Horizon Threshold Project

The Horizon Threshold Project is a research programme devoted to the geometric, perturbative, and formal analysis of curvature-defined transition regimes in black-hole interiors.

The project investigates where the classical spacetime description ceases to provide a controlled physical model in regions of extreme curvature, and how such a transition may be formulated invariantly rather than by introducing an arbitrary microscopic cutoff. Its central construction is a curvature-threshold prescription in which the transition scale is determined from the exterior black-hole geometry itself.

In the spherically symmetric case, the transition radius is defined by imposing the invariant condition

KSch(rc) = η KP,

where KSch is the Schwarzschild Kretschmann scalar, KP = ℓP⁻⁴ is the reference Planck-curvature scale, and η > 0 is a dimensionless threshold parameter. This yields a mass-dependent transition scale,

rc³ = √(12/η) rs ℓP²,

and therefore rc ∝ M¹ᐟ³. The transition scale is thus fixed by the exterior curvature and by the asymptotic mass, rather than being introduced as an independent regularization length.

The project does not aim merely to construct a single regular black-hole metric. Its purpose is to develop a systematic framework for distinguishing:

which properties of regular black-hole interiors follow from the curvature-threshold prescription itself;
which properties depend on the chosen regularizing profile;
whether bounded curvature is sufficient for geodesic completeness and maximal extension;
how inner horizons behave under perturbative diagnostics;
how the threshold construction must be reformulated in rotating geometry;
which further dynamical questions require an effective source, coupled perturbations, and backreaction.

The programme is developed through a sequence of analytic manuscripts, computational packages, reproducibility records, and selected formal-verification components.

Scientific Motivation

Classical black-hole solutions contain regions in which curvature invariants indicate a breakdown of the classical geometric description. In the Schwarzschild solution this breakdown appears as a central curvature singularity. In rotating Kerr geometry, the high-curvature obstruction is associated with the ring-singularity structure and with an angular dependence absent in spherical symmetry.

Regular black-hole models replace the singular region by an effective finite-curvature interior. However, regularity alone does not determine the transition scale, the global causal structure, the stability of inner horizons, or the dynamical behaviour of the effective source. A finite curvature scalar is therefore only the first layer of the problem.

The Horizon Threshold Project separates these layers explicitly. The programme begins with curvature boundedness, then examines geodesic extendibility, perturbative viability, rotating threshold surfaces, effective stress-energy structure, backreaction, and possible external signatures.

Research Programme

The project is organized into two broad stages.

Stage I — Curvature-threshold geometry and static diagnostic structure

Stage I develops the geometric and diagnostic foundation of the programme. It establishes the curvature-threshold prescription in spherical symmetry, separates threshold-driven from profile-dependent consequences, analyzes geodesic completeness and maximal extensions, introduces perturbative diagnostics of regular interiors, and extends the threshold concept to rotating Kerr geometry.

This stage currently includes five completed preprints and further planned technical studies.

Stage II — Effective source, coupled dynamics, backreaction, and observable consequences

Stage II will address the dynamical completion of the programme. Its aim is to move beyond static effective geometries by reconstructing the effective stress-energy tensor, studying the response of the source to perturbations, analyzing backreaction at the inner horizon, and examining whether curvature-threshold regular black holes can give rise to observable external signatures.

Verification and Reproducibility

The project uses a layered reproducibility strategy. Where applicable, analytic results are accompanied by computational packages, numerical diagnostics, symbolic checks, CSV data, figures, and instructions for reproduction. Selected results are also accompanied by Lean formal-verification packages.

Current verification status:

Study 1: analytic preprint; full Lean verification; computational and reproducibility package available.
Study 2: analytic preprint; full Lean verification; computational and reproducibility package available.
Study 3: analytic preprint; purely geometric and analytic; no separate computational package required.
Study 4: analytic preprint; computational and reproducibility package available; Lean verification planned.
Study 5: analytic preprint; computational and reproducibility package available; Lean verification planned.

The long-term objective is to convert the manuscript series into a stabilized, independently reviewable, reproducible, and partially mechanically verified research programme.

