REVIEW 4 major objections 5 minor 1 cited by
Scalar field source Teleparallel Robertson-Walker F(T)-gravity solutions
T0 review · 4 major / 5 minor · reviewed 2026-08-10 · deepseek-v4-flash
Pith's one-line read This paper derives master integral formulas that yield exact teleparallel $F(T)$ functions for scalar-field Robertson--Walker cosmologies, including new non-flat analytic cases.
desk verdict A useful k=0 reconstruction formula and a promising framework, but the non-flat solutions are unverified and at least one explicit formula is non-real on its domain; needs revision before use. read the letter →
The pith
A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.
The reading
What carries the argument
The load-bearing object is the unified field equation obtained by combining the two Friedmann-like TRW equations; for $k=0$ it takes the form $-\sqrt6\,\kappa\dot\phi^2/2 = \partial_t(\sqrt T\,F_T)$, which integrates to the master formula (21). For $k=\pm1$, the analogous combinations (29) and (52) integrate to the general solutions (30) and (53). The practical machinery is the power-law ansatz $a(t)=a_0 t^n$, the torsion-scalar relations (19), (28), and (51) that convert $t$ into $T$, and the characteristic equations (32) and (55) that decide which values of $n$ allow the integrals to close in elementary or special functions. The scalar potential never enters the construction; Equation (15) is reserved for reconstructing $V(\phi)$ afterwards.
What would settle it
Take one of the displayed solutions, for example the logarithmic field $\phi(t)=p_0\ln t$ in Equation (24), substitute it into the conservation equation (15), integrate to obtain $V$ as a function of $t$, then invert $t$ in terms of $\phi$. If the resulting $V(\phi)$ is multi-valued on the relevant domain, or if the pair $(V,F)$ fails to satisfy the original field equations (17)--(18), then the listed solution is not a valid scalar-field model.
Extended reading notes
Core claim
On the paper's own terms, the discovery is that a scalar-field source combines with the teleparallel Robertson--Walker field equations into a single differential relation between $F(T)$ and $\dot\phi^2$, namely Equations (20), (29), and (52), so that $F(T)$ is determined by an iterated integral whose integrand contains only the kinetic term $\dot\phi^2(t(T))$. With the power-law scale factor $a(t)=a_0 t^n$, the flat case yields the closed two-integral formula (21), which reproduces the known perfect-fluid solution plus a scalar-field contribution. For $k=\pm1$, the same combination produces the general formulas (30) and (53), and for the specific exponents $n=\frac12,1,2$ and the very large-$n$ limit the associated characteristic equations become solvable, yielding explicit analytic $F(T)$ for power-law, logarithmic, and exponential scalar fields. The paper states that these solutions go beyond previous TRW results and are intended as building blocks for quintessence, phantom, and quintom dark-energy models.
Load-bearing premise
The construction assumes one may prescribe any time profile $\phi(t)$ and then define the potential $V(\phi)$ through the conservation equation, without checking whether that potential is single-valued and physically admissible, or whether it is consistent with the resulting $F(T)$.
Editorial extensions
If this is right
- For flat TRW cosmology, any scalar-field profile combined with a power-law scale factor yields an explicit $F(T)$ from Equation (21), so model builders can generate teleparallel dark-energy models without fixing the potential first.
- In the very large-$n$ limit, the $k=-1$ and $k=+1$ solutions reduce to the flat $k=0$ solutions, so fast-expansion non-flat universes are governed by the same $F(T)$ as flat ones.
- The analytic non-flat solutions exist only for the special values $n=\frac12,1,2$ and for specific $\phi(t)$ profiles, giving a concrete catalogue of exact open and closed scalar-field teleparallel cosmologies.
- Each solution carries a quintessence index $\alpha_Q$ from Equation (16), so the paper's models can be sorted into quintessence, phantom, cosmological-constant, or quintom regimes for later comparison with data.
