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Higher Order Rigidity and Energy

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Pith's one-line read For every rigid bar-and-joint framework, the rigidity order is fixed by any stiff-bar energy and equals a maximum over flex orders.

desk verdict A strong, clean contribution that resolves the energy-dependence question for rigidity order; fix the definition of tight growth order and send it out. read the letter →

arxiv 2506.03108 v2 pith:OXE3MXQ5 submitted 2025-06-03 math.MG math.OC

Steven J. Gortler , Miranda Holmes-Cerfon , Louis Theran This is my paper
classification math.MGmath.OC MSC 52C25
keywords bar-and-jointframeworkrigidityorderstiff-barenergy(jk)-flexhigher-ordergrowthfourth-derivativetestdegeneratecriticalpoint
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

This paper proposes that every rigid bar-and-joint framework has a numerical rigidity order measuring how fast its energy must grow when the framework is displaced from its pinned configuration. The authors show that any stiff-bar energy — harmonic springs, algebraic length-squared potentials, Lennard-Jones, Morse — gives the same tight growth order $s$, so the rigidity order $\nu=s/2$ is an intrinsic invariant of the framework, not a feature of the chosen energy. They prove the formula $\nu=\max\{(k+1)/j : (G,p)\text{ has a }(j,k)\text{-flex}\}$, tying the invariant directly to higher-order flexes. The same energy viewpoint yields new proofs of two classical rigidity certificates: absence of a second-order flex forces rigidity, and when the space of first-order flexes is one-dimensional, absence of a $k$th-order flex forces rigidity. If the paper is right, a single rational number organizes the hierarchy of rigidity tests and explains why some rigid frameworks feel softer than others.

What carries the argument

For an edge-based energy $E(q)=\sum E_{ij}(|q_i-q_j|)$ whose edge terms are analytic with a strict local minimum and positive curvature at the resting lengths, the paper proves through Faà di Bruno's formula that a trajectory is a $(j,k)$-flex of the framework if and only if it is a $(j,2k)$-flex of $E$ (Theorem 3.13). This doubling of vanishing order is what links energy growth order to flex order. The second engine is a family of indicative test trajectories at a degenerate critical point: lines $x_0t$ for second order, parabolas $(x_0t^2,y_0t)$ for fourth order, and higher-degree families of the form $y_0p't+y_0^2p''t^2+\cdots+p^{(k)}_0t^k$ when the Hessian kernel is one-dimensional. A general proposition shows that if all test trajectories have leading coefficients of one sign after $2k$ derivatives, the critical point is a strict local minimum; this turns the absence of flexes into energy certificates.

What would settle it

Take a rigid framework and compute the tight growth order of two different stiff-bar energies, for instance harmonic spring and algebraic squared-length energies, by sampling energy along radial and curved paths; if the two exponents differ, or if any found $(j,k)$-flex gives $(k+1)/j$ strictly larger than half the measured exponent, Theorem 4.1 is false. Alternatively, search numerically for a cusp mechanism that has a $(1,1)$-flex but no $(1,2)$-flex and yet is flexible; such an example would be a counterexample to Theorem 5.2.

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Extended reading notes

Core claim

This paper's central claim is that the old distinction between rigid and flexible can be refined by a rational order computed from energy growth. For a rigid framework $(G,p)$, every stiff-bar energy $E$ has the same tight growth order $s$ at the pinned configuration, and defining the rigidity order as $\nu=s/2$ gives $\nu=\max\{(k+1)/j : (G,p)\text{ has a }(j,k)\text{-flex}\}$ (Theorem 4.1). In particular, a framework with a $(1,1)$-flex but no $(1,2)$-flex has rigidity order exactly 2 and is therefore rigid (Theorem 5.2), and a framework with a one-dimensional space of first-order flexes that has a $(1,k-1)$-flex but no $(1,k)$-flex has rigidity order exactly $k$ and is therefore rigid (Theorem 5.3). The proofs run through a new fourth-derivative test for degenerate critical points and a family of $2k$-derivative tests for stiff-bar energies, and they show why the known cusp mechanisms do not contradict the classical certificates.

Load-bearing premise

The load-bearing premise imported from outside the paper is an analytic growth theorem: at a strict local minimum, a real analytic function has a rational tight growth order and there is some trajectory along which the growth is exactly that order; if this theorem failed, the definition of rigidity order and the flex formula would have nothing to attach to.

Editorial extensions

If this is right

  • Second-order rigidity is now a statement about energy: a framework with a $(1,1)$-flex but no $(1,2)$-flex has rigidity order 2, meaning every stiff-bar energy grows at least as fast as quartic in distance from the pinned configuration.
  • In the one-dimensional-flex case, rigidity order is an integer, and it can be computed by iteratively solving linear systems: when a $(1,k-1)$-flex exists but no $(1,k)$-flex does, the order is exactly $k$.
  • Frameworks that are rigid but feel floppy are quantified: a rigid framework with a $(j,k)$-flex has an energy that grows sometimes-$s$-slowly for $s=(2k+2)/j$, so larger $k$ relative to $j$ means a genuinely smaller energy growth exponent.
  • The general rigidity order can be fractional, so the classical integer-order hierarchy of rigidity tests is replaced by a rational scale that has a physical meaning.
  • The fourth-derivative test gives a new sufficient condition for rigidity that can be applied beyond rigidity theory to any optimization problem with a positive-semidefinite degenerate Hessian.

