Starobinsky Inflation with T-Model Kaehler Geometries

C. Pallis

REVIEW 3 major objections 5 minor 3 cited by

Starobinsky Inflation with T-Model Kaehler Geometries

T0 review · 3 major / 5 minor · reviewed 2026-08-09 · deepseek-v4-flash

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Pith's one-line read This paper shows that Starobinsky inflation can be implemented in supergravity using T-model Kähler geometries, not only the usual E-model kinetic mixing.

desk verdict A credible SUGRA extension of Starobinsky inflation to T-model Kähler geometries, but the printed analytic section contains typos that must be fixed before the derivation is trustworthy. read the letter →

arxiv 2502.00636 v2 pith:H52VNOT5 submitted 2025-02-02 hep-ph astro-ph.COhep-th

C. Pallis This is my paper · ORCID

classification hep-phastro-ph.COhep-th PACS 98.80.Cq12.60.Jv95.30.Cq95.30.Sf

keywords StarobinskyinflationT-modelSupergravityKählerpotentialsalpha-attractorsshiftsymmetrytensor-to-scalarratioU(1)_X

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

Starobinsky inflation is usually embedded in supergravity through an E-model kinetic pole, where the inflaton approaches its plateau exponentially. This paper claims that the same plateau can be produced by a T-model geometry, whose kinetic term has a pole of order two and the field relation $\phi = \tanh(\hat\phi/\sqrt{2N})$, provided the scalar potential is $V_I = \lambda^2(\phi^{n/2}-M^2)^2/(1+\phi)^{n_d}$. The author constructs two supergravity realisations — one with a gauge-singlet inflaton and one with a Higgs-like pair that breaks a $U(1)_X$ symmetry — and shows that along D-flat trajectories they yield exactly this potential with a stabilized companion field. If the construction is correct, Starobinsky predictions are not tied to the E-model form: the spectral index can be $0.961$–$0.969$ and the tensor-to-scalar ratio up to $0.032$, matching Planck, BICEP/Keck, BAO and lensing data at 95% confidence. That widens the class of viable supergravity models that can mimic Starobinsky inflation and gives future CMB experiments concrete targets to discriminate among them.

What carries the argument

The central object is the T-model Kähler normalization: $K_T = -N\ln F_T$ with $F_T = 1-|\Phi|^2$ for the singlet case (or the two-field versions $F_T = ((1-2|\Phi|^2)(1-2|\bar\Phi|^2))^{1/2}$ and $F_T = 1-|\Phi|^2-|\bar\Phi|^2$ for the Higgs case). This yields the kinetic metric $\langle K_{\Phi\Phi^*}\rangle = N/f_T^2$ with $f_T = 1-\phi^2$, producing the second-order pole and the canonical relation $\phi=\tanh(\hat\phi/\sqrt{2N})$. The new ingredient is the addition of holomorphic and anti-holomorphic logarithmic terms $K_{sh}$ and $K_d$ whose mixed derivatives vanish, so the kinetic metric is untouched while the $e^K$ prefactor becomes exactly $(1+\phi)^{-n_d}$. Combined with a superpotential $W=\lambda S F_W(\phi)$ and with $S$ stabilized at zero, the only surviving F-term, $e^K|W_{,S}|^2$, turns into $V_I = \lambda^2(\phi^{n/2}-M^2)^2/(1+\phi)^{n_d}$.

What would settle it

Evaluate the complete Kähler metric including $K_{sh}$ and $K_d$ away from the inflationary trajectory, or compute the one-loop corrected potential with $\Lambda$ fixed by renormalization-group running instead of the endpoint conditions $\Delta V_I(\phi_\star)=0$ or $\Delta V_I(\phi_f)=0$; if the second-order pole is deformed or $n_s$ shifts outside $0.961$–$0.969$ with $r\le 0.032$ at 95% c.l., the central claim fails.

