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REVIEW 5 major objections 4 minor 28 references

Uniqueness of star central configurations in the $5$-body problem

T0 review · 5 major / 4 minor · reviewed 2026-08-07 · deepseek-v4-flash

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Pith's one-line read This paper proves that the unique star central configuration for five equal masses in the planar Newtonian problem is the regular pentagon.

desk verdict A serious attempt at a human-checkable proof of a known result, but the proof is incomplete and contains a false numerical comparison in the only worked-out region. read the letter →

arxiv 2505.16101 v1 pith:HMKKEJ6A submitted 2025-05-22 math-ph math.MP

Leasly A. Campa-Raymundo , Luis Franco-Pérez This is my paper · ORCID
classification math-phmath.MP MSC 70F1070F15
keywords centralconfigurationsfive-bodyproblemstarregularpentagonequalmassesuniquenessplanarn-bodypolygonal
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 aims to prove that, among configurations of five equal masses with the bodies placed at equal angular steps around the center of mass (star configurations), the only central configuration is the regular pentagon. It reduces the central-configuration equations to a two-variable system in the radial parameters $r_3$ and $r_5$, then splits the admissible domain into sixteen regions and rules out every non-regular point by showing that two components of the multiplier vector cannot coincide. A reader should care because this gives an algebraic uniqueness statement for a geometrically constrained subclass of the five-body problem, where prior classifications were computer-assisted and covered a larger class. If the proof is correct, the regular pentagon is not just one solution but the unique solution under this symmetry.

What carries the argument

The load-bearing object is the reduced system (4.2), obtained after fixing $q_1=(1,0)$ and equal angular positions $0,2\pi/5,\ldots$; it expresses the central-configuration multipliers as functions $\lambda_{ik}(r_3,r_5)$ and asks when all components agree, normalized by $\lambda_{12}=0$. The admissible domain is $\hat S=\{(r_3,r_5)\in\mathbb R^2: r_3>0,\ r_5>0,\ r_5>r_3-b/2,\ r_5>(ar_3-a)/2\}$ with $a=\sqrt5+1$, $b=\sqrt5-1$. The proof machinery is the split of $\hat S$ into sixteen regions $J_1,\dots,J_{16}$, together with auxiliary function families (written as $e_{i\eta}$ and $f_{i\eta}$) whose monotonicity, convexity, and crossing properties yield strict inequalities between distinct $\lambda$ components on each region.

What would settle it

Evaluate the reduced system (4.2) on a fine grid over $\hat S$; any point with $(r_3,r_5)\neq(1,1)$ where all required $\lambda$-components coincide would falsify uniqueness. More narrowly, checking the asserted bounds in Properties 5.4 to 5.9, such as $\max e_3 \le 1.0696$ or $r_5^*(0)=0.417957$, with an independent calculation would settle whether the $J_1$ argument and its deferred siblings are sound.

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Extended reading notes

Core claim

The central claim is Theorem 5.2 combined with Theorem 5.1: for five equal masses, any star central configuration in the plane must have $r_3=r_5=1$, i.e. the bodies form a regular pentagon, and that configuration indeed minimizes the configuration measure $IU^2$. The proof writes positions in polar coordinates with $q_1=(1,0)$ and angular spacing $2\pi/5$, derives the reduced system (4.2) in the two variables $r_3,r_5$, and then partitions the domain $\hat S$ into sixteen regions. In each region the authors exhibit a pair of $\lambda$-components that cannot be equal; the region $J_1$ is shown in detail through monotonicity and convexity properties of auxiliary functions, while the other fifteen regions are said to follow from the same strategy with details in supplementary material.

Load-bearing premise

The conclusion depends on the correctness and completeness of the fifteen case analyses ($J_2$ through $J_{16}$) that are deferred to a supplementary file not included with the paper, together with the asserted numerical bounds used for region $J_1$; if any of those deferred estimates is wrong or missing, the uniqueness theorem is not established.

