The Schwarzschild solution

with $f_{\rm Schw}(r)=1-2M/r$, exhibits a curvature singularity at $r=0$, signalled by the divergence of all curvature scalars, $K_{\rm Schw}(r)=R_{abcd}R^{abcd}=48M^{2}/r^{6}$. The Penrose (1965) and Hawking–Penrose (1970) theorems show that this singularity is generic under classical Einstein gravity supplemented with the null and strong energy conditions: every Killing horizon enclosing a trapped surface admits at least one incomplete causal geodesic.

A wide programme has been devoted to the question of whether, and how, this singularity should be regularized once gravity is treated either as an effective theory (Hayward, Bardeen, Dymnikova [1, 2, 3]) or as a sector of a more fundamental, ultraviolet-complete theory (Modesto, Koshelev–Marto–Mazumdar, Buoninfante et al.[4, 5, 6]). The pattern emerging from these efforts is that linearized analyses on Schwarzschild backgrounds typically deliver a smooth conformally-flat soliton that lacks a horizon, while genuine horizon-preserving regular solutions of the full nonlinear vacuum equations remain elusive without the ad hoc insertion of an exotic matter sector.

We work within Spectral Causal Theory (SCT, the strict gravitational sector of the parent Causal Architecture Theory (CAT) [19]), a specific instance of the broader class of nonlocal quadratic gravity (NQG) theories. The one-loop effective action contains the spin-2 form factor $F_{1}(z)=\alpha_{C}^{-1}h_{C}(z)=(\alpha_{C}\pi)^{-1}\int_{0}^{\infty}e^{-zu}\rho_{C}(u)\,\mathrm{d}u$ with $\alpha_{C}=13/120$, computed from the Standard Model + three right-handed-neutrino particle content via the spectral heat-kernel expansion. The resulting transverse–traceless propagator $\Pi_{TT}(z)$ is an entire function whose first positive real zero we label $z_{1}\approx 2.41483889$. We adopt the Anselmi fakeon prescription [20, 15] at this zero, which fixes a spin-2 fakeon-pole mass $m_{2,\,{\rm pole}}=\Lambda\sqrt{z_{1}}\approx 1.554\Lambda$. The full action is

where $S_{\rm scalar}$ contains the analogous scalar block $RF_{2}(\Box/\Lambda^{2})R$, which is inactive on the vacuum Bach-flat backgrounds we consider in this work (since $R=0$ on Schwarzschild and on the leading effective Hayward metric except in the deep core where $C^{2}=0$).

We impose a single restrictive postulate: the no-modulus principle, that no new continuous low-energy parameter beyond the ADM mass $M$ and the cutoff scale $\Lambda$ be introduced in the black-hole sector. This restricts the de Sitter core length to one of the finitely many length scales canonical to the framework ($1/\Lambda$, $1/m_{2,\,{\rm pole}}$, $1/m_{2,\rm loc}$, etc.), of which we identify $1/m_{2,\,{\rm pole}}$ as the natural choice on physical grounds developed in Sec. III.

The present work establishes that this framework, supplemented by the no-modulus postulate, admits a parameter-free horizon-preserving regular black hole whose lapse takes the Hayward form

where the core length $l_{\rm can}$ is fixed by the model’s spin-2 pole and the ADM mass alone, without any free integration constant. We prove that this geometry resolves the central singularity while preserving an event horizon at $r_{+}\approx 2M$ for every astrophysical mass; that the construction respects the null and dominant energy conditions globally, violating only the strong energy condition in a finite ball inside the inner Cauchy horizon; that the leading correction to the Schwarzschild lapse arises at $1/r^{4}$, leaving all post-Newtonian parameters at their GR values $\beta=\gamma=1$ exactly; and that the de Sitter core is conformally flat to all orders in $r/l_{\rm can}$, guaranteeing that the leading vacuum residue $\Theta^{(C)}$ vanishes at the centre.

The construction is positioned in three respects against the prior art. With respect to the Koshelev–Marto–Mazumdar negative theorem [5], which forbids the Schwarzschild $1/r$ branch as a vacuum solution of ghost-free infinite-derivative gravity, our construction provides a constructive replacement parameter-free in the model’s branch. With respect to Buoninfante–Koshelev–Lambiase–Marto–Mazumdar linearized soliton results [6], which produce a horizonless compact object lacking a Killing horizon, our construction preserves a horizon by nonlinear resummation tied to a specific spectral datum. With respect to the asymptotic-safety regular black holes of Bonanno–Malafarina–Panassiti [7] (dust collapse) and Harada–Chen–Mandal [8] (perfect-fluid collapse), which build the regular interior from a collapsing matter source matched to an exterior, our construction is fully vacuum: the de Sitter core is generated by the model’s effective stress tensor itself, not by added matter.