- The plotted $F(T)$ curves can be fitted to cosmological observations, which would determine the parameters $n$, $p$, and $p_0$ of the scalar-field source.
Reading between the lines
- Editorial extension: because the derivation only needs $\dot\phi^2$, the same integral formulas should extend to non-canonical or multi-field sources, making a two-field teleparallel quintom model a direct next target.
- Editorial extension: the paper never verifies $V(\phi)$ for its examples, so checking that the reconstructed potential is single-valued and admissible should be the first test before any listed function is used in a cosmological fit.
- Editorial extension: the large-$n$ equivalence suggests curvature corrections to $F(T)$ are suppressed at late times, which if true would make flat-universe observational constraints approximately valid for near-flat fast-expanding universes.
Signed reviews
Editorial analysis
A structured set of objections, weighed in public.
Referee Report
Summary. The paper studies teleparallel Robertson-Walker (TRW) F(T) gravity with a minimally coupled scalar field source. For the flat case k=0, the author combines the two Friedmann equations into a single equation for F(T) and gives a general quadrature formula (Eq. 21) that yields F(T) for any prescribed scalar field profile under a power-law scale factor. Explicit solutions are then displayed for power-law, special power-law, logarithmic, and exponential scalar fields (Eqs. 22-25). For the non-flat cases k=-1 and k=+1, the paper states general integral formulas (Eqs. 30 and 53) and derives many analytic F(T) solutions for special values of the scale-factor exponent n and special scalar-field profiles (Eqs. 34-48 and 57-72). The author emphasizes that the solutions are independent of the explicit form of the scalar potential V(φ) and discusses their possible relevance for quintessence, phantom, and quintom dark-energy models.
Significance. If all displayed formulas were correct and validated, the k=0 quadrature formula (Eq. 21) would be a useful and easy-to-evaluate tool for constructing scalar-field F(T) cosmologies, and the non-flat solutions would supply a library of exact models for future dark-energy studies. The paper has the merit of making the flat-case derivation explicit and of presenting a wide range of analytic examples with the stated n-parameter interpretation. However, the non-flat library is currently not established: the general formulas (30) and (53) are asserted without derivation, none of the displayed formulas is verified by substitution, and at least one announced k=-1 solution (Eq. 36) is not real on the domain T>=0 required by the geometry. The significance of the claimed library therefore remains uncertain until these issues are resolved.
major comments (4)
- [Section IV, Eq. (36)] For k=-1, the torsion scalar T defined in Eq. (28) is 6(H+δ√(-k)/a)^2 and is therefore nonnegative for all t and for both sign choices δ. The displayed n=1/2, p=1 solution (Eq. 36) contains the term ln(-2T), which is not real for any T>0, and the accompanying logarithmic ratio is undefined on part of the admissible T-range as well. Consequently Eq. (36) is not a real F(T) solution of the stated k=-1 scalar-field system. The author must correct this expression or explicitly restrict to a domain and branch on which the formula is real, and then verify that it satisfies the original field equations on that domain.
- [Sections IV and V, Eqs. (30) and (53)] The general solutions (30) and (53) are introduced without derivation from the first-order linear ODEs (29) and (52). It is not shown how the integrating factor is computed, how the inverse relation t(T) from the characteristic equations (32) and (55) is inserted, or how the displayed special cases follow. Since every subsequent non-flat formula depends on these two expressions, a full derivation and at least one worked substitution check for each of k=-1 and k=+1 are needed before the claimed solution library can be assessed.
- [Section II.C and Sections III-V] The method eliminates V by combining the two Friedmann equations. For a prescribed φ(t), the existence of a single-valued potential V(φ) requires φ(t) to be monotonic on the interval of interest, and the Klein-Gordon equation (15) must be satisfied. The paper never states the monotonicity condition, never computes V for any of the listed examples, and never checks Eq. (15) explicitly. Although solving the summed equation is a necessary step and can be made sufficient by defining V from one of the original equations, the author should prove this consistency for at least one representative solution in each k-sector and specify the t- and T-domains for which the solution is valid.