Editorial extensions

If this is right

  • Every star central configuration of five equal masses in the plane is homothetic and rotationally equivalent to the regular pentagon.
  • The uniqueness statement is an analytic counterpart to the earlier computer-assisted classification of equal-mass five-body central configurations, covering exactly the star subclass without interval-arithmetic computations.
  • Combined with the known three- and four-body cases, the result completes the pattern that star configurations are regular polygons for $n\le 5$, whereas the same uniqueness is false for $n\ge 6$.
  • The regular pentagon is a genuine solution (Theorem 5.1), so existence and uniqueness coincide for this constrained family.

Reading between the lines

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

  • Because the text explicitly says that only region $J_1$ is worked out and that the rest is deferred, a careful reader should treat Theorem 5.2 as conditional on the promised supplementary case analyses; the uniqueness claim is not fully self-contained in the submitted text.
  • The same two-variable reduction could be run for six equal masses with equal angular spacing, where known nested-triangle configurations suggest uniqueness will fail; a direct analogue of the $J_1$ inequality argument could locate where the pattern breaks.
  • If the deferred regions are supplied and verified, the algebraic approach would give a template for proving uniqueness in other constrained families, such as equilateral pentagons or cyclic equal-edge chains, without computer assistance.
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Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

5 major / 4 minor

Summary. The paper claims a fully analytical proof that, for five equal masses in the Newtonian planar five-body problem, the only star central configuration—five bodies at equal angular steps around the center of mass—is the regular pentagon. The authors reduce the governing equations to a two-variable system (4.2), split the domain into sixteen regions, and aim to show by case analysis that no non-pentagonal solution exists. Only region J1 is treated in detail in the submitted text; the remaining regions are deferred to supplementary material that is not present in the submission.

Significance. If the main theorem were established rigorously, it would be a valuable analytic complement to the computer-assisted classification of Moczurad and Zgliczynski and would strengthen the known result that for n≤5 the only star central configuration is the regular polygon. The intended result is plausible and consistent with existing literature. However, the submitted proof is not verifiable in its current form: the central reduction is not derived, the auxiliary functions are not defined, most of the sixteen-region case analysis is absent, and the one fully displayed argument contains a numerically contradictory comparison. As submitted, the paper does not meet the standard of a rigorous analytical proof.

major comments (5)
  1. [Section 4, Eq. (4.2)] The derivation of the reduced system (4.2) is not presented. The functions λik(r3,r5) are not defined explicitly, the condition λ12=0 is introduced without justification, and the equivalence between solutions of (4.2) and central configurations of the original five-body problem is not proved. The domain S-hat is declared with inequalities involving a=√5+1 and b=√5−1, but no derivation of these inequalities is given. Because all later estimates operate on this system and domain, the proof is not self-contained.
  2. [Section 5.1, definition before Properties 5.4] The parameterization λikη(r5):=λik(b/2+η, r5) with η∈(0,∞) does not cover the stated region J1={0<r3≤b/2, 0<r5≤b/2}; for η>0 one has r3=b/2+η>b/2. Throughout Section 5.1 the proofs use η∈[0,0.02] or η∈(0.02,∞), yet the propositions are asserted to hold for region J1. This mismatch means the displayed J1 analysis is not applicable to the region it claims to treat.
  3. [Section 1 and Section 5.2, Theorem 5.2] Theorem 5.2 requires excluding solutions in all sixteen regions J1–J16, but the text states that only J1 is shown and that the details for J2–J16 are in supplementary material; no such supplement is included in the submission. The abbreviated arguments for Propositions 5.17–5.29 contain unsupported assertions such as "We can verify" and "strictly positive" without explicit computations or derivations. The central uniqueness claim therefore rests on absent material and is not established as submitted.
  4. [Proposition 5.13, Part IIb] The printed numerical comparison contradicts the claimed inequality. For r5∈(0.5,0.54), the proof states that the maximum value of L166η(0.5) is L1660.037(0.5)=18.40326 and that (dηf20.037)''(0.54)=9.60098. Since 18.40326>9.60098, these numbers cannot support the asserted conclusion L166η(r5)<(dηf2η)''(r5). The subsequent sentence "In all cases, L166η(r5)<(dηf2η)''(r5)" is therefore unsupported. This proposition is used to establish convexity, which feeds into Propositions 5.14 and 5.16, so the J1 exclusion is not proved.
  5. [Properties 5.4–5.9] The properties are asserted "through straightforward computation" with decimal bounds such as e3η(r5η)≤1.0696, r*5(0)=0.417957, and r̂5(0)=0.156497, but no derivations or interval certificates are provided. The underlying functions e_iη and f_iη are never defined in the paper. Since the paper claims an entirely analytical proof but relies on these unverified decimal assertions, the rigor of the subsequent estimates cannot be assessed.
minor comments (4)
  1. [Introduction and Section 1] The phrase "can be consulted oin the supplementary material" contains a typo; it should read "in the supplementary material."
  2. [Section 6] The conclusion states that "The proof of Theorem 5.1 follows from the propositions discussed in Subsection 5.2," but the propositions in Subsection 5.2 concern the uniqueness assertion of Theorem 5.2, not the existence statement of Theorem 5.1.
  3. [References] The citation numbering is inconsistent: Euler's work is cited as [28] in the introduction, while [28] in the reference list is a 2015 paper by Zhao and Chen; the reference for Euler's paper should be corrected.
  4. [Abstract] The abstract states the approach is "entirely analytical, relying on algebraic techniques rather than numerical approximations," but the proof uses many decimal numerical bounds without rigorous interval arithmetic or rational-certified estimates; the wording should be clarified.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: the uniqueness claim is derived from the Newtonian central-configuration equations, not from its own conclusion; deferred verifications and numerical gaps are correctness concerns, not circularity.

