A computer assisted existence proof for a non-Schwarzschild black hole in Einstein--Weyl gravity

Kevin Goldstein; Vishnu Jejjala

REVIEW

A computer assisted existence proof for a non-Schwarzschild black hole in Einstein--Weyl gravity

Kevin Goldstein, Vishnu Jejjala

General Relativity and Quantum Cosmology · Submitted 18 Sep 2026

Pith's verdict: Rigorous existence proof for a non-Schwarzschild black hole in quadratic gravity — the computer-assisted method is careful and the result is important.

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arXiv ID2609.21797 (version 2)

Pith IDNKLQXNV6, Pith's permanent record of this paper and its review

CitationKevin Goldstein, Vishnu Jejjala. "A computer assisted existence proof for a non-Schwarzschild black hole in Einstein--Weyl gravity." arXiv:2609.21797 (2026). https://pith.science/paper/NKLQXNV6

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Editor's note

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This paper delivers what the title promises: a computer-assisted proof that a static, spherically symmetric, asymptotically flat non-Schwarzschild black hole exists in Einstein-Weyl gravity. The gap between numerical evidence and analytic proof is finally closed for this system. The method is the main contribution: combininig convergent horizon series, validated interval integration for a compact core, a centered stable manifold fixed poimt at infinity, and a Pincaré-Miranda matching certificate. The proof is self-contained, the computations are fully documented and reproducible (source code provided, directed rounding, exact rational postprocessing), and the authors are upfront about limitations: no uniqueness, single horizon radius, dependence on computational layer. The comparison with earlier numerical results shows agreement to eight digits, which is reasuring but not part of the proof. The only real vulnerability is the reliance on the correctness of the interval arithmetic libray and hardware. Thata's inherent to any computer-assisted proof, and he authors mitigate it as well as one can. The proof does not cover a range of horizon radii, but that's a next step, not a weakness of this result. The parameter intervals (radii like 10^-45) look absurdly small, but the authors explain they reflect exponential amplification in the core; they are conditioning numbers, not precision targets. This is a serious piece of work. The mathematics is sound. The presentation is clear, with fifteen appendices that readers who need to check details can consult. Thia absolutely deserves peer review and publication. I'd bring it to a reading group on rigorous methods in GR or computer-assisted proofs. I'd cite it if I worked in higher-derivative gravity or validated numerics.

Referee report

Minor objections (5)

[Abstract] In the phrase 'm2¯rh = 1probes the higher derivative scale' a space is missing after the equals sign; similarly on p. 1 and elsewhere. The notation should be rendered as 'm_2 r_h = 1' for clarity.
[Section 2.3 (Eq. 1.2, 1.3)] The displayed intervals contain many digits. It would be helpful to add a parenthetical remark (e.g., in the caption of Table 5) that these are rigorous outward enclosures, not merely approximate decimal values.
[Section 5.2] The comparison with Ref. [8] in Table 4 is useful, but the text does not explicitly state the normalization of the comparison quantities (e.g., that both use r_h = -1, k=1/2). A brief sentence clarifying the conversion would aid readers.
[Appendix E] The description of the reproducibility infrastructure is thorough, but the caption of Table 5 refers to 'the regenerated core' without earlier defining 'regeneration'. The text in Appendix E explains the makeflow, but cross‑referencing the table to the pipeline step would prevent confusion.
[Throughout] The hyphenation of 'non-Schwarzschild' is sometimes 'non‑Schwarzschild' and sometimes 'non‑Schwarzschild'. Please standardize.

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The paper

The paper claims

Computer-assisted proof finds black hole beyond Schwarzschild in Einstein–Weyl gravity

The reading

This paper presents a computer-assisted proof that a static, spherically symmetric, asymptotically flat vacuum solution of Einstein–Weyl gravity exists with a non-zero Bach parameter, i.e., a non-Schwarzschild black hole. The proof splits the exterior into three domains: a convergent horizon series, a validated Taylor model integration of the compact core with directed rounding, and a Banach fixed-point argument for the asymptotic tail. A five-dimensional matching map between core and tail is shown to have a zero via the Poincaré-Miranda theorem, using exact rational arithmetic. The solution has a regular nondegenerate horizon, vanishing Ricci scalar, nonvanishing Ricci tensor, and no additional exterior Killing horizon. The proof establishes existence but not uniqueness.

What carries the argument

The key machinery is a domain decomposition combined with validated numerics: (1) a majorant series for the horizon, (2) a parameter-dependent Taylor model with directed rounding for the compact core, (3) a centered stable manifold fixed point argument on a weighted function space for the asymptotic tail, and (4) the Poincaré-Miranda theorem applied to a five-dimensional preconditioned matching map to certify the existence of a parameter tuple linking the three pieces.

Extended reading notes

Core claim

The central discovery is that a non-Schwarzschild black hole exists in Einstein–Weyl gravity at dimensionless horizon radius m2 * r_h = 1, with Bach parameter b in the interval (1.2) and compactified endpoint R in (1.3). The metric is not Ricci-flat and carries massive spin-two hair. The construction is explicit but the metric is given by a convergent horizon series, a validated ODE flow, and a stable asymptotic fixed point, all rigorous.

Load-bearing premise

The proof relies on the premise that interval arithmetic operations with directed rounding produce correct enclosures of true real results, and that hardware executes these operations without error; the paper mitigates this with consistency checks and exact rational postprocessing, but the validity depends on the computational layer being correct.