- [Sections IV and V, Eqs. (34)-(48) and (57)-(72)] Many non-flat formulas contain square roots of expressions such as 1+δ1√(2T/3) or 1+δ2√(1+T/6) and logarithms of quantities that can vanish or change sign on the geometric T-domain. For example, for k=+1 Eq. (51) gives T=6(H^2-1/a^2), which can be negative for small a, while Eq. (68) contains (-T)^(-1/2) and presumes T<0. The paper should state the admissible range of T for every displayed solution and confirm that the function is real, finite, and differentiable on that range; without this, the classification into quintessence/phantom models is premature.
minor comments (5)
- [Throughout] The manuscript contains many typesetting artifacts, including 'TR W' instead of 'TRW', stray spaces in 'F (T )', 'Th e' in the abstract, 'f or' in the caption of Figure 1, and inconsistent spacing in references. A careful copy-edit is needed.
- [Equation (32)] The symbol δ1 is used without prior definition; δ2 is defined as ±1 in Eq. (33), but δ1 also appears to be a sign choice and should be defined at first use.
- [Figure 2 caption] The caption states δ√(-k)/a0=2, whereas the n=1/2 and n=2 subcases in the text (Eqs. 35-36 and 47-48) are derived under δ√(-k)/a0=1. The parameter values used in the plots should be made consistent with the text.
- [Section III, Eq. (21)] Equation (21) is said to apply for 'any scalar field source', but the double integral requires the prescribed φ(t) to be invertible as a function of T and the integrals to converge; these conditions should be stated explicitly.
- [Section VI] The concluding remark that the solutions are 'new' is not supported by a comparison with the existing reconstruction literature beyond refs. [7,9]; a short comparison would help the reader judge the novelty.
Circularity Check
No significant circularity: F(T) is obtained by solving merged field equations, and the author's prior work supplies only the input geometry/equations, not the claimed solutions.
full rationale
The derivation chain is self-contained in the relevant sense: the new F(T) formulas are obtained by merging the two Friedmann equations to eliminate V and then solving the resulting first-order differential equations, Eqs. (20), (29), and (52), for the unknown function F. The integration constants are arbitrary constants of integration, not fitted parameters that are later relabeled as predictions. The scalar potential is not an independently specified input; the paper explicitly states that Eq. (15) 'yields to a scalar potential V(phi(t))' for any chosen phi(t), so the method is a reconstruction from a prescribed phi(t) rather than a prediction from a specified V. That raises validity and domain questions, for instance the apparent ln(-2T) issue in Eq. (36) on the T >= 0 domain of the k=-1 case, but this is a correctness/reality concern, not circularity: the output F(T) is not used to define phi or V, and the merged equations are not identical to the final formulas. The TRW field equations and coframe/spin-connection pairs are cited from refs. [8,9], which include the author, but these are prior derivations of the input geometry and field equations, not of the F(T) solutions claimed here, and the same equations appear in the broader teleparallel literature. No uniqueness theorem is invoked, no ansatz is smuggled in via citation, and no known result is merely renamed. The central claim is therefore independent of the cited results, and no circular step is present.