Watch

Extended reading notes

Core claim

The paper's central message is that Starobinsky inflation is not exclusively implemented by the E-model kinetic mixing in Eq. (1.1); it is also attainable via T-model normalization in Eq. (1.2) in conjunction with the potential in Eq. (1.3). Concretely, taking the Kähler potential as $K = K_2 + \tilde{K}_T + K_d$, where $K_2 = N_S\ln(1+|S|^2/N_S)$ stabilizes the goldstino-like field $S$ at zero, $\tilde{K}_T$ parameterizes the hyperbolic T-model manifold, and $K_d$ contributes the prefactor $(1+\phi)^{-n_d}$, together with superpotentials $W = \lambda S \Phi^{n/2}$ for the gauge-singlet case (CSI) and $W = \lambda S((2\bar\Phi\Phi)^{n/4}-M^2)$ for the Higgs case (HSI), the supergravity F-term potential reduces exactly to $V_I$ along the D-flat trajectory $\langle S\rangle = \langle \Phi-\Phi^*\rangle = 0$ (or the corresponding Higgs-direction condition). The extra holomorphic and anti-holomorphic terms in $K_{sh}$ and $K_d$ are engineered to leave the Kähler metric unchanged, so the T-model pole of order two survives. The mass spectrum shows all non-inflaton scalars and fermions heavy, with $N_S<6$ ensuring stability of $S$, so that only the canonically normalized inflaton $\hat{\phi}$ generates the observed curvature perturbations.

Load-bearing premise

The construction assumes that the extra pieces added to the Kähler potential change only the overall exponential factor, never the kinetic geometry of the inflaton, and that the one-loop quantum corrections are fully controlled by choosing the renormalization scale at either the start or the end of inflation.

Editorial extensions

If this is right

Starobinsky inflation can be reproduced with simple monomial superpotentials respecting R and U(1)_X symmetries, without invoking induced gravity or higher-order curvature terms.
The predicted observables are $n_s\simeq 0.961$–$0.969$, $r\lesssim 0.032$, and $\alpha_s\simeq -(5.3$–$8.2)\times 10^{-4}$, consistent with PR4+BK18+BAO+lensing at 95% confidence, with $N_\star\simeq 50$–$60$ e-folds.
Allowed parameter regions are ample: for CSI with $n=2$, $1\lesssim N\lesssim 180$ and $0\le n_d\le 3.99$; for HSI, $1\lesssim N\lesssim 165$ (for $n=4$) and $1\lesssim N\lesssim 152$ (for $n=8$), with $n_d<2n$.
In HSI the inflaton can be a Higgs-like pair whose vacuum breaks $U(1)_X$ at a scale compatible with MSSM gauge-coupling unification, and cosmic strings are avoided because the symmetry is already broken during inflation.
Since $r$ grows with the curvature parameter $N$ while the spectral index stays close to its central value, future CMB polarization measurements can narrow the allowed $(n_d, N)$ regions and test whether this T-model implementation is realized in Nature.

Reading between the lines

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

If the construction is right, the E-model versus T-model distinction for Starobinsky-like inflation becomes a choice of kinetic normalization rather than a physical observable, because both can produce the same plateau potential.
The same Kähler-engineering trick — adding holomorphic and anti-holomorphic terms with vanishing mixed derivatives — could be carried over to Palatini-$R^2$ supergravity or D-brane-motivated models to generate other desired prefactors without disturbing the geometry.
A direct testable extension would be to fix the one-loop scale $\Lambda$ from a full renormalization-group running instead of setting it at $\phi_\star$ or $\phi_f$; if the resulting shift in $n_s$ or $r$ exceeds current 95% contours, the quoted parameter ranges would need revision.
The dependence of $r$ on the Kähler curvature suggests that precise measurements of $r$ below about 0.03 could be used to infer the geometry of the inflaton-sector Kähler manifold.

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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

This is a legitimate, checkable model-building contribution to the α-attractor/Starobinsky program. Pallis gives two SUGRA embeddings that produce the T-model kinetic term plus a pole-of-order-two potential VI = λ²(φ^{n/2} − M²)²/(1+φ)^{n_d}, with a stabilizer field and a clean shift-symmetry breaking mechanism. The central construction checks out: the mixed derivatives vanish as claimed, so the T-model metric is preserved, and the prefactor engineering in Eqs. (2.6)–(2.8) works at tree level. The mass spectra in Tables 1 and 2 support trajectory stability, and the ns–r predictions (ns ≈ 0.961–0.969, r ≲ 0.032) sit in the observationally allowed region. The real value is showing that Starobinsky-like inflation does not require the E-model kinetic mixing, and giving a concrete, explicit way to build the T-model version.