Research Outputs

Each study developed within the Horizon Threshold Project is presented in a common format:

explanatory video or public summary, where available.

scientific question;
principal result;
publication or preprint record;
verification and reproducibility status;
supplementary material, where applicable;

Study 1

Planck-Curvature Thresholds and the Limits of Geometric Description in Regular Black Hole Interiors

Status: Completed preprint
DOI: 10.5281/ZENODO.20681533
Verification status: Lean verified
Supplementary status: Computational and reproducibility package available

This study introduces the curvature-threshold prescription in the spherically symmetric setting. The transition radius rc is determined directly from the invariant exterior Schwarzschild curvature condition KSch(rc) = η KP, rather than being introduced as an independent microscopic parameter.

The prescription is applied to a known exponential regular black-hole metric. The resulting geometry has a mass-dependent transition scale rc ∝ M¹ᐟ³, a finite de Sitter-like central region, and a global curvature bound independent of the asymptotic mass. The analysis proves that the Kretschmann scalar is globally bounded by

K(r) ≤ K(0) = 2η KP

with equality only at the regular center.

The study also derives the exact horizon classification: a two-horizon geometry above a critical mass, a degenerate horizon at the critical mass, and a regular horizonless geometry below it. It examines the relation between the transition radius and the inner horizon and determines the behaviour of the inner-horizon surface gravity across the supercritical branch.

The computer-assisted component verifies the global monotonicity of the dimensionless curvature profile and supports the exact curvature bound established in the paper.

Publication:
https://doi.org/10.5281/zenodo.20681533

Research report:
A structured, source-based summary of the paper and its principal results.

English version

Polish version

Explanatory video:

Supplementary material:
Available within the Zenodo record.

Study 2

Common and Profile-Dependent Consequences of Planck-Curvature Scale Setting in Exponential and Hayward-Type Regular Black Holes

Status: Completed preprint
DOI: 10.5281/ZENODO.20767326
Verification status: Lean verified
Supplementary status: Computational and reproducibility package available

This study determines which results follow from the curvature-derived scale relation itself and which remain dependent on the selected regularizing profile.

The work compares two regular black-hole interiors constructed with the same exterior mass, the same threshold parameter, and the same transition radius fixed by KSch(rc) = η KP: an exponential profile and a Hayward-type profile. Both profiles have the same leading central behavior and the same Schwarzschild limit, but differ in their radial interpolation and asymptotic corrections.

The analysis shows that several properties are threshold-driven: the scaling rc ∝ M¹ᐟ³, the mass-independent central curvature normalization, the common critical-mass exponent Mmin ∝ η⁻¹ᐟ⁴ MP, and the existence of a global curvature bound K(r) ≤ K(0) = 2η KP in the representative models considered.

Other features are profile-dependent, including the numerical value of the critical mass, detailed horizon locations, energy-condition transition surfaces, large-radius corrections, and some surface-gravity properties.

Publication:
https://doi.org/10.5281/zenodo.20767326

Explanatory video:
[to be added]

Supplementary material:
Available within the Zenodo record.

Study 3

Geodesic Completeness and Maximal Extensions of Curvature-Bounded Regular Black Hole Interiors

Status: Completed preprint
DOI: 10.5281/ZENODO.21038157
Verification status: Lean verification planned
Supplementary status: No separate computational package required; the work is purely geometric and analytic.

This study addresses whether bounded curvature is sufficient to establish full spacetime regularity.

The analysis separates three distinct notions: curvature regularity, geodesic extendibility, and global maximal extension. It shows that a finite curvature scalar removes the local Schwarzschild-type curvature obstruction, but does not automatically determine whether all geodesics can be extended or whether the spacetime has a unique maximal extension.

The study examines radial null geodesics, radial timelike geodesics, non-radial geodesics, the angular-momentum barrier near the regular center, horizon crossing as a coordinate-extension problem, and the status of the regular center as an extension surface.

The principal conclusion is that curvature boundedness is necessary but not sufficient for full spacetime regularity. Geodesic completeness and maximal extension constitute an independent layer of the regular-black-hole problem.