Assumptions & free parameters
free parameters (7)
- scale factor exponent n =
1/2, 1, 2, large n (20 in plots)
- scale factor amplitude a0 =
fixed through ratios such as δ√-k/a0 = 1 or 2, k/a0^2 = 0.25, 1, 2
- scalar field amplitude p0 =
κ p0^2/(2B) = 1 in plots
- scalar field exponent p =
0.75, 1, 1.3, -0.5, -1, -1.5
- exponential rate p for φ = p0 exp(p t) =
not numerically fixed
- integration constants Λ0, B, C1 =
free
- sign choices δ, δ1, δ2 =
±1
assumptions (5)
- domain assumption Teleparallel F(T) field equations (2)-(3) with null hypermomentum and standard scalar-field energy-momentum (13)-(14)
- domain assumption TRW coframe/spin-connection pair (9)-(10) with W1,W2 assignments per k is the correct proper frame
- domain assumption Scalar field conservation law Eq (15) is valid for any φ(t) and can be used to define V(φ)
- ad hoc to paper Power-law ansatz a(t) = a0 t^n and prescribed φ(t) forms
- ad hoc to paper For k=-1 and +1, analyticity requires restricting n and ratios like δ√-k/a0 and k/a0^2 to chosen values
Cite this review
Pith. "Pith review of Scalar field source Teleparallel Robertson-Walker F(T)-gravity solutions." pith.science (2026). https://pith.science/paper/K4NXW26F
@misc{pith2026250113895,
author = {Pith},
title = {Pith review of: Scalar field source Teleparallel Robertson-Walker F(T)-gravity solutions},
year = {2026},
howpublished = {\url{https://pith.science/paper/K4NXW26F}},
note = {Machine review of arXiv:2501.13895}
}
abstract
This paper investigates the teleparallel Robertson--Walker (TRW) $F(T)$ gravity solutions for a scalar field source. We use the TRW $F(T)$ gravity field equations (FEs) for each $k$-parameter value case added by a scalar field to find new teleparallel $F(T)$ solutions. For $k=0$, we find an easy-to-compute $F(T)$ solution formula applicable for any scalar field source. Then, we obtain, for $k=-1$ and $+1$ situations, some new analytical $F(T)$ solutions, only for specific $n$-parameter values and well-determined scalar field cases. We can find by those computations a large number of analytical teleparallel $F(T)$ solutions independent of any scalar potential $V(\phi)$ expression. The $V(\phi)$ independence makes the FE solving and computations easier. The new solutions will be relevant for future cosmological applications in dark matter, dark energy (DE) quintessence, phantom energy and quintom models of physical processes.
Figures
Forward citations
Cited by 1 Pith paper
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Traversable Wormhole Solutions in massive $F(T)$ gravity
In massive F(T) gravity, exact traversable wormhole solutions exist whose throats are supported by the graviton mass term, with matter that satisfies standard energy conditions in selected parameter ranges.
Reference graph
Works this paper leans on
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[1]
Quintessence −1 <αQ< − 1 3 : This describes a controlled accelerating universe expansion where energy conditions are always satisfied, i.e., Pϕ + ρϕ > 0 [29–31, 36]
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[2]
k = +1: W1 = 0 and W2(t) = ± √ k a(t) ,
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[3]
For any W1 and W2, we will obtain the same symmetric FEs set to solve for each subcase depending on the k-parameter
k = −1: W1(t) = ± √ −k a(t) and W2 = 0. For any W1 and W2, we will obtain the same symmetric FEs set to solve for each subcase depending on the k-parameter. The previous coframe and spin-connection expressions were found by solving Equation (8) and imposing the null Riemann curvature condition (i.e., Ra bµν = 0, as stated in ref. [10]). These solutions we...
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[4]
The energy condition is violated, i.e., Pϕ + ρϕ ≱ 0
Phantom energy αQ< −1: This usually describes an uncontrolled universe expansion accelerating toward a Big Rip event [42–44]. The energy condition is violated, i.e., Pϕ + ρϕ ≱ 0
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[5]
A constant scalar field source ϕ = ϕ0 will directly lead to this case, according to Equation (16)
Cosmological constantαQ= −1: This is an intermediate limit between the two previous and main types of DE, where Pϕ + ρϕ = 0. A constant scalar field source ϕ = ϕ0 will directly lead to this case, according to Equation (16)
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[6]
Z dT ′ T ′−1/2
Quintom models: This is a mixture of previous DE types, usually described by some double scalar field models [49–53]. By using Equation (16), we can find the perfect fluid equivalent for any new teleparallel F (T ) solution, potential V (ϕ) or ansatz. Equation (16) is useful for making new teleparallel F (T ) solution classifications in terms of DE quinte...