The novelty is modest but real. The potential and the Kähler potentials are drawn from earlier work (Refs. [56,60,66], including the author's own), and the construction is reverse-engineered: the observables are inherited from a potential chosen by hand. That is a structural feature of the approach rather than a fatal flaw, but it does mean the headline numbers are not predictions in a strong sense.

Soft spots, in order of seriousness. First, the analytic slow-roll formulas in Sec. 5.1 contain typos that matter. Eq. (5.1b) has a factor of 2 too small in the denominator of ε; the correct expression for n=2, n_d=1 is (1−φ)²(2+φ)²/(4Nφ²), not (…)/(2Nφ²). And Eq. (5.4) has ln(n f_T − n_d φ) where the integral actually gives ln(n f_d − n_d φ) = ln(2+φ) for the same case; the printed argument becomes negative in the inflationary domain. These are not cosmetic: a reader following the printed derivation cannot reproduce Table 3. The numerics in the table appear internally consistent with the corrected formulas, so this is an error in presentation rather than in the underlying computation, but it still needs fixing. Second, the one-loop treatment sets Λ by requiring ΔV = 0 at φ⋆ or φf; the sensitivity of the quoted observables to that choice is not quantified. Minor-to-moderate omission. Third, no code or data files are provided, so the figures cannot be independently reproduced.

On citation practice: the self-citation is heavy, but it is directly relevant prior work and the dependence is disclosed. I would not call it a flaw.

Who should read this: inflation model-builders, especially those working on SUGRA embeddings of attractor models. It deserves a serious referee. I would recommend major revision: fix the analytic formulas, add a sensitivity check on Λ, and ideally release the numerics. The central embedding is sound, so the paper should not be rejected on the typos alone, but the text as it stands is not a reliable guide to the derivation.

Referee Report

3 major / 5 minor

Summary. The paper constructs explicit N=1 supergravity embeddings of Starobinsky-like inflation based on T-model (pole-of-order-two) Kähler geometries rather than the usual E-model kinetic mixing. For a gauge-singlet inflaton (CSI) and a gauge non-singlet Higgs pair (HSI), the author selects superpotentials W=λ S F(Φ) and Kähler potentials K=K2+K~T+Kd, where the holomorphic and anti-holomorphic additions Ksh and Kd are engineered to leave the T-model Kähler metric unchanged while generating the potential VI=λ²(φ^{n/2}−M²)²/(1+φ)^{n_d}. The paper verifies D-flatness, stabilizer stabilization, and mass spectra along the inflationary trajectory, discusses one-loop corrections, and compares the resulting ns and r with PR4+BK18+BAO+lensing data. It finds ns≈0.961–0.969 and r≲0.032 for ranges of N, nd, and n, and concludes that Starobinsky inflation can be implemented with T-model normalization in conjunction with the potential in Eq. (1.3).

Significance. If the construction is correct, this is a genuinely new route to Starobinsky inflation in supergravity: it uses the T-model pole of order two, simple monomial superpotentials, and fully symmetric Kähler manifolds, while avoiding the more complicated superpotentials often needed in such frameworks. The paper explicitly acknowledges that the potential is reverse-engineered from the desired observable predictions, so the limited predictive novelty is a feature of the method rather than an internal inconsistency. The mass-spectrum analysis and the explicit statement of the symmetries are useful. However, the analytical slow-roll section currently contains internal inconsistencies that prevent the printed derivation from reproducing the quoted observables; this must be repaired before the paper can be fully relied upon.

major comments (3)