Publication:
https://doi.org/10.5281/zenodo.21038157

Explanatory video:
[to be added]

Supplementary material:
Not applicable as a separate computational package.

Study 4

Perturbative Stability of Curvature-Bounded Regular Black Hole Interiors with Planck-Threshold Scale Setting

Status: Completed preprint
DOI: 10.5281/ZENODO.21038220
Verification status: Lean verification planned
Supplementary status: Computational and reproducibility package available

This study examines whether curvature-bounded and geodesically extendible regular interiors remain dynamically viable under a controlled linear perturbative diagnostic.

The work focuses on the axial perturbation sector as a first metric-sector test. It introduces a Regge–Wheeler-type diagnostic equation on fixed effective regular-black-hole backgrounds and analyzes the behavior of the regular core, the outer horizon, the inner horizon, and the near-degenerate regime.

The study emphasizes that the diagnostic axial potential is not presented as a complete source-coupled perturbation theory. Instead, it is used to isolate the metric-sector behavior of the representative regular geometries.

The principal conclusion is that regularity of the core does not remove the independent problem of the inner horizon. The inner horizon remains dynamically delicate because perturbative modes may undergo blueshift amplification controlled by the corresponding surface gravity. This identifies inner-horizon dynamics as a separate obstruction beyond curvature boundedness and geodesic extendibility.

Publication:
https://doi.org/10.5281/zenodo.21038220

Explanatory video:
[to be added]

Supplementary material:
Available within the Zenodo record.

Study 5

Curvature-Threshold Surfaces in Rotating Black Hole Geometry

Status: Completed preprint
DOI: 10.5281/ZENODO.21038253
Verification status: Lean verification planned
Supplementary status: Computational and reproducibility package available

This study extends the curvature-threshold programme beyond spherical symmetry.

In the Schwarzschild case, the threshold condition selects a single transition radius. In Kerr geometry, curvature depends on both the Boyer–Lindquist radial coordinate and the polar angle. The threshold condition therefore defines a family of curvature-threshold surfaces rather than a single radius.

The work introduces an invariant curvature diagnostic for Kerr geometry using the scalar and pseudoscalar curvature invariants, equivalently the modulus of the Petrov type D Weyl curvature structure. The resulting threshold set is written as
Ση(M, a) = {(r, θ): Keff(r, θ; M, a) = η KP}

In the non-rotating limit, this construction reduces to the spherical Schwarzschild threshold. For nonzero spin, the threshold becomes an axially deformed surface depending on mass, angular momentum, and polar angle.

The study does not propose a complete regular Kerr metric. Its purpose is diagnostic: to identify the invariant threshold geometry that any threshold-adapted regular rotating black-hole model would have to respect.

Publication:
https://doi.org/10.5281/zenodo.21038253

Explanatory video:
[to be added]

Supplementary material:
Available within the Zenodo record.

Planned Research Sequence

The following studies are planned as part of the continuation of the Horizon Threshold Project. Their titles and scopes may be refined as the programme develops.

Study 6

A Threshold-Adapted Regular Rotating Black Hole Metric

Status: Planned

This study will develop a controlled ansatz for a regular rotating black-hole geometry adapted to the Kerr curvature-threshold surface. The goal is to replace the naive radial regularization strategy by a construction sensitive to the Kerr structure Σ = r² + a² cos²θ, the angular deformation of the threshold surface, and the ring-singularity geometry.

The study will examine the non-rotating limit, the exterior Kerr limit, curvature invariants, horizon structure, and the conditions under which the proposed metric remains a threshold-adapted effective model rather than a purely formal deformation.

Study 7

Matching Conditions and Transition Layers for Curvature-Threshold Regular Black Hole Interiors

Status: Planned

This study will analyze the transition between the classical exterior geometry and the effective regular interior. The central question is whether the transition can be represented by a smooth profile alone or whether a distinct effective transition layer must be introduced.

The work will examine continuity of the metric, differentiability conditions, curvature behavior across the transition region, possible effective source layers, and the relation between spherical and rotating threshold geometries.