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[7]
Power law general:For ϕ(t)= p0 tp and p ̸= 1 2 , we find F (T ) = −Λ0 + B √ T + √ 6 κ p2 0 p2(6n2)p− 1 2 2(2p − 1)(p − 1) T 1−p. (22)
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[8]
Power law special:For ϕ(t)= p0 t 1 2 , we find F (T ) = −Λ0 + B √ T + √ 6 κ p2 0 8 √ T ln (T ). (23)
Show all 86 references
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[9]
Logarithmic: For a field defined as ϕ(t)= p0 ln (t), we find F (T ) = −Λ0 + B √ T + κ p2 0 2n T. (24)
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[10]
There are several other possible ϕ(t) source terms which may lead to additional new teleparallel F (T ) solutions by using Equation (21)’s general formula
Exponential: For ϕ(t)= p0 exp (p t), we find F (T ) = −Λ0 + B √ T − √ 6κ 2 p2 0 p " 2 √ 6 np Ei1 − 2p √ 6 n√ T ! + √ T exp 2p √ 6 n√ T !# , (25) where Ei1(x) is an exponential integral function. There are several other possible ϕ(t) source terms which may lead to additional ne...
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[11]
This is a ”Big Crunch”-type scenario for a large |n|-parameter scenario
n < 0: Contracting universe. This is a ”Big Crunch”-type scenario for a large |n|-parameter scenario
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[12]
This is the limit between expanding and contracting universe sce- narios
n = 0: Static universe. This is the limit between expanding and contracting universe sce- narios
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[13]
0 < n < 1: This is slow and controlled universe expansion, but a non-inflationary scenario
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[14]
n = 1: This is linear universe expansion and the limit between slow and fast universe ex- pansion scenarios
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[15]
This is a plausible dark energy quintessence case because it is an inflationary scenario
1 < n < ∞: The is fast, inflationary and controlled universe expansion. This is a plausible dark energy quintessence case because it is an inflationary scenario
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[16]
This strong inflationary case is so far the best phantom dark energy scenario description and leads to ”Big Rip” singularity after a determined time delay
n ≫ 1 or n → ∞: This is very fast and uncontrolled universe expansion. This strong inflationary case is so far the best phantom dark energy scenario description and leads to ”Big Rip” singularity after a determined time delay. For F (T ) FEs solutions for Equations (22)–(24), ...
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[17]
n = 1 2 : Equation (32) becomes 0 =t−1 + 2δ √ −k a0 t− 1 2 − δ1 r 2T 3 , ⇒ t− 1 2 (T ) = − δ √ −k a0 + δ2 s − k a2 0 + δ1 r 2T 3 , (33) 11 where δ2 = ±1. Then, Equation (31) for the F (T ) solution is F (T ) = − Λ0 + Z dT " C1 − κ δ1 r 3 2 Z t(T ) dt′ ˙ϕ2(t′) exp 2δ √ −k a0 t′...
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[18]
Z t(T ) dt′ exp 2p t′ t′ δ √−k a0 # . (42) There is no general solution for Equation (42). However, there are specific solutions: • δ √ −k a0 = 1: F (T ) = − Λ0 + B T − κ p2 0 16
n = 1: Equation (32) becomes 0 =δ1 r T 6 − 1 + δ √ −k a0 t−1, ⇒ t(T ) = δ1 1 + δ √ −k a0 √ 6√ T , (37) Equation (31) for the F (T ) solution becomes F (T ) = − Λ0 + B T δ √−k 2a0 + 1 2 − δ1 √ 6κ 2 δ1 √ 6 1 + δ √ −k a0 δ √−k a0 Z dT T δ √−k 2a0 − 1 2 "Z t(T ) dt′ ˙ϕ2(t′) t′ δ √...