Editorial extensions

If this is right

The existence of this black hole confirms the numerical branch found earlier by shooting, providing a rigorous foundation for its geometry and thermodynamics.
The same domain decomposition strategy can be extended to other horizon radii along the non-Schwarzschild branch, provided the matching Jacobian remains nondegenerate.
The work demonstrates a template for computer-assisted existence proofs in higher-derivative gravity and other theories with exponentially growing modes.
The proof clarifies why the asymptotic expansion does not reveal the hair: it is beyond all algebraic orders in 1/r.
The black hole has well-defined mass, temperature, and entropy enclosures, enabling rigorous comparisons with numerical data.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

The same method could be adapted to asymptotically AdS black holes in quadratic gravity, where the tail problem involves exponentially growing modes from the negative cosmological constant.
The constructive representation may allow rigorous computation of quasinormal modes or stability via interval arithmetic on linearized equations.
Proposition 6.1, which shows no regular extremal horizon of finite area exists in this theory, suggests a possible upper bound on temperature or a minimum mass for the branch.

Original abstract

Static, asymptotically flat, non-Schwarzschild black holes in four-dimensional Einstein--Weyl gravity have previously been constructed by numerical shooting, while the local field equations admit horizon and asymptotic expansions to all orders. We give a computer assisted existence proof for a non-Schwarzschild solution whose dimensionless horizon radius is one in units of the massive spin-two scale, with parameters close to the previously reported numerical branch. The exterior is divided into a convergent horizon series domain, a compact core enclosed by a parameter dependent Taylor model with directed rounding, and an infinite asymptotic domain treated as a centered stable manifold fixed point problem. The resulting five-dimensional matching map between the core and tail is continuous on a rigorously specified box. After exact rational preconditioning, the signs on opposite faces satisfy the hypotheses of the Poincar\'e--Miranda theorem and therefore certify a matching zero. The solution has a regular nondegenerate horizon, no additional exterior horizon, vanishing Ricci scalar, and nonvanishing Ricci tensor. The strict bounds also imply existence at every sufficiently nearby horizon radius and a local continuous family at fixed coupling. The proof establishes existence, but not uniqueness of the matching zero.

The compiled record

The paper's own sentences, arranged, kept beside the review and separate from it. Each label is drawn from a closed vocabulary; the number beside it is the probability the classifier gave that label. Numbers in the tables are the authors', checked by code. No sentence in this record was written by a model.

Compiled by classifier-1.13.0 via TypeSafe API, question bank abstract-p1-20260923, 2026-09-23: 1 request, 5,692 input tokens.

What the abstract itself states, as probabilities: claims first 0.09 · states limitation 0.96 · human participants 0.01 · controlled comparison 0.03 · randomized assignment 0.01 · states quantitative result 0.84 · compares with prior results 0.93 · mentions code or data availability 0.03.

The abstract, sentence by sentence

arXiv listing abstract v1. Results and the main claim are underlined; show every role.

Static, asymptotically flat, non-Schwarzschild black holes in four-dimensional Einstein--Weyl gravity have previously been constructed by numerical shooting, while the local field equations admit horizon and asymptotic expansions to all orders. We give a computer assisted existence proof for a non-Schwarzschild solution whose dimensionless horizon radius is one in units of the massive spin-two scale, with parameters close to the previously reported numerical branch. The exterior is divided into a convergent horizon series domain, a compact core enclosed by a parameter dependent Taylor model with directed rounding, and an infinite asymptotic domain treated as a centered stable manifold fixed point problem. The resulting five-dimensional matching map between the core and tail is continuous on a rigorously specified box. After exact rational preconditioning, the signs on opposite faces satisfy the hypotheses of the Poincar\'e--Miranda theorem and therefore certify a matching zero. The solution has a regular nondegenerate horizon, no additional exterior horizon, vanishing Ricci scalar, and nonvanishing Ricci tensor. The proof establishes existence, but not uniqueness of the matching zero.

background contribution method result interpretation main claim

What the classifier reports for each sentence. The number is the probability it gave the label.
#	sentence	role	the authors' own work	states the main conclusion	strength
s0	Static, asymptotically flat, non-Schwarzschild black holes in four-dimensional Einstein--Weyl gravity have previously been constructed by numerical shooting, while the local field equations admit horizon and asymptotic expansions to all orders.	background 1.00	0.04	0.04	asserted 1.00
s1	We give a computer assisted existence proof for a non-Schwarzschild solution whose dimensionless horizon radius is one in units of the massive spin-two scale, with parameters close to the previously reported numerical branch.	contribution 0.97	0.96	0.79	asserted 1.00
s2	The exterior is divided into a convergent horizon series domain, a compact core enclosed by a parameter dependent Taylor model with directed rounding, and an infinite asymptotic domain treated as a centered stable manifold fixed point problem.	method 0.99	0.87	0.05	asserted 0.99
s3	The resulting five-dimensional matching map between the core and tail is continuous on a rigorously specified box.	method 0.81	0.93	0.07	asserted 1.00
s4	After exact rational preconditioning, the signs on opposite faces satisfy the hypotheses of the Poincar\'e--Miranda theorem and therefore certify a matching zero.	method 0.95	0.93	0.19	asserted 1.00
s5	The solution has a regular nondegenerate horizon, no additional exterior horizon, vanishing Ricci scalar, and nonvanishing Ricci tensor.	result 1.00	0.90	0.23	asserted 1.00
s6	The proof establishes existence, but not uniqueness of the matching zero.	limitation 1.00	0.90	0.17	asserted 1.00

What kind of paper the classifier reports this is: theoretical result 1.00 · evidence from proof or derivation 0.91.

What was examined, and what was not

sources read

arXiv listing abstract v1

not examined

full text
figures and tables
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relations to other papers

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S sections
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Every quoted sentence on this record is the authors'. Every label is from Pith's closed vocabulary, with the probability shown beside it. Every number in the measurements table is the authors', checked by code. Record as JSON

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