The only open question we leave behind concerns the dynamical stability of the inner Cauchy horizon under generic perturbations, an issue common to every two-horizon regular black hole and elucidated for general templates by Carballo-Rubio et al. [13]. We delineate the issue explicitly and discuss two scenarios, evaporation toward a stable extremal endpoint, and possible suppression of the mass-inflation mode by the fakeon prescription, neither of which we settle in the present paper.

The plan of the paper is as follows. Section II summarizes the canonical inputs of the model. Section III introduces the Hayward ansatz and derives the canonical core length $l_{\rm can}$. Section IV establishes finiteness of all curvature invariants and the vanishing of the vacuum residue of the model. Section V examines the energy conditions. Section VI maps the horizon structure. Section VII addresses geodesic completeness. Section VIII treats the post-Newtonian limit and external consistency. Section IX discusses inner-horizon stability and the attendant open question. Section XI compares the construction with the existing regular-black-hole literature. Section XII states explicitly what this paper does NOT show, in the spirit of honest framing. Section XIII discusses the physical interpretation and future research directions. Section XIV concludes.

The fragment of the model relevant for this work is the effective gravitational action of the spectral causal sector,

with $\alpha_{C}=13/120$, $\alpha_{R}(\xi)=2(\xi-1/6)^{2}$, and the form factors $F_{1},F_{2}$ entire functions of their argument fixed by the model’s one-loop spectral expansion. The transverse–traceless propagator denominator

is an entire function whose first positive real zero we denote $z_{1}$. Numerical verification at quadruple precision and mpmath dps=100 yields

This is the canonical fakeon-pole position. We do not reproduce its derivation here; the relevant background appears in the model linearized field-equation block [24], and the value (6) is fixed across model outputs.

We additionally adopt a single restrictive selection principle: the no-modulus condition, that no new continuous low-energy parameter beyond the ADM mass $M$ and the cutoff scale $\Lambda$ be introduced in the black-hole sector. Every numerical input must descend from canonical spectral data or from boundary conditions specified by $M$. This restricts admissible core-scale choices to a finite set of canonical lengths $\{1/\Lambda,1/m_{2,\,{\rm pole}},1/m_{2,\rm loc},1/m_{0}(\xi)\}$ generated by the propagator-pole structure of the model; we identify $1/m_{2,\,{\rm pole}}$ in Sec. III as the unique choice consistent with vanishing of the leading vacuum residue $\Theta^{(C)}$ at the centre.

The Einstein tensor for (7) reads

so that the static-observer energy density $\rho$, radial pressure $p_{r}$, and tangential pressure $p_{T}$ of the effective source are

The anisotropic-vacuum signature $p_{r}=-\rho$ is immediate.

For Hayward, $m^{\prime}(r)=3Ml^{3}r^{2}/(r^{3}+l^{3})^{2}$ and $m^{\prime\prime}(r)=6Ml^{3}r(l^{3}-2r^{3})/(r^{3}+l^{3})^{3}$, so the energy density and tangential pressure read in closed form

Horizons of (7) – (8) are the positive real roots of $f(r)=0$, equivalently of $r^{3}-2Mr^{2}+l^{3}=0$. The discriminant of this cubic is $\Delta=l^{3}(32M^{3}-27l^{3})$, so it has three real roots (of which two positive) for $l<l_{c}$ and one positive root for $l>l_{c}$, with

For astrophysical applications, $M\Lambda\gg 0.836$ always holds (since $\Lambda>2.565\,\text{meV}$ [25] and even one solar mass gives $M\Lambda>10^{30}$); the two-horizon regular black hole is therefore the generic vacuum object of the model.

The surface gravities at the two horizons are $\kappa_{\pm}=|f^{\prime}(r_{\pm})|/2$, with Hawking temperature $T_{H}^{+}=\kappa_{+}/(2\pi)$ matching the Schwarzschild value $1/(8\pi M)$ up to $\mathcal{O}((M\Lambda)^{-2})$ corrections, as confirmed by direct numerical evaluation.

For a radial timelike geodesic on (7) – (8) with conserved energy parameter $E$, $(\mathrm{d}r/\mathrm{d}\tau)^{2}=E^{2}-f(r)$. Since $f$ is bounded between $0$ and $1$ (in the exterior and on the core where $f\to 1$, attaining negative values only in the trapping trapped region $r_{-}<r<r_{+}$), a geodesic with $E^{2}>1$ admits everywhere a non-vanishing radial velocity, including across $r=0$.