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[19]
C1 − δ1 κ √ 6 2 Z t(T ) dt′ ˙ϕ2(t′) exp − δ √ −k a0 t′−1 ! × exp
n = 2: Equation (32) becomes 0 =t−2 + 2δa0√ −k t−1 − δ1δa0 r − T 6k , ⇒ t−1(T ) = − δa0√ −k + δ2 s δ1δa0 r − T 6k − a2 0 k , (45) where δ2 = ±1. Equation (31) becomes F (T ) = − Λ0 + Z dT " C1 − δ1 κ √ 6 2 Z t(T ) dt′ ˙ϕ2(t′) exp − δ √ −k a0 t′−1 ! × exp " −1 + δ2 r 1 + δ1 δ a...
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[20]
From this point, we obtain Equation (21)’s exact formula and then Equations (22)–(25) as F (T ) solutions under the large n limit
n ≫ 1: Equation (32) leads to t(T ) = n q 6 T with δ1 = 1. From this point, we obtain Equation (21)’s exact formula and then Equations (22)–(25) as F (T ) solutions under the large n limit. The n ≫ 1 solutions are the same as for the k = 0 flat cosmological case and the graphs...
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[21]
1 + δ2 r 1 + T 6 ! exp 4 1 + δ2 r 1 + T 6 !−1 + Ei1 −4 1 + δ2 r 1 + T 6 !−1 # + 3κ p2 p2 0 2
n = 1 2 : Equation (55) becomes 0 =t−2 − 4k a2 0 t−1 − 2T 3 , ⇒ t−1(T ) = 2k a2 0 + 2δ2 s k2 a4 0 + T 6 , . (56) where δ2 = ±1. Equation (54) becomes F (T ) = − Λ0 + Z dT " C1 − κ a2 0 2k Z t(T ) dt′ ˙ϕ2(t′) exp − 2k t′ a2 0 # exp 4 1 + δ2 q 1 + a4 0 6k2 T −1! 1 + δ2 q 1 + a4 ...
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[22]
(61) 18 Equation (54) becomes F (T ) = − Λ0 + B T 1 2 − k 2a2 0 + κ 2 6 1 − k a2 0 k 2a2 0 + 1 2 × Z dT " Z t(T ) dt′ ˙ϕ2(t′) t ′− k a2 0 # T − 1 2 − k 2a2 0
n = 1: Equation (55) becomes 0 =T − 6 1 − k a2 0 t−2, ⇒ t2(T ) = 6 1 − k a2 0 T . (61) 18 Equation (54) becomes F (T ) = − Λ0 + B T 1 2 − k 2a2 0 + κ 2 6 1 − k a2 0 k 2a2 0 + 1 2 × Z dT " Z t(T ) dt′ ˙ϕ2(t′) t ′− k a2 0 # T − 1 2 − k 2a2 0 . (62) Equation (62) yields new analy...
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[23]
1 + δ2 s 1 − k 24a2 0 T # , (69) where δ2 = ±1. Equation (54) becomes F (T ) = − Λ0 + Z dT
n = 2: Equation (55) becomes 0 = a2 0T 6k − 4a2 0 k t−2 + t−4, ⇒ t−2(T ) = 2a2 0 k " 1 + δ2 s 1 − k 24a2 0 T # , (69) where δ2 = ±1. Equation (54) becomes F (T ) = − Λ0 + Z dT " C1 − κ 4 Z t(T ) dt′ ˙ϕ2(t′) exp k t′−2 4 a2 0 # " 2a2 0 k " 1 + δ2 s 1 − k 24a2 0 T ##−1/2 × exp −...
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[24]
From this point, we obtain the exact Equation (21) formula and then Equations (22)–(25) as F (T ) solutions for the large values of n
n ≫ 1: Equation (55) leads to t(T ) = n q 6 T with δ2 = 1. From this point, we obtain the exact Equation (21) formula and then Equations (22)–(25) as F (T ) solutions for the large values of n. The n ≫ 1 solutions are the same as for the k = 0 and −1 cases and the graphs are s...
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