A polynomial affine model of gravity: after ten years

Aureliano Skirzewski; Bastian Grez; Jefferson Vaca; Jose Perdiguero; Manuel Morocho-L\'opez; Nicolas Zambra-G\'omez; Oscar Castillo-Felisola

REVIEW 4 major objections 3 minor 1 cited by

A polynomial affine model of gravity: after ten years

T0 review · 4 major / 3 minor · reviewed 2026-08-10 · deepseek-v4-flash

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗

Pith's one-line read The polynomial affine model of gravity claims that a metric is unnecessary: a diffeomorphism-invariant polynomial action built only from an affine connection and torsion reproduces General Relativity's successes on Einstein spaces and…

desk verdict Useful ten-year status report on the authors' own affine gravity model; the central completeness claim rests on an unverified enumeration and a misprinted counting condition, so treat the rigidity claim as provisional. read the letter →

arxiv 2412.21122 v1 pith:YR6ETRL3 submitted 2024-12-30 gr-qc hep-thmath-phmath.MP

Oscar Castillo-Felisola , Bastian Grez , Manuel Morocho-López , Jose Perdiguero , Aureliano Skirzewski , Jefferson Vaca , Nicolas Zambra-Gómez This is my paper · ORCID

classification gr-qchep-thmath-phmath.MP MSC 83D0553B0583F05 PACS 04.50.Kd04.20.Cv02.40.Hw

keywords affinegravitymetric-freetorsionpolynomialactioncosmologicalsolutionsemergentmetricconnectionEinsteinspaces

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

The paper reviews ten years of work on the polynomial affine model of gravity, a theory whose only fundamental field is the affine connection and whose action contains no metric tensor. Its central claim is that this metric-free model is viable: the field equations are second order, Einstein spaces are among their solutions, and in the torsion-free, volume-preserving sector the equations reduce to the Codazzi condition on the Ricci tensor. Because the action is assembled from every diffeomorphism-invariant polynomial term the connection and its torsion can form, the model is rigid rather than arbitrary. The review reports exact cosmological solutions, emergent metric structures on the space of solutions, a perturbation framework, and scalar-field couplings that reproduce Einstein–Klein–Gordon dynamics. If the model is right, it offers a way to keep General Relativity's successes while absorbing dark-sector and quantum-gravity effects into purely geometrical degrees of freedom.

What carries the argument

The load-bearing object is the action in Eq. (15), obtained by counting indices and tensor-density weights with the operators $N$ and $W$ (Eqs. 11–14) and then pruning the candidate list with the algebraic and differential Bianchi identities (Eqs. 7–9). This procedure leaves fifteen independent interaction terms, which the paper treats as exhaustive. The action's structure does the conceptual work: every term contains the torsion fields $A$ or $B$, so the torsion-free limit exists only at the level of the field equations; all couplings are dimensionless, so the model is power-counting renormalisable and has no explicit three-point graviton vertices. The covariant field equations follow from varying with respect to $\Gamma$, $B$, and $A$, and they are attacked with spherical and cosmological ansätze that reduce the sixty-four components of the connection to a few functions of time or radius.

What would settle it

Run an independent, computer-assisted enumeration of all diffeomorphism-invariant polynomial 4-forms constructible from a symmetric connection, a traceless torsion, and a torsion vector in four dimensions, subject to the same Bianchi identities; if it produces any term not equivalent modulo boundary and topological terms to a combination of the terms in Eq. (15), then the assumed exhaustive action is incomplete and the field equations built from it would need revision.

Watch

Extended reading notes

Core claim

On the paper's own terms, the result of a decade of work is that a purely affine, polynomial theory of gravity is a working alternative to metric theories. The action in Eq. (15) is presented as the most general diffeomorphism-invariant 4-form built from the decomposition of the connection into a symmetric part $\Gamma$, a traceless torsion $B$, and a torsion vector $A$, up to boundary and topological terms. Varying it yields second-order covariant field equations, and Einstein spaces are a subset of their solutions. In the torsion-free sector with a volume-preserving connection, the equations become $\nabla_{[\mu} R_{\nu]\lambda} = 0$, the condition that the Ricci tensor be a Codazzi tensor; this links the solution space to statistical manifolds and to the pure Yang–Mills-type gravitational action, while avoiding the metric-based objections raised against that action in metric theories. Exact solutions are exhibited in cosmology and in static spherical symmetry, including the Schwarzschild–(anti-)de Sitter geometry, with the cosmological constant appearing as an integration constant.

