal effective continuation) can be assessed independently of any echo phenomenology. In particular, the absence of detected echoes does not constrain the core framework, which does not require horizon-scale departures from semiclassical GR. If echoes were detected, they would constrain transition-region microphysics (e.g., ) rather than the global identification itself [8, 38, 39].

4.6.1 Baseline GR behavior: quasi-normal modes and late-time tails

Echo searches must be interpreted against standard GR expectations: ringdown quasi-normal modes followed by late-time tails sourced by backscattering off the exterior potential. Putative echoes must be separated from imperfect quasi-normal mode (QNM) modeling, power-law tails, and instrumental systematics; within LSC, echoes occur only if the transition region contributes an additional effective reflectivity beyond exterior barrier physics [40–42].

4.6.2 Physical picture

After the merger, a fraction of the perturbation propagates inward. If the bounded-curvature transition region induces a nonzero, frequency-dependent reflectivity , the inward component can be partially reflected and re-scatter off the exterior barrier, producing a sequence of late-time attenuated copies of the ringdown [38]. In LSC, the existence, magnitude, and spectral dependence of are not fixed by topology and encode realization-specific microphysics.

4.6.3 Characteristic timescale

To leading order, the echo delay time for a remnant of mass and dimensionless spin can be written in a near-horizon approximation aswhere is the outer horizon radius, denotes an effective radius at which the transition region significantly modifies propagation, and encodes the location of the exterior potential barrier in tortoise coordinates. The logarithmic dependence reflects near-horizon redshift; physically, corresponds to an effective round-trip time between the exterior barrier and the transition region [38, 39]. Figure 5 shows the time- and frequency-domain signatures alongside an optional echo train.

FIGURE 5

4.6.4 Phenomenological interpretation

Because a first-principles computation of from a fully inhomogeneous rotating quantum-gravity interior is not available in this effective treatment, is treated as a phenomenological parameter (potentially frequency-dependent) summarizing transition-region physics. Echo amplitudes scale approximately as a geometric series in after propagation through the exterior barrier, so upper limits on can strongly suppress detectability. This diagnostic role is logically downstream of Tier I: null results constrain for specific realizations without challenging BHBB identification. Figure 6 shows the associated mass scaling of the echo delay.

FIGURE 6

4.6.5 Detection prospects and multiband leverage

Space-based GW observatories (mHz band) observe long inspirals of massive black hole binaries. They can improve pre-merger parameter inference, thereby sharpening post-merger/ringdown residual tests when combined with ground-based detections in a multiband context [43, 44].

4.6.6 Observational status

Searches in Advanced LIGO–Virgo–KAGRA data have examined post-merger echo signatures using template-based and morphology-independent methods [13, 15, 45]. Reported candidates have not met robust detection thresholds, with statistics consistent with noise and systematics in current catalogs [45]. Within LSC, these non-detections translate into upper limits on for specific realizations.

4.7 Master test matrix

Table 1 summarizes the empirical gates and discriminators used throughout the article. The matrix is structured to keep LSC decisively vulnerable at the framework level (Tier I) while cleanly separating realization-dependent phenomenology (Tier II) and implementation-dependent microphysics/optional add-ons (Tier III).

7.1 How to read the matrix

Crit. (H/M) indicates observational priority. Tier encodes logical role. Tier I rows are framework-level rule-out gates: failure falsifies the minimal LSC framework outright, independent of any dark-sector or near-horizon modeling. Tier II rows are correlated-realization diagnostics that can rule out specific implementations (especially those that adopt optional benchmarks) without negating the global BHBB identification. Tier III rows are implementation-dependent probes or optional add-ons that constrain transition-region microphysics (e.g., ) or additional components. Null results in Tier II/III are interpreted as parameter constraints on those realizations, not as a direct falsification of the core BHBB identification. In this sense, the present manuscript defines framework-level falsification gates and correlated observational discriminants rather than a fully parameterized precision-fit cosmological model.