[Sec. 5.1, Eq. (5.1b)] The slow-roll parameter ε is printed with denominator 2Nφ². A direct computation from VI in Eq. (1.3) and J in Eq. (3.8), using f_d=1+φ and f_T=1−φ², gives ε=(φ−1)²(n f_d−n_d φ)²/(4N φ²) for M=0. The missing factor of 2 propagates through Eq. (5.8) into r and into the bracketed analytic values in Table 3. For the CSI row with N=10, φ⋆=0.943, n=2, n_d=1, the printed formula gives r≈0.025, whereas the listed r=0.013 is reproduced by the corrected denominator. This equation must be fixed and the affected entries in Table 3 and Eqs. (5.9) rederived.
[Sec. 5.1, Eq. (5.4)] The e-fold integral I_N is internally inconsistent. Combining J from Eq. (3.8) with the derivative of VI in Eq. (1.3) gives the partial fraction φ/[(1−φ)(n f_d−n_d φ)], so the resulting logarithm is ln(n f_d−n_d φ), not ln(n f_T−n_d φ). For the CSI parameters n=2, n_d=1, the printed argument n f_T−n_d φ=2(1−φ²)−φ becomes negative for φ>0.781, making I_N complex in the inflationary domain. The coefficients in Eq. (5.4) also do not match direct partial-fraction integration. Since Eq. (5.5) and the approximate formulas in Eqs. (5.9) are built on this expression, the printed analytic derivation cannot reproduce the quoted observables.
[Sec. 5.1, Eqs. (5.5)–(5.9)] The approximate expressions for φ⋆, ns, r, and αs inherit the errors in Eqs. (5.1b) and (5.4). These formulas are used in Sec. 5.2 to claim agreement with the data and to delimit the allowed parameter regions. The analytic derivation should be corrected and the numerical comparison repeated, or the paper should clearly state that only the numerical results are definitive and remove the incorrect analytic support.

minor comments (5)

[Sec. 4.3, Eq. (4.12)] The expression for DX under K=~K2(11)2d appears to contain a typo: the two identical factors (1−2|¯Φ|²)^{-1} should likely be (1−2|Φ|²)^{-1}(1−2|¯Φ|²)^{-1}, since the printed expression is not symmetric in Φ and ¯Φ.
[Sec. 5.2.1, Eq. (5.16a)] The inequality '1 ≲ Δ⋆/100 ≲ 53' is inconsistent with the definition Δ⋆=1−φ⋆ and with Table 3, where Δ⋆ is listed in percent (e.g., 5.7 for the first CSI column). The notation should be harmonized to avoid confusion.
[Conclusions, p. 15] The sentence 'The present data on δ21' appears to contain a typo: δ21 is not defined anywhere and from context should be δn or nd.
[Abstract] The abstract contains the typo 'Starobisky-like inflation'; it should read 'Starobinsky-like inflation'.
[Sec. 5.1, Eq. (5.1a)] The sentence 'which and can be estimated' contains a grammatical error; it should be 'which can be estimated'.

Circularity Check

1 steps flagged · score 2.0 of 10

Transparent Kähler-potential engineering rather than a hidden fit or self-citation loop; the potential is an input ansatz, while the existence claim is independently checked.

self definitional [Sec. 2.2.3, Eqs. (2.6a)-(2.7b), applied in Secs. 3.3 and 4.3]
"It has to generate the denominator of VI in Eq. (1.3). To achieve this, we focus on the exponential prefactor of VI in Eq. (2.3) and we demand ⟨eKd⟩I = (1 + φ)−nd, where Kd has the following structure (similar to that of Ksh) Kd = −(nd/2) ln Fd − (nd/2) ln F ∗ d with ⟨Fd⟩I =: fd = 1 + φ."

Kd is not fixed by an independent physical principle; it is defined by the target denominator of VI. With ⟨Fd⟩I = 1 + φ, the definition yields e^{Kd} = |1 + φ|^{−n_d} on the trajectory, so substituting into Eq. (2.3) returns VI in Eq. (1.3) with the denominator (1 + φ)^{−n_d} by construction. Similarly, Ksh is fixed by the requirement e^{KT+Ksh} = 1, cancelling the T-model prefactor. Thus the 'derivation' of the inflationary potential is a restatement of the chosen ansatz. The paper is explicit about this ('we demand'), so it is a stated model-building construction rather than a concealed fit, but in the derivation chain the potential is an input, not an output.