Study 8

Effective Stress-Energy Tensor for Threshold-Adapted Rotating Black Hole Geometry

Status: Planned

This study will reconstruct the effective stress-energy tensor associated with the threshold-adapted rotating geometry. The analysis will examine the resulting density, pressures, fluxes, anisotropy, behavior near the axis and equatorial region, behavior near horizons, and the dependence of the effective source on the angular structure of the rotating threshold surface.

Study 9

Energy Conditions, Limiting Curvature, and Effective Matter in Threshold-Regular Black Hole Models

Status: Planned

This study will synthesize the energy-condition structure across the threshold-regular black-hole programme. It will compare spherical and rotating models, distinguish violations required by regularization from those introduced by specific profiles, and analyze the relation between limiting curvature, anisotropic effective matter, and the maintenance of a regular core.

Study 10

Linear Response of the Effective Source in Regular Black Hole Interiors

Status: Planned

This study will move from static reconstruction of the effective source to its linear response under perturbations. It will examine possible phenomenological response laws, anisotropic response matrices, local stability conditions, causality constraints, and the coupling between metric perturbations and perturbations of the effective stress-energy tensor Teff.

Study 11

Backreaction at the Inner Horizon of Curvature-Bounded Regular Black Holes

Status: Planned

This study will address when the fixed-background approximation breaks down near the inner horizon. It will examine the growth of perturbative energy, mass-inflation-type mechanisms, possible weak null singularities, the fate of the regular core after backreaction, and the limits of static regular-black-hole models.

Study 12

Polar and Coupled Perturbations of Regular Black Hole Interiors

Status: Planned

This study will extend the perturbative analysis beyond the axial diagnostic sector. It will examine polar perturbations, additional degrees of freedom, coupling to the effective source, boundary conditions at the core and horizons, and the differences between axial and polar stability.

Study 13

Global Causal Structure After Inner-Horizon Amplification

Status: Planned

This study will return to the problem of maximal extension after including the effects of inner-horizon amplification and backreaction. It will examine whether geodesic extendibility survives dynamical testing, whether the inner horizon becomes a weak null singularity, and how the global causal diagram is modified.

Study 14

External Signatures of Curvature-Threshold Regular Black Holes

Status: Planned

This study will investigate whether curvature-threshold regular black holes can be externally distinguished from classical black holes. The analysis may include ringdown behavior, quasi-normal modes, possible echoes, shadow structure, accretion signatures, gravitational-wave constraints, and the observational relevance of regularized interiors.

Study 15

Semiclassical Horizon Formalism and the Failure of Black Hole Disappearance Claims

Status: Planned final synthesis

This final study will examine the limits of the semiclassical horizon-emission formalism as a basis for claims of complete black-hole disappearance. The study will not treat the formal semiclassical calculation as equivalent to a complete dynamical theory of black-hole evolution.

The central question is whether a result obtained within quantum field theory on a restricted classical background can justify the stronger claim that a black hole disappears completely as a physical object. The analysis will focus on the role of backreaction, the breakdown of fixed-background assumptions, the high-curvature end regime, inner-horizon dynamics, and the distinction between formal emission and complete dynamical disappearance.

The guiding thesis will be:

A formally consistent calculation performed within a restricted semiclassical background does not, by itself, establish a complete dynamical disappearance of a black hole.

Programme Summary

The Horizon Threshold Project develops a layered approach to the black-hole interior problem.

The first layer establishes curvature-threshold scale setting and global curvature bounds.
The second layer separates universal threshold-driven consequences from profile-dependent effects.
The third layer examines geodesic completeness and maximal extension.
The fourth layer introduces perturbative diagnostics and the inner-horizon problem.
The fifth layer extends the threshold construction to rotating geometry.
The later layers address regular rotating metrics, effective stress-energy tensors, energy conditions, coupled perturbations, backreaction, external signatures, and the limits of semiclassical disappearance claims.

The project is therefore not a single-metric construction. It is a structured programme for determining where classical geometry loses applicability, how a regular effective description may be introduced, and what further dynamical conditions must be satisfied before a regular black-hole model can be regarded as physically viable.