Load-bearing premise

The load-bearing premise is that the index-counting rules and Bianchi identities used to enumerate the action catch every independent diffeomorphism-invariant polynomial 4-form that can be built from $\Gamma$, $B$, and $A$; if any term was missed or misclassified as a boundary or topological term, the field equations and all derived solutions would change.

Editorial extensions

If this is right

In the volume-preserving torsion-free sector the field equations reduce to $\nabla_{[\mu} R_{\nu]\lambda} = 0$, so every Einstein space solves the model and the solution space contains statistical manifolds.
In four-dimensional cosmology the field equations admit exact analytical power-law and exponential solutions parameterised by the torsion function $\psi(t)$, with the cosmological constant entering as an integration constant rather than as a fundamental parameter.
The symmetrised Ricci tensor on shell gives an emergent metric, so distances and causal structure become definable even though the action never contains a fundamental metric.
Coupling a scalar field through a density-scaled kinetic term produces field equations equivalent to General Relativity with a massless scalar, with the scalar equation emerging from symmetry rather than from a least-action principle.
The affine perturbation framework yields 60 physical components after gauge fixing and 165 possible gauge choices, giving a route to structure formation and stability analysis.

Reading between the lines

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

If the enumeration behind Eq. (15) is truly exhaustive, then any metric theory that emerges on shell is tightly constrained: the same coupling constants govern all non-Riemannian corrections, so a measurement in one sector would fix predictions in another.
The 64 connection components act as hidden fields, and under cosmological symmetry they collapse to the functions $f(t), g(t), h(t), \psi(t), \eta(t)$; this suggests that dark-sector phenomenology could in principle be fitted from background cosmology without introducing new matter fields.
The affine Kaluza–Klein foliation outlined in Section VII E, if completed, would make gauge fields and scalar matter projections of a single connection, turning the model into a unified field theory; the review only sketches the bundle formalism.
Because the action has dimensionless couplings and no explicit three-graviton vertices, a direct computation of graviton scattering would be the natural next test of quantum viability, but that test lies beyond what the review establishes.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

The thing to know: this is a ten-year retrospective on the authors' own polynomial affine gravity model, not a new result. Read it as a status report. It is well organized and unusually candid about what is not solved. The main thing I'd flag is the completeness claim for the action in Eq. (15): it is load-bearing and, as printed, not verifiable. Eq. (14) gives m+n+p+q=1, but every row of Table I has m+n+p+q=5; the intended condition is clearly q=1 and m+n+p=4. That is a simple typo, but it matters because the whole rigidity argument rests on that enumeration, and the enumeration is not re-derived here.

What is genuinely good: the paper gives the full covariant field equations in Appendix A, works out cosmological ansatze and branches in detail, summarizes the perturbation formalism, and presents new material on static spherical connections and affine foliations. The authors are honest about underdetermined systems: they say plainly when equations do not close and when a branch is not solvable without assumptions. The scalar-coupling section is a nice example of how to make the model talk to standard cosmology. If you want a single entry point to this program, this is the paper.

The soft spots are real but proportionate. The completeness of the action is the foundation of the model's main selling point, and the paper asserts it rather than proves it. For a review, importing a proof from earlier papers is acceptable, but the authors should at least point to the exact reference and state that the proof is not reproduced. Right now the text says the enumeration follows from index counting and Bianchi identities, and the printed counting condition is wrong, so a referee cannot check the central claim without going back to the original papers. The strong statements about power-counting renormalizability and bypassing graviton no-go theorems are also asserted without derivation or citation to a detailed argument; in a review they should be labeled as expectations or supported by references.

The self-review aspect is not a flaw by itself—the model is theirs and the literature is mostly theirs—but it means there is no independent validation of the key steps. The GR-enclosing sector is derived under explicit assumptions (e.g., existence of a parallel nondegenerate tensor), which the paper does state, so the claim is not circular. I do not see a fatal problem; I see a useful status report that needs a corrected equation, clearer pointers to where completeness is proved, and tempering of the quantum-gravity claims.

Who is this for: researchers working on affine or metric-affine gravity who want a compact overview of this model and its solution space. It deserves a serious referee. I would send it, with a request that the authors fix Eq. (14), add a precise citation for the completeness proof, and soften or support the renormalizability and no-go statements.

Referee Report

4 major / 3 minor

Summary. This paper is a ten-year retrospective review of the polynomial affine model of gravity, a metric-free theory whose fundamental variable is an affine connection decomposed into a symmetric connection Gamma, a traceless torsion B, and a torsion vector A. The central object is the action in Eq. (15), claimed to be the most general diffeomorphism-invariant polynomial 4-form built from these fields up to boundary and topological terms. The paper reviews the construction, derives covariant field equations (Appendix A), presents par-spherical and cosmological connection ansaetze, solves the cosmological field equations analytically in several branches, outlines a cosmological perturbation formalism, discusses emergent metrics and scalar-field coupling, examines spherically symmetric solutions, and closes with future perspectives.