5 Comparison with alternative theories

5.1 Loop quantum cosmology

Loop quantum cosmology (LQC) provides the primary technical inspiration for the bounded-curvature continuation used in LSC. Both frameworks employ polymer (holonomy-based) quantization ideas in symmetry-reduced gravitational settings, leading in the effective description to modified Friedmann dynamics with a universal critical density in standard treatments (cf.Equation 2.7) [1, 4]. In homogeneous cosmological settings, LQC replaces the classical Big Bang singularity with a quantum bounce at , and related constructions can be extended to certain anisotropic (Bianchi) models [2, 46, 47]. In this sense, the effective minisuperspace regularization adopted in LSC is closely aligned with standard LQC at the level of local high-curvature dynamics.

The crucial differences are conceptual and topological. In most LQC applications, the bounce is treated as a local feature of a single FLRW (or Bianchi) universe, and black holes are either not modeled explicitly or are treated in separate symmetry-reduced toy models. LSC instead adopts the same bounded-curvature, polymer-regulated continuation as a local mechanism that prevents terminal boundaries and then adds an independent global postulate: all black hole interior continuations are topologically identified with a single Big Bang hypersurface within one closed spacetime. Once singularities are replaced by consistent continuations rather than endpoints, this global identification is introduced to avoid proliferating disconnected “origins” and formulate a unified accounting of information flow on the closed manifold.

These similarities and differences are summarized in Table 2. Both LSC and LQC share polymer-inspired bounded-curvature effective dynamics and a universal critical density in standard effective treatments, but they diverge in how black holes are incorporated, whether a global BHBB identification is postulated, and how topology enters. Correspondingly, LSC emphasizes global structure tests (near-closure curvature and topology consistency) and population-level gates (high-spin/strong-anisotropy viability), while treating near-horizon gravitational-wave phenomenology (e.g., echoes) as an optional, model-dependent Tier III diagnostic of transition-region microphysics rather than a defining prediction.

5.2.1 String theory and Anti-de Sitter/conformal field theory (AdS/CFT)

String theory offers a candidate ultraviolet completion of gravity in which classical singularities can be softened by extended objects (strings, branes) and duality symmetries. It has provided microscopic accounts of Bekenstein–Hawking entropy for certain supersymmetric and near-extremal black holes [48]. Via the AdS/CFT correspondence, it realizes black hole formation and evaporation as manifestly unitary dynamics in a dual conformal field theory [49, 50]. In asymptotically AdS spacetimes, this gives a compelling resolution of the information paradox [51]. However, these results rely on special charge configurations, supersymmetry, and AdS boundary conditions; a unique, observationally sharp cosmological model for our approximately de Sitter universe has not yet emerged from string theory or AdS/CFT [52].

LSC should, therefore, be viewed as complementary rather than competing. It does not attempt a full ultraviolet (UV) completion or derive its bounded-curvature continuation from a specific compactification. Instead, it works at the level of a four-dimensional effective description, assuming a compact topology, an LQC-style bounded-curvature continuation in high-density regions, and the global BHBB identification described in Section 5.1. Within this minimalist setup, LSC offers a clear global picture of information preservation and turns its structural assumptions into a small set of linked observational gates: near-closure with sign-fixed negative curvature, compact topology consistency tests in the CMB, and population-level constraints on the high-spin/strong-anisotropy regime. Optional near-horizon gravitational wave phenomenology, if present, is treated as a Tier III diagnostic of transition-region microphysics rather than a defining pillar of the framework. Such an effective description could, in principle, arise as the low-energy limit of some string construction, but the phenomenology can be tested directly without committing to a specific UV embedding.

Compared to string theory, LSC works directly in four dimensions without requiring extra dimensions or supersymmetry. While string theory aims for a UV-complete framework, LSC prioritizes near-term falsifiability through a small set of structurally linked gates.

UV-embedding outlook (non-committal)

Although LSC is formulated as a four-dimensional effective framework, it is natural to ask whether its two structural ingredients, (i) bounded-curvature continuation in high-density regions and (ii) a global BHBB identification on a compact manifold, could emerge from a more fundamental ultraviolet completion. At present, we do not propose (and do not require) such a derivation. Nevertheless, one can identify concrete routes that would qualify as a successful embedding: (1) a mechanism that replaces would-be singular endpoints by a controlled high-curvature phase with a well-posed perturbation problem (so that the effective continuation used here is recovered in an appropriate limit) and (2) a globally consistent identification (quotient) structure that realizes the BHBB map without introducing additional dynamical degrees of freedom or pathological causality violations. String-theoretic constructions and holographic models provide examples of unitary black hole evolution in special settings (notably asymptotically AdS), but connecting those settings to an observationally sharp, approximately de Sitter cosmology while accounting for the present topological constraints remains an open problem. Accordingly, we treat UV completion as an important direction for future work rather than an input assumption of the present phenomenological program.