full rationale

The potential VI is engineered rather than predicted from first principles: Eq. (2.7a) literally demands e^{Kd} = (1 + φ)^{−n_d}, and Eq. (2.7b) then defines Kd to satisfy that demand, while Ksh is chosen so that e^{KT+Ksh} = 1. This is a genuine self-definitional element, but the paper is transparent about it, even calling the procedure 'Kähler potential engineering'. The central existence claim retains independent content: the paper explicitly verifies that the Kähler metric is unchanged, derives the mass spectrum and stability along the inflationary trajectory, and computes the canonical normalization. The observables ns, r and αs are evaluated from the assumed VI and kinetic term and compared with external PR4+BK18+BAO+lensing data; no observational quantity is used to define Ksh or Kd. Self-citations to Refs. [58,66,86] provide provenance for known hyperbolic Kähler geometries, but the needed metric, curvature and shift-symmetry properties are re-derived in the text and Appendix A, so the citations are not load-bearing. The printed analytic slow-roll formulas in Sec. 5.1 appear to contain internal inconsistencies (e.g., the ε denominator and the logarithm argument in IN), but these concern correctness and reproducibility, not circularity. Overall, the circularity is minor and constructional, not a logical loop, and the paper is self-contained against external benchmarks.

Assumptions & free parameters 8 free parameters · 8 assumptions · 0 invented entities

The central claim rests on standard SUGRA formalism, the holomorphic/anti-holomorphic completion trick (an ad hoc construction choice), stability assumptions verified at tree level by the paper itself, and standard slow-roll/reheating machinery with hand-set reheating inputs. Eight inputs (λ, φ⋆, N, nd, n, NS, Λ, Trh) enter the final observable ranges; M for HSI is fixed by the GUT condition (Eq. 4.15) and is the only parameter derived from an external scale. No new particles, forces, or dimensions are introduced.

free parameters (8)

λ (superpotential coupling) = 2.9e-5 to 4.7e-5 (CSI), 2.8e-5 to 23.2e-5 (HSI); Table 3
Fitted to the curvature perturbation amplitude As via Eq. (5.7) and the N⋆ constraint (5.11). It sets the overall scale of VI but not its shape.
φ⋆ (field value at horizon exit) = 0.88 to 0.98 mP (Table 3)
Chosen to satisfy the e-fold constraint N⋆ ≈ 50-60 (Eq. 5.11); equivalently selects the initial condition near the kinetic pole. The required tuning is quantified by Δ⋆ = 1-53% (Eq. 5.16a).
N (Kähler curvature parameter) = 1-180 (CSI), 1-165 (HSI n=4), 1-152 (HSI n=8)
Scanned over the data-allowed range; controls the pole strength, the Kähler curvature R (Eqs. 3.7, 4.8b) and the tensor-to-scalar ratio r.
nd (shift-symmetry breaking exponent) = 0-3.99 (CSI), 0-7.99 (HSI n=4), 0-15.99 (HSI n=8)
Scanned free parameter that shapes the denominator (1+φ)^(-nd) of VI (Eq. 1.3) and hence shifts ns from 0.961 to 0.969.
n (potential exponent) = 2 (CSI), 4 and 8 (HSI)
Chosen by hand; fixed by symmetry and holomorphy considerations (Secs. 3.1, 4.1).
NS (stabilizer manifold curvature) = 0 < NS < 6 (chosen, not fitted)
Restricted by the stability requirement m²s > 0 (Tables 1-2); not fit to cosmological data.
Λ (Coleman-Weinberg RG scale) = Λ⋆ ≈ 0.93-2.8 x 10^-5 mP (Table 3)
Determined post hoc by imposing ΔVI(φ⋆) = 0 or ΔVI(φf) = 0 (Eq. 3.10); a scale-choice freedom that removes the one-loop correction at an endpoint.
Trh (reheat temperature) = 1 EeV
Set by hand in Eq. (5.11), motivated by non-thermal leptogenesis; for CSI no decay sector is specified to justify this value.