Significance. If the completeness of Eq. (15) and the reported exact solutions hold, the model would be a genuinely rigid alternative to metric gravity: a fundamental connection without a metric, dimensionless couplings, second-order field equations, Einstein spaces as a subset of solutions, and emergent metric structures on-shell. The review is useful in collecting a decade of the authors' work, including explicit exact cosmological solutions, a detailed perturbation scheme, and machine-checked tensor computations with SAGE/Cadabra. However, several load-bearing technical claims are either asserted without proof or contain concrete algebraic misprints, so the paper in its present form is not yet a reliable review of the model's viability.

major comments (4)

[§II, Eq. (14) and Table I] The printed counting constraints are internally inconsistent. Eq. (14) states m+n+p+q=1 and m+n+p=4q; for q=1 the first condition gives total exponent 1, while every row of Table I has m+n+p+q=5. More importantly, the exhaustiveness of the fifteen index-structure classes and their reduction to the twenty terms of Eq. (15) via Bianchi identities and discarding boundary/topological terms is asserted rather than demonstrated. Because this enumeration is the foundation of the field equations and of the model's rigidity claim, the review should either repair and present the counting algorithm or give a precise reference to a derivation that independently establishes completeness. As it stands, the central claim is conditional on an unverified enumeration.
[§V, Eqs. (75)–(76)] Substituting g(t) = psi(t)(g0 - alpha int psi dtau) from Eq. (73) into Eq. (70) with kappa = 0 yields (3B3 - 2B4) psi_dot (g0 - alpha int psi dtau) = psi^2 [2 alpha (B3 - 2B4) + 4F3], which after division by 2 becomes beta psi_dot(...) = psi^2 [alpha (B3 - 2B4) + 2F3]. The printed definition gamma = (beta - 2B3) alpha + 2F3 in Eq. (76) equals alpha[-(B3 + 2B4)/2] + 2F3, not alpha(B3 - 2B4) + 2F3. Hence Eq. (77) and the resulting exact solution (80)–(83) solve a different differential equation. The branch-one cosmological solution must be re-derived with the correct coefficient.
[§VII.D, Eq. (164)] For the metric p(r) = 1 + c1/r + c2 r^2, the second equation of Eq. (164) as printed, r p'' + 2 - 2p/r = 0, evaluates to 2 - 2/r, not zero. The equation actually satisfied by the Schwarzschild-(A)dS metric is r^2 p'' + 2 - 2p = 0, equivalently p'' + 2(1-p)/r^2 = 0. With the printed equation, the claimed Schwarzschild-(A)dS solution does not solve the field equations, so the derivation in §VII.D needs to be corrected.
[§II, features (ii) and (iii)] The feature list states that dimensionless couplings 'ensure that the model is power-counting renormalisable' and that the absence of explicit three-point graviton vertices 'allows to bypass' the no-go theorems of Refs. [91,92]. Neither statement is derived or referenced in this review. Dimensionless couplings are necessary but not sufficient for power-counting renormalizability, and the bypass claim requires at least a linearized amplitude argument or a citation to a place where it is established. These items should be either substantiated or explicitly labeled as conjectures.

minor comments (3)

[Throughout] There are numerous typographical errors, including 'repect' in Sec. III, 'pertubation' in Sec. VI, 'lake of predictability' in Sec. VII.C, and 'The table I' in Sec. II. A careful proofreading pass is needed.
[§V] The statement that the first branch is solved 'without any type of assumption on the functions' is too strong: the derivation sets kappa = 0 and assumes B3 != 2B4, and the second branch explicitly requires kappa != 0. These qualifications should be stated up front.
[§VII.E] Section VII.E presents a general fiber-bundle formalism that is not yet applied to the polynomial affine action; it should be labeled explicitly as an outlook or research program rather than as a result of the model.

Circularity Check

2 steps flagged · score 4.0 of 10

Self-review leans on the authors' own action-completeness count and on a self-cited Einstein–Klein–Gordon equivalence; the GR-enclosing claim has independent derivations, but the advertised 'all possible terms' uniqueness is asserted rather than proved, and Eq. (14) is internally inconsistent with Table I.

self citation load bearing [Sec. VII C (Coupling scalar matter), after Eq. (150)]
"In Ref. [144], we shown that the case of parallel S-tensor is (somehow) equivalent to the minimally coupled Einstein–Klein–Gordon system. The equivalence is ensured by the existence of a symmetric, nondegenerate, and parallel (02)-tensor, say gμν."