full rationale

The derivation is not circular in any of the relevant senses. Star central configurations are defined in Definition 2.2 by equal angular spacing 2π/n with variable radii, and the reduced system (4.2) is obtained by substituting the polar-coordinate ansatz (2.3) into the Newtonian equations (4.1); no parameter is fitted to the target result and no prediction is renamed from data. The regular pentagon is checked locally by an explicit Hessian computation in Theorem 5.1, and the uniqueness claim is benchmarked against the independent, external computer-assisted classification of Moczurad and Zgliczynski [15], which is cited for context rather than used as the load-bearing argument. The paper's genuine weaknesses are incompleteness and possible numerical error, not circularity: Section 1 states 'the details for the rest of the subregions can be consulted in the supplementary material' while only J1 is shown; Properties 5.4–5.9 are asserted 'through straightforward computation' without derivations or interval certificates; and Proposition 5.13 contains a suspicious comparison in which L1660.037(0.5) = 18.40326 is used to bound (dηf2 0.037)''(0.54) = 9.60098, with 18.40326 > 9.60098. These are gaps in rigor or possible errors, but they do not amount to the conclusion being assumed in the hypotheses. There are no self-citations, no imported uniqueness theorem from the authors' prior work, and no definition that encodes the regular pentagon by construction. Hence the circularity score is 0.

Assumptions & free parameters 3 free parameters · 4 assumptions · 0 invented entities

The paper introduces no invented entities and fits no data. The load-bearing structure is: (i) standard Newtonian model and central configuration definitions; (ii) the star ansatz of equal angular spacing; (iii) an unshown reduction to two variables (r3, r5) with a domain S-hat asserted without derivation; (iv) a long chain of numerical bound properties (Properties 5.4-5.9) and sixteen region verifications, fifteen of which are deferred to missing supplementary material. The decimal constants throughout Section 5 function as ad hoc bound certificates with no exact arithmetic backing, which is what makes the proof unverifiable as submitted.