5.3 Conformal cyclic cosmology

Conformal cyclic cosmology (CCC), proposed by Penrose, envisions the Universe as an infinite sequence of “aeons.” Each aeon begins with a Big Bang and ends in an exponentially expanding, asymptotically dilute future in which, in the CCC hypothesis, effectively only massless degrees of freedom remain relevant [53]. A conformal rescaling is then used to identify the future null infinity of one aeon with the Big Bang of the next, producing a cyclic picture without a conventional bounce in proper time.

LSC differs from CCC both structurally and in observational commitments. Structurally, LSC posits a single closed spacetime with topology , not an infinite chain of distinct aeons. The BHBB identification is an internal identification within one global manifold, and the high-curvature transition is modeled (in this article) by an LQC-inspired bounded-curvature continuation, rather than a conformal matching of asymptotic boundaries. Phenomenologically, LSC does not adopt any CCC-specific CMB “feature” claims. Dedicated searches for CCC-motivated signatures (e.g., low-variance circle and Hawking-point statistics) have not found statistically robust evidence once look-elsewhere effects and map-level systematics are controlled, so these do not provide supporting evidence for LSC [54]. The leading observational priorities in LSC are instead the near-term falsification gates emphasized throughout this work: sign-fixed near-closure curvature, consistency of topology bounds with the implied compact scale, and population-level constraints on the high-spin/strong-anisotropy sector.

6 Discussion

6.1 Interpretive scope and potential

6.2 Open challenges

Several theoretical issues require further investigation before LSC can be regarded as a fully robust alternative to CDM or other quantum-gravity inspired cosmologies. We highlight three of the most important ones: (i) the viability of bounded-curvature continuation in highly rotating/anisotropic interiors, (ii) the global consistency of the BHBB identification when many black holes form, and (iii) the quantum-to-classical transition across the bounded-curvature regime.

A further class of potential objections concerns “baby-universe” or holography-motivated constraints, which are typically formulated in asymptotically AdS settings or within specific AdS/CFT realizations of black hole unitarity [51, 56]. The present LSC framework is not posed in asymptotically AdS spacetimes, and its global identification is stated at the level of an effective Lorentzian manifold and induced data on , rather than via an AdS boundary dual. For that reason, such arguments are not automatically decisive for LSC, as written here. However, they do motivate an important open question: whether an explicit UV completion of LSC can be made compatible with relevant holographic consistency conditions in settings in which a boundary description exists.

6.2.1 High-spin/strong-anisotropy viability

The bounded-curvature continuation adopted in LSC is best controlled in regimes that are approximately homogeneous on the relevant interior scales, where effective LQC-inspired dynamics provides a concrete and well-studied template. For moderately spinning remnants, , it is plausible that parts of the interior evolution can be approximated by a symmetry-reduced effective description; however, quantitative control of anisotropies and the continuation in genuinely Kerr-like interiors remains an open problem. The most severe stress test is the strongly rotating, strongly anisotropic sector: for , the interior geometry is expected to develop large shear and strong BKL-like behavior [11], and it is not yet established that the LQC-inspired effective continuation remains valid or stable in that regime.

A convenient diagnostic is the shear scalar of an interior congruence. In generic anisotropic effective cosmologies, one may write schematicallywhere is an effective interior expansion rate and encodes anisotropy sourcing from curvature terms, rotation-induced distortions, and matter/field gradients. In LQC-inspired effective descriptions, holonomy corrections typically bound the energy density and can also bound certain combinations of anisotropy variables in Bianchi models [46, 47]; however, the quantitative behavior in the near-extremal Kerr interior is not yet derived from first principles. The role of this gate in LSC is, therefore, clear: the high-spin/strong-anisotropy regime is the most important target for future numerical validation of the transition-region continuation.