Central feature (viii) and the abstract's claim that the model 'encloses the successes of General Relativity' rest on this equivalence, but the review does not derive it: it delegates to the authors' own arXiv:2312.07312 [144]. The stated sufficient condition, existence of a symmetric, nondegenerate, parallel two-tensor, is the defining datum of a Riemannian geometry, so the 'equivalence' is conditional on importing a metric-compatible structure that the model claims to do without. No independent check (external benchmark, machine verification, or derivation in this paper) is supplied, so the load-bearing GR-limit claim reduces to a self-citation with an assumption close to the conclusion.
self definitional [Sec. II, Eqs. (11)–(14) and Table I]
"In order to define an action functional on a four-dimensional manifold M, we write a linear combination of all possible 4-forms that can be made that are linearly independent. ... Since the action has to be a scalar, the action of the operators N and W on the term in Eq. (13) yield the restrictions m +n +p +q = 1, and m +n +p = 4q. ... The table I shows all the solutions to the constraints in Eq. (14)."

The advertised 'milestone' of the model, that the action 'contains all possible combinations of the fields compatible with the covariance under diffeomorphisms', is established only by the N/W index-counting that is used to build the action. A term is included exactly when it satisfies Eq. (14), so 'all possible combinations' is a restatement of the construction rule rather than an independently derived theorem. The printed first condition m+n+p+q=1 is inconsistent with Table I, in which every row has m+n+p+q=5 with q=1, so the enumeration is not even self-consistent as printed. All downstream results, field equations, torsion-free sector, Einstein-space subset, and cosmological solutions, inherit this unverified 20-term list; if a term was missed or misclassified, the dynamics change.

full rationale

This is a self-review of the authors' own polynomial affine model. Much of the paper is a summary of prior work by the same group: the action Eq. (15) comes from refs. [72,73]; the connection ansätze from [119]; the perturbation formalism from [128]; the scalar-field/GR equivalence from [144]. These self-citations are numerous. The most load-bearing is the Einstein–Klein–Gordon equivalence, which is not re-derived here and is conditional on the existence of a parallel nondegenerate tensor. The 'all possible terms' rigidity claim is presented as a derived milestone, but it is generated by the same index-counting that defines the model, and the printed counting condition m+n+p+q=1 contradicts Table I. That is a correctness and omitted-proof concern as much as a circular one. On the other hand, substantial parts of the derivation chain are internal and non-circular: the torsion-free field equations are derived from the action in Sec. III A; the cosmological solutions in Sec. V are obtained by solving the stated ODE system with integration constants; the Schwarzschild-(A)dS solution in Sec. VII D follows from inserting the Levi-Civita ansatz into the Codazzi equations. No quantity is fitted to external data and then renamed a prediction; the DESI comment is explicitly a program for future use, not a retrospective fit. Overall, the central GR-enclosing claim has independent content in the paper, but the completeness of Eq. (15) and the scalar equivalence are sustained by the authors' own construction and citations rather than by an independent derivation.

Assumptions & free parameters 10 free parameters · 5 assumptions · 0 invented entities

The central claim rests on the assumption that Eq. (15) exhausts all diffeomorphism-invariant polynomial terms. The model also depends on the variational principle and the existence of nondegenerate emergent metrics. The large number of unspecified coupling constants and arbitrary functions (e.g., V(phi), h(t)) are free parameters, not fitted to data.

free parameters (10)

B1,B2,B3,B4,B5 (5 couplings)
Couplings of curvature-torsion terms in Eq. (15); not fixed by data.
C1,C2 (2 couplings)
Couplings of torsion-derivative terms; C2 defines the ratio C=C1/C2 used in the torsion-free field equations.
D1...D7 (7 couplings)
Couplings of cubic torsion terms in Eq. (15).
E1,E2 (2 couplings)
Couplings of derivative-squared torsion terms.
F1...F4 (4 couplings)
Couplings of quartic torsion terms.
Scalar coupling constant C
Coupling to scalar field kinetic term in Eqs. (144)-(148).
Scalar potential V(phi)
Arbitrary function introduced via scaling E -> E/V(phi); not determined by the model.
R0 (metric normalization)
Constant with curvature dimensions used to normalize the emergent Ricci metric in Sec. VII.B.
alpha,beta,gamma (scalar coupling tensor coefficients)
Arbitrary constants in Eq. (142) defining the emergent inverse metric used to couple a scalar field.
h(t) (Hubble function)
In the scalar-field cosmological solutions of Sec. VII.C, the h function is undetermined and must be fixed by observations; the model then determines the rest.