free parameters (3)
  • Monotonicity splitting points r*_5(eta) and r-hat_5(eta) = r*_5(0)=0.417957, r*_5(0.02)=0.397798, r-hat_5(0)=0.156497, r-hat_5(0.02)=0.148828
    Numerically computed thresholds in Properties 5.5 that split the r5 domain into monotonicity regimes for the auxiliary functions f3 and f4; used in the proof of Proposition 5.15.
  • Decimal bound constants for auxiliary functions = For example, max e3 <= 1.0696; 3.413203, 1.90132, 0.83961, 3.51384, 1.92656, 1.05376 in Proposition 5.10
    Decimal bounds asserted 'by straightforward computation' in Properties 5.4 and Propositions 5.10-5.16; they anchor the inequality comparisons but have no exact algebraic verification in the text.
  • Lower bound constant in Proposition 5.28 = 0.00093769
    A tight numerical lower bound for L127(r3)-lambda_52(r3); its verification is not shown and it is used to conclude lambda_11 > lambda_52 on a region.
assumptions (4)
  • standard math Newtonian gravitational equations and central configuration definitions (Eqs. 2.1-2.2)
    Background model for the problem; not in question.
  • domain assumption Star ansatz: equal angular spacing theta_i = 2*pi*(i-1)/n with free radial distances
    Definition 2.2 restricts the studied subclass of configurations.
  • domain assumption Center-of-mass reduction to two variables (r3, r5) and the domain S-hat with inequalities r5 > r3 - b/2 and r5 > (a/2)(r3 - 1)
    Section 4 asserts (without derivation) that the 5-body equations reduce to the pair (r3, r5) and states the domain S-hat; the elimination of the remaining radii and the origin of the domain inequalities are not shown.
  • ad hoc to paper Correctness of Properties 5.4-5.9 and of the J2-J16 region verifications
    The main theorem's proof depends on these unproved monotonicity claims and on 15 region proofs deferred to missing supplementary material.

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Cite this review

Pith. "Pith review of Uniqueness of star central configurations in the $5$-body problem." pith.science (2026). https://pith.science/paper/HMKKEJ6A

@misc{pith2026250516101,
  author       = {Pith},
  title        = {Pith review of: Uniqueness of star central configurations in the $5$-body problem},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/HMKKEJ6A}},
  note         = {Machine review of arXiv:2505.16101}
}
read the original abstract

In this study, we present a rigorous analytical proof of the uniqueness of central configurations for the five-body problem, assuming that all five masses are equal and positioned at the vertices of a planar polygon. We consider configurations in which the bodies are equally spaced in angular position relative to the center of mass, and aim to determine whether a central configuration arises under these constraints. We prove that the only central configuration that satisfies these conditions occurs when the five bodies form a regular pentagon. Our approach is entirely analytical, relying on algebraic techniques rather than numerical approximations. By transforming the governing equations into a reduced system involving only two variables, we analyze the solution space over a significant and carefully bounded domain. This domain is divided into sixteen disjoint regions, within which we rule out additional solutions through explicit algebraic arguments. Our results confirm that the regular pentagonal configuration is the only central configuration in this symmetric five-body scenario.

Figures

Figures reproduced from arXiv: 2505.16101 by the authors.