assumptions (8)

standard math N=1 Poincaré supergravity action (Eq. 2.1) with minimal Einstein gravity and no higher-order R terms
The framework of the whole paper; standard textbook SUGRA.
ad hoc to paper Holomorphic and anti-holomorphic additions Ksh, Kd leave the Kähler metric unchanged while contributing to the e^K prefactor (Sec. 2.2.3, Eqs. 2.6-2.8)
This is the paper's central engineering trick; if the mixed derivatives of these terms were not exactly vanishing, the pole structure and VI would differ. Verified at tree level only along the trajectory.
domain assumption Stabilizer S is confined to the origin by K2 = NS ln(1+|S|²/NS) with 0 < NS < 6 (Eq. 2.2)
Taken from Ref. [35]; ensures ⟨KSS*⟩I = 1 and m²s > 0. Higher-order S terms are assumed negligible.
domain assumption The inflationary trajectory (Eqs. 3.2, 4.2) is stable and D-flat, with all non-inflaton masses above H²I (Tables 1-2)
Checked by the paper's own mass-spectrum computation along the trajectory; off-trajectory and late-time dynamics are not fully analyzed.
domain assumption Effective field theory validity for φ < 1 even though the canonical field φ̂ exceeds mP (Eq. 1.2, Sec. 5.2)
The model relies on transplanckian canonical excursions with subplanckian original field; the paper argues this is safe because VI is exponentially flat, but this is an assumption about the UV cutoff.
domain assumption R symmetry with U(1)X (plus Zn/4 for HSI with n>4) fixes W uniquely (Secs. 3.1, 4.1)
Symmetry choices are inputs to the model; they justify the monomial form of W and protect against higher-order terms.
standard math Standard slow-roll and perturbation-spectrum formulas (Eqs. 5.1, 5.8)
Single-field slow-roll inflation with canonical perturbation theory; universally accepted.
domain assumption Reheating assumptions: Trh = 1 EeV, grh* = 228.75, wrh from Eq. (5.12)
Reheating parameters are inputs (Eq. 5.11), not derived from a specific decay Lagrangian.

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Cite this review

Pith. "Pith review of Starobinsky Inflation with T-Model Kaehler Geometries." pith.science (2026). https://pith.science/paper/H52VNOT5

@misc{pith2026250200636,
  author       = {Pith},
  title        = {Pith review of: Starobinsky Inflation with T-Model Kaehler Geometries},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/H52VNOT5}},
  note         = {Machine review of arXiv:2502.00636}
}

read the original abstract

We present novel implementations of Starobisky-like inflation within Supergravity adopting Kaehler potentials for the inflaton which parameterize hyperbolic geometries known from the T-model inflation. The associated superpotentials are consistent with an R and a global or gauge U(1)_X symmetries. The inflaton is represented by a gauge-singlet or non-singlet superfield and is accompanied by a gauge-singlet superfield successfully stabilized thanks to its compact contribution into the total Kaehler potential. Keeping the Kaehler manifold intact, a conveniently violated shift symmetry is introduced which allows for a slight variation of the predictions of Starobinsky inflation: The (scalar) spectral index exhibits an upper bound which lies close to its central observational value whereas the constant scalar curvature of the inflaton-sector Kaehler manifold increases with the tensor-to-scalar ratio.

Discussion (0). Continue with ORCID to comment.

Forward citations

Cited by 3 Pith papers

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

Kinetically Modified Palatini Inflation Meets ACT Data
hep-ph 2025-05 conditional novelty 6.0 of 10

Palatini chaotic inflation with kinetic mixing f_K = f_R^m can shift the predicted spectral index up to the ACT DR6 value n_s = 0.974 while keeping r below current bounds.
Induced-Gravity Palatini-Like Higgs Inflation in Supergravity Confronts ACT DR6
hep-ph 2026-02 unverdicted novelty 5.0 of 10

A Palatini-supergravity Higgs-inflation model with induced gravity predicts a scalar spectral index ns≈0.972-0.974, consistent with ACT DR6, and favors split supersymmetry with gravitino mass 40-60 PeV.
Shifted Hybrid Realization of Non-Minimal Higgs Inflation in Light of ACT DR6 and Planck Data
hep-ph 2026-07 conditional novelty 4.0 of 10

A shifted-hybrid non-minimal Higgs inflation model deforms the Starobinsky attractor via a leading non-renormalizable superpotential operator to raise ns into the ACT-preferred range while keeping r ~ 10^{-3}-10^{-2}.

Reference graph

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