For $r\gg l_{\rm can}$, the lapse function admits the asymptotic expansion

The leading correction to Schwarzschild appears at order $1/r^{4}$, three powers beyond Schwarzschild, and entirely beyond the parametrized post-Newtonian (PPN) expansion $f=1-2M/r+(2\beta-\gamma)M^{2}/r^{2}+\ldots$.

A regular black hole with two horizons admits, in addition to the standard external Hawking radiation, an inner Cauchy horizon at $r_{-}$ whose dynamical stability under generic perturbations is governed by Ori-style mass inflation [12, 13]. The relevant quantity is the inner-horizon surface gravity $\kappa_{-}$, which sets the exponential build-up rate of the Misner–Sharp mass at the inner Cauchy horizon under generic ingoing fluxes. The asymptotic value in the limit $M\Lambda\to\infty$ is, by direct computation,

i.e. the inner Cauchy horizon sits at the de Sitter scale $L_{\rm dS}$ independently of $M$, with surface gravity equal to the fakeon mass. Numerically at $M=\Lambda=1$ we find $\kappa_{-}=0.4295$. Fig. 4 displays $\kappa_{-}(M\Lambda)$ and the e-folding count $\kappa_{-}\,\tau_{\rm Hawk}$ across mass-product range $M\Lambda\in[1,10^{3}]$.

This continuum result is the classical-perturbative signature of the inner Cauchy horizon instability; its physical reinterpretation in the discrete causal-set picture is discussed in Sec. X.

The mass-inflation $e$-folding rate $\kappa_{-}$ describes the exponent of an unbounded perturbative singularity build-up at $r_{-}$; it is not a timescale comparable to Hawking evaporation. For an astrophysical black hole of solar mass with $\Lambda$ at the Planck scale, $\kappa_{-}^{-1}\sim m_{2,\,{\rm pole}}^{-1}\sim 10^{-44}\,\text{s}$, against a Hawking lifetime $\tau_{\rm Hawk}\sim M^{3}\sim 10^{67}\,\text{yr}$: the mass-inflation singularity formally builds up “instantaneously” against the evaporation clock. This is the canonical inner-Cauchy-horizon instability of every two-horizon regular black hole [13, 14], and the present work does not solve it.

Two physically distinct stabilisation scenarios remain open in the present setting:

1. 1.

   Outer-extremal endpoint of Hawking evaporation. Hawking radiation removes mass and drives the BH toward $M\Lambda\to\sqrt{27/(16z_{1})}\approx 0.836$ from above. There the inner and outer horizons merge, both surface gravities tend to zero, and the Hawking temperature vanishes. The BH ceases to evaporate, leaving an extremal stable remnant of mass $M_{\rm rem}=\sqrt{27/(16z_{1})}/\Lambda$. Distinct from the Carballo-Rubio inner-extremal cure [14], which makes $\kappa_{-}=0$ at fixed $r_{+}$; here the entire two-horizon structure collapses.

2. 2.

   Non-perturbative back-reaction in the model. The leading $O(1/\Lambda^{2})$ correction to the metric is contained in the Hayward ansatz; higher-order corrections in $1/(\Lambda r_{-})$ are unbounded near $r_{-}$ and may qualitatively change the inner-horizon geometry. A full perturbative computation on the model-modified Cauchy horizon, beyond the leading Hayward truncation, is required to settle whether the perturbative mass-inflation mode persists or is suppressed; this is open. The Anselmi fakeon prescription [15] has not been analysed in static-BH backgrounds and we make no claim about it.

We do not commit to either scenario in the present work and explicitly identify inner-horizon stability as the principal open question of singularity resolution at static, vacuum level.

The previous sections developed the proposed regular black hole as a continuum effective metric. The framework, however, is fundamentally causal-set theories [21, 22]: the underlying degree of freedom is a finite locally-finite partially ordered set $(\mathcal{C},\prec)$, of which the Lorentzian continuum metric is the coarse-grained large-$N$ limit by Poisson sprinkling. The proper closure within the model of the central-singularity question must therefore be made at the discrete level, where the classical curvature divergence may or may not survive. A standard formal discrete curvature observable is the Benincasa–Dowker scalar [23]; in this work we use a simpler operational proxy (26) chosen for clarity and report a first-principles BD-action evaluation as the natural next step.