Figure 1
Figure 1. domain Sˆ 5. Uniqueness The main objective of this paper is to demonstrate that a solution to the SCC within the domain Sˆ, which satisfies equation (4.2), corresponds exclusively to a regular pentagon and not to any other configuration. This finding leads to two important conclusions: first, it confirms that the regular pentagon is indeed a valid solution, and second, it establishes the uniqueness of this solution.… view at source ↗
Figure 2
Figure 2. regions J1, ..., J16. 5.1. Properties of auxiliary functions of J1 Definition 5.3. A family of functions fα(r) with α ∈ I is strictly increasing with respect to the parameter α if for α1 < α2, it satisfies fα1 (r) < fα2 (r) for all r in the appropriate domain. A family of functions fα(r) with α ∈ I is strictly decreasing with respect to the parameter α if for α1 < α2, it satisfies fα1 (r) > fα2 (r) for all r in the … view at source ↗
Figure 4
Figure 4. functions (dηf2η ) ′′(0.4), −(dηf3η ) ′′(0.2), L156(η) and L158(η). Clearly, −(dηf3η ) ′′(0.2) < L158(η) < L156(η) < (dηf2η ) ′′(0.4) . Therefore, it follows that −(dηf3η ) ′′(r5) < (dηf2η ) ′′(r5) . Hence, we have (dηf2η ) ′′(r5) + (dηf3η ) ′′(r5) > 0. • Part Ic: for r5 ∈ (0.4, b/2] . The functions from the families (dηf2η ) ′′(r5) and −(dηf3η ) ′′(r5) are convex, as stated in Properties 5.9 . To analyze these func… view at source ↗
Figures from the paper (26 more)
Figure 5
Figure 5. Figure 5: proof parts of Proposition 5.14. into (0.02, 0.37] ∪ (0.37, 0.66] . It is important to note that the functions of the family [PITH_FULL_IMAGE:figures/full_fig_p016_5.png]
Figure 6
Figure 6. Figure 6: functions − [PITH_FULL_IMAGE:figures/full_fig_p016_6.png]
Figure 7
Figure 7. Figure 7: ). 0.15 0.20 0.25 0.30 0.35 η 1.0 1.2 1.4 1.6 L9η pn1 η)) dη f1η  0.45) + dη f2η (0.45) + dη f3η (0.45) L12(η) L13(η) [PITH_FULL_IMAGE:figures/full_fig_p017_7.png]
Figure 8
Figure 8. Figure 8: functions L10η (p2(η)), dηf1η  (0.45) + dηf2η  (0.45) + dηf3η  (0.45), L14(η) and L15(η) . L14(η) :=    (dηf10.02 + dηf20.02 + dηf30.02 )(0.45) = 1.13542, 0.02 ≤ η < 0.08, (dηf10.08 + dηf20.08 + dηf30.08 )(0.45) = 1.26796, 0.08 ≤ η < 0.14, (dηf10.14 + dηf…
Figure 9
Figure 9. Figure 9: functions − [PITH_FULL_IMAGE:figures/full_fig_p018_9.png]
Figure 10
Figure 10. Figure 10: functions − [PITH_FULL_IMAGE:figures/full_fig_p018_10.png]
Figure 11
Figure 11. Figure 11: functions − [PITH_FULL_IMAGE:figures/full_fig_p019_11.png]
Figure 12
Figure 12. Figure 12: functions −dηf41.16249 (r5), dηf10.66 (r5)+dηf20.66 (r5)+dηf30.66 (r5) and −dηf41.16249 (0.09455), with η = 1.16249 . Therefore, its true that − [PITH_FULL_IMAGE:figures/full_fig_p019_12.png]
Figure 13
Figure 13. Figure 13: functions − [PITH_FULL_IMAGE:figures/full_fig_p020_13.png]
Figure 14
Figure 14. Figure 14: functions dηf1η + dηf2η + dηf3η  (0.55), − [PITH_FULL_IMAGE:figures/full_fig_p022_14.png]
Figure 15
Figure 15. Figure 15: functions −f ′ 2η (b/2) and f ′ 3η (b/2). 0.005 0.010 0.015 0.020 η 14.6 14.7 14.8 14.9 15.0 f4η ′ 0 - f2η ′ + f3η ′  0.05 L72 η L73 η [PITH_FULL_IMAGE:figures/full_fig_p025_15.png]
Figure 16
Figure 16. Figure 16: functions −f ′ 4η (0), −(f ′ 2η + f ′ 3η )(0.05), L72(η) and L73(η) . L73(η) := ( f ′ 40.11 (0) = 14.7469, 0 < η ≤ 0.011 , f ′ 40.02 (0) = 14.927, 0.011 < η ≤ 0.02 , It follows that L72(η) > L73(η), thus, −f ′ 2η (r5) − f ′ 3η (r5) > f′ 4η (r5). – For r5 ∈ (0.05, 0.09…
Figure 17