assumptions (5)

ad hoc to paper The action Eq. (15) is the most general diffeomorphism-invariant polynomial action built from Gamma, B, A up to boundary terms.
This completeness is asserted from index counting (N, W operators) and Bianchi identities, but not independently proved in this review; it is the foundation of all field equations.
standard math The decomposition of the affine connection into symmetric part, traceless torsion B, and trace torsion A (Eq. 4) is complete and the fields are independent.
Standard decomposition of a connection in 4D; used to build the action.
domain assumption Variation of the action with respect to Gamma, B, A yields the field equations, with B constrained traceless as in Eq. (30).
The variational principle for a purely affine theory without a metric is assumed valid; note the mismatch in number of equations (20) vs unknowns (40) is acknowledged.
domain assumption The emergent metric interpretation requires that on-shell the symmetrized Ricci tensor (or other candidates) is symmetric and nondegenerate with Lorentzian signature.
In Sec. VII.B the paper notes the signature depends on explicit solutions and coupling constants; no guarantee for generic solutions.
ad hoc to paper Power-counting renormalizability follows from dimensionless couplings.
Asserted in Sec. II feature (ii) without a detailed proof.

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Cite this review

Pith. "Pith review of A polynomial affine model of gravity: after ten years." pith.science (2026). https://pith.science/paper/YR6ETRL3

@misc{pith2026241221122,
  author       = {Pith},
  title        = {Pith review of: A polynomial affine model of gravity: after ten years},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/YR6ETRL3}},
  note         = {Machine review of arXiv:2412.21122}
}

read the original abstract

The polynomial affine model of gravity was proposed as an alternative to metric and metric-affine gravitational models. What at the beginning was thought as a source of unpredictability, the presence of many terms in the action, turned out to be a milestone, since it contains all possible combinations of the fields compatible with the covariance under diffeomorphisms. Here, we present a review of the advances in the analysis of the model after ten years of its proposal, and sketch the guideline of our future perspectives.

Discussion (0). Continue with ORCID to comment.

Forward citations

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Field Equations for Γ B1: ∇µ ( 2δ[µ λ Bα ν] βBγ ρ τ + 2δ[µ λ Bα ρ] βBγ ν τ ) Eαβγτ (A1a) (A1b)B2: ∇µ ( 4Bγ σ δBσ(ρλ ) Eν)µγδ (A1c)B3: ∇µ ( 2δ[µ λ Bβ ν] γAδEρβγδ + 2δ[µ λ Bβ ρ]γAδEνβγδ ) (A1d)B4: ∇µ ( −4Bγ (ρδAλ ) Eν)µγδ (A1e)B5: ∇µ ( −4Bγ σδAσδ(ρ λ ) Eν)µγδ (A1f) C1: ∇µ ( 2∇βBγ ρδδ[µ λ Eν]βγδ + 2∇βBγ ν δδ[µ λ Eρ]βγδ ) + 2RµαµλBγ (ρδEν)αγδ (A1g)C2: ∇µ ( −4...

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Field Equations for B (A2a) B1: − 4Rµ(σ µ λ)Bγ σ δEνργδ − 4 3 Rµτ µ σBγ σ δδ[ν λ Eρ]τ γδ − 4 3 Rµσ µ τ Bγ σ δδ[ν λ Eρ]τ γδ (A2b) B2: − 2Rαβ µσBµσ λEνραβ − 2Rαβ [ν λBγ ρ]δEαβγδ − 4 3 Rαβ µσBµσ τδ[ν λ Eρ]τ αβ − 2 3 Rαβ τ τ Bγ [ν δδρ] λ Eαβγδ + 2 3 Rαβ [ν τδρ] λ Bγ τ δEαβγδ B3: − 2Rµλ µ αAβ Eνραβ − 4 3 Rµτ µ αAβδ[ν λ Eρ]τ αβ (A2c) B4: − 2Rαβ σ λAσEνραβ − 4 3...

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From this, it is straightforward to recover the original function ψ(t) = βg0 2γ(t −t0) h(t) = − γ 2γ(t −t0) (99) g(t) = g1 γ(t −t0) (100) where g1 is another integration constant

to (97)¨φg0β − ˙φ2γ = 0, which can be solved exactly by φ(t) = φ0 + βg0 γ log (γ(t −t0)), (98) where φ0 and t0 are integration constants. From this, it is straightforward to recover the original function ψ(t) = βg0 2γ(t −t0) h(t) = − γ 2γ(t −t0) (99) g(t) = g1 γ(t −t0) (100) where g1 is another integration constant. The latter constraint leads to the foll...

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