For phenomenology, we, therefore, adopt an indicative stability window (Section 2.5) in which the minimal realization is expected to fail if a model-robust population of long-lived Kerr black holes with is established. This is not claimed as a theorem but as a crisp empirical exposure point: either future theoretical work extends the continuation to genuinely near-extremal interiors, or such a population would falsify the minimal LSC realization as it is presently formulated.

6.3 Observational status

At present, LSC is broadly consistent with existing observational data, but several of its core falsification gates lie close to current sensitivity limits. This is precisely the regime in which the framework is most useful as an idea-level proposal: modest improvements in a small set of measurements can either further constrain the allowed parameter space or decisively rule out the minimal realization. We summarize the most relevant channels below, using the master test matrix (Table 1) as the organizing structure.

6.3.1 Black hole spins as an empirical gate

A central gate for the minimal realization is the absence of a model-robust population of long-lived Kerr black holes with . Current constraints from both gravitational wave inference and electromagnetic modeling are compatible with this requirement, but they do not yet exclude rare, near-extremal candidates.

6.3.2 Spatial curvature and topology

Current CMB + BAO analyses constrain the curvature parameter to be small, with bounds at the level – depending on dataset combinations and modeling assumptions [35, 57, 58]. The fiducial LSC expectation with – remains compatible with these limits and defines a sign-fixed falsification target.

6.3.3 Optional transition-region diagnostics (Tier III)

Dedicated searches for post-merger ringdown anomalies, including possible echo trains, have not yielded a statistically robust detection [8, 59, 60]. Within the LSC framing adopted here, such effects are treated as optional diagnostics rather than required consequences of the BHBB identification.

7 Future tests and timeline

Falsification hierarchy (single sentence). Tier I gates test the global identification and its hard consistency conditions; Tier II/III tests constrain realization-dependent phenomenology conditional on Tier I survival (Table 1).

Near-term posture. The most decisive near-term exposures are (i) the sign-fixed curvature/topology consistency gate and (ii) the population-level high-spin/anisotropy stress test. Correlated dark-sector benchmarks are treated as Tier II diagnostics, while near-horizon/ringdown anomalies remain Tier III probes of transition-region microphysics (e.g., constraints on ) rather than requirements of the core BHBB identification.

7.1 Near-term (2025–2030)

7.2 Medium-term (2030–2035)

7.3 Long-term (mid-2030s and beyond)

8 Conclusion

LSC is a falsifiable, effective framework in which black hole interiors are topologically identified with a unique Big Bang hypersurface on a single closed spacetime. In this picture, what appears on local three-dimensional slices as the “end” of gravitational collapse and the “beginning” of cosmological expansion are complementary views of one four-dimensional, globally identified geometry. The construction does not posit a sequence of independent cycles; instead, the Universe is described as one closed spacetime, with the familiar forward arrow of time attributed to the perspective of embedded observers on local foliations rather than to periodic proper time or traversable closed timelike curves.

The status of LSC is, therefore, an empirical question. The near-term decisive exposures are the Tier I gates: population-level constraints on the high-spin tail (and associated strong-anisotropy viability), and sign-fixed tests of near-closure spatial curvature together with topology consistency bounds [12, 14, 33, 35, 57, 63]. Tier II channels provide correlated consistency tests within Tier I-surviving realizations, and Tier III channels constrain optional transition-region microphysics (including any echo-like late-time features) without being required by the core BHBB identification [8, 34, 37–39, 64, 65].

This work does not present a final theory of quantum gravity or a derived solution to the cosmological constant problem. It instead provides an explicit, logically tiered example of how a global topological identification and an effective bounded-curvature continuation can be combined into a coherent and testable spacetime framework. If future theoretical work extends the high-spin/strong-anisotropy continuation, and future observations favor the Tier I sign-fixed curvature/topology gate, the result would motivate a picture in which black hole interiors and the Big Bang are linked within one closed four-dimensional geometry, and in which the apparent arrow of time reflects the perspective of embedded observers within that structure. If the Tier I gates fail, the minimal LSC realization is falsified. Either outcome is scientifically useful: it forces sharp contact between global structure hypotheses and a small number of prioritized observational tests.

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