Figure 17. Figure 17: functions f ′ 4η (0.05) and −(f ′ 2η + f ′ 3η )(0.09). – For r5 ∈ (0.09, rˆ5(η)] . We compare the families −(f ′ 2η+f ′ 3η )(rˆ5(η)) and f ′ 4η (0.09), which we 24 [PITH_FULL_IMAGE:figures/full_fig_p025_17.png]
Figure 18
Figure 18. Figure 18: functions f ′ 4η (0.09) and −(f ′ 2η + f ′ 3η )( ˆr5(η)). Therefore, the functions of the family λ31η (r5) are monotonically decreasing. 5.2. Proofs of the regions Jn Proposition 5.16. For the region J1, is true that λ11η (r5) < λ31η (r5). Proof. To present a more leg…
Figure 21
Figure 21. Figure 21: b. Therefore, λ11ζ (r3) < λ52ζ (r3). 0.63 0.64 0.65 0.66 0.67 0.68 r3 2.2 2.4 2.6 2.8 3.0 3.2 3.4 λ11ζ (r3) λ52ζ (r3) (a) Part I 0.70 0.75 0.80 0.85 0.90 0.95 1.00 r3 3 4 5 6 7 λ11ζ (r3) λ52ζ (r3) (b) Part II [PITH_FULL_IMAGE:figures/full_fig_p029_21.png]
Figure 22
Figure 22. Figure 22: functions λ11ζ (r3) and λ52ζ (r3) for ζ = 1.23607 . 1.05 1.10 1.15 1.20 r3 4 6 8 10 λ11ζ (r3) λ52ζ (r3) [PITH_FULL_IMAGE:figures/full_fig_p030_22.png]
Figure 23
Figure 23. Figure 23: functions λ11ζ (r3) and λ52ζ (r3) for ζ = 1.25 . Proof. We realize that the family of functions λ52ζ (r3) is increasing, then the function λ520 (r3) is a lower bound function and this is monotonically increasing with a minimum value λ520 (1) = 4.4042 . On the other ha…
Figure 24
Figure 24. Figure 24: functions λ11µ (r3) and λ31µ (r3) for µ = 0.1543145 . function H1ι (r3) − G1ι (r3), H3ι (r3) − G3ι (r3) and H2ι (r3) + H4ι (r3) − G2ι (r3) − G4ι (r3) are strictly positive. Therefore λ52ι (r3) > λ42ι (r3). See [PITH_FULL_IMAGE:figures/full_fig_p031_24.png]
Figure 25
Figure 25. Figure 25: functions λ52ι (r3) and λ42ι (r3) for ι = 0.232235 . Proposition 5.23. For the region J8, is true that λ22ι (r3) > λ32ι (r3). Proof. We can write λ22ι (r3) as follow λ22ι (r3) = K1ι (r3) + ... + K4ι (r3) and λ32ι (r3) as λ32ι (r3) = z1ι (r3) + ... + z4ι (r3). We defin…
Figure 26
Figure 26. Figure 26: functions λ22ι (r3) and λ32ι (r3) for ι = 0.237965 . 30 [PITH_FULL_IMAGE:figures/full_fig_p031_26.png]
Figure 27
Figure 27. Figure 27: functions L47ξ (r3), L48ξ (r3), L49ξ (r3) and L50ξ (r3) for ξ = 0.189. • Part II: for ξ ∈ [0.99412, ∞) . Since the family λ11ξ (r3) is strictly decreasing, we will analyze the function obtained by calculating limξ→∞ λ11ξ (r3), which is a monotonically decreasing funct…
Figure 28
Figure 28. Figure 28: functions λ11ξ (r3), λ41ξ (r3), L91ξ (r3), L92ξ (r3) y L93ξ (r3) for ξ = 2.09. The minimum values of r ∗ 1ξ and r ∗ 2ξ are 1.13653 and 1.44536, respectively. On the other hand, since the family λ41ξ (r3) is strictly decreasing and the functions of the family are incre…
Figure 32
Figure 32. Figure 32: functions λ11b (r3) − λ311.05 (r3), L98(r3) and L99(r3). Proof. First, we divide the interval ξ in [2, 036, 3) ∪ [3, ∞). • Part I: for ξ ∈ [2.036, 3) . The family of functions λ11ξ (r3) is strictly decreasing, so we can consider the function λ113 (r3), which is convex…
Figure 33
Figure 33. Figure 33: functions λ113 (r3), α5, L123(r3), L124(r3), L125(r3) and λ522.036 (r3). 34 [PITH_FULL_IMAGE:figures/full_fig_p035_33.png]
Figure 34
Figure 34. Figure 34: function L127(r3) − λ523 (r3) and 0.00093769. Proposition 5.29. For the region J16, the system (4.2) is not satisfied. Proof. There exists a function r¯5(r3) such that λ21(r3, r¯5(r3)) = 0. Then, we divide the interval of r3 into (1, 1.152781] ∪ (1.152781, 1.201923] ∪…
Figure 35
Figure 35. Figure 35: function λ21(r3, r5) for r3 ∈ (1, 1.3] and r5 ∈ (1, 1.4]. Is true that (See Figs. 36a, 36b and 36c), 35 [PITH_FULL_IMAGE:figures/full_fig_p036_35.png]

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