Reading between the lines

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

  • A testable consequence the paper leaves implicit is that rigidity order should predict, at least qualitatively, the exponent of force-displacement response of a real elastic structure near the pinned configuration; comparing measured force curves with the predicted energy order would test the framework's physical relevance.
  • The same energy-growth construction could be applied to vertex models, packing contact constraints, or rigid-body contacts, assigning each rigid state an order and potentially ordering the softness of jammed packings.
  • The paper's discussion of why a general sixth-derivative test fails suggests that a complete higher-order rigidity theory may need to track not just the Hessian kernel but the geometry of the $(j,k)$-flexes themselves, possibly through the Puiseux expansion of the configuration-space variety.
  • One could connect rigidity order to numerical stability: if a rigid framework has high rigidity order, equilibrium solvers may converge more slowly or be more sensitive to perturbations, a prediction that could be checked in existing numerical experiments.
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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

0 major / 7 minor

Summary. The paper studies higher-order rigidity of bar-and-joint frameworks by connecting the existence of (j,k)-flexes to the growth order of stiff-bar energy functions. It proves that a trajectory is a (j,k)-flex if and only if it is a (j,2k)-energy-flex for any stiff-bar energy (Theorem 3.13), defines the rigidity order of a rigid framework as half the tight growth order of any such energy, and shows in Theorem 4.1 that this order is independent of the chosen energy and equals the maximum of (k+1)/j over all (j,k)-flexes. The paper then develops a fourth-order derivative test for degenerate critical points and uses it to give new proofs that second-order rigidity implies rigidity (Theorem 5.2) and that, when the space of first-order flexes has dimension one, absence of a (1,k)-flex implies rigidity with rigidity order k (Theorem 5.3). Several examples with explicit coordinates illustrate the results, and an extension to general measurement constraints is outlined.

Significance. If the results hold, the paper provides a physically motivated and energy-independent definition of rigidity order, addressing previously noted pathologies in higher-order rigidity. The main theorems are proved carefully, with a clean use of Faà di Bruno formulas to relate flex existence to energy growth. The fourth-derivative test for degenerate critical points is a useful stand-alone contribution that generalizes ideas of Cushing. The paper is well situated relative to prior work by Salerno, Stachel, Nawratil, Tachi, and others, and it includes concrete numerical examples. The principal vulnerability is the reliance on the external result Theorem 3.6 (Barone-Netto et al.); the statement is plausible and standard in analytic geometry, but the paper would benefit from a more precise citation and a discussion of how the analytic-trajectory requirement in Definition 2.6 is met.

minor comments (7)
  1. [Definition 3.5] The phrase 'In this case we say that s is the tight growth order' could be misread as allowing multiple s satisfying both growth conditions. The uniqueness is a consequence of Theorem 3.6 together with the leading-exponent description of m(r), and an explicit sentence stating this uniqueness would prevent potential confusion.
  2. [Section 3.1 (stress-test concern)] The concern that Definition 3.5 admits multiple tight growth orders does not land. For E(x,y)=x^2+y^4, the 'always-s-quickly' condition must hold for all q, and along the y-axis it forces s≥4; the 'sometimes-s-slowly' condition along the y-axis forces s≤4, so the only tight order is s=4. In general, for a strict local minimum with m(r)~c r^a, the 'always' condition holds exactly for s≥a and the 'sometimes' condition exactly for s≤a, yielding a unique s=a. Adding this observation would strengthen the presentation.
  3. [Theorem 4.1 proof] The proof does not explicitly handle the case k0=0, which occurs when the tight-growth trajectory has its first nonzero energy term at t^2 (e.g., for first-order rigid frameworks). In that case the argument appeals to a (j0,0)-flex, which is outside Definition 2.8. The value (k0+1)/j0 = 1 is still attained by any (2,1)-flex, so the theorem remains correct, but a short additional sentence is needed to close this gap.
  4. [Theorem 3.6] Please specify the exact result in [4] that guarantees the existence of an analytic trajectory (in the sense of Definition 2.6) along which E has the optimal growth order, and clarify that the lower bound E(q)-E(p) ≥ c|q-p|^s holds on a full neighborhood of p. This would remove ambiguity about the match between the cited theorem and the paper's definitions.
  5. [Lemma 3.10 and Theorem 4.1] The statement that an odd-order leading term is impossible because it would make the function decrease should mention that the analytic trajectory extends to negative t, since trajectories are defined only for t∈[0,ε]. With this extension, the local-minimum property rules out sign changes.
  6. [Theorem 5.3] The statement should specify k≥2 or treat k=1 separately, since for k=1 the hypothesis asserts a (1,0)-flex, which is not defined in Definition 2.8. The case k=1 follows from Theorem 5.1.
  7. [Typos] There are a few typographical errors: in the proof of Theorem 4.1, 'j+1/2 = sE/2' should read '(k+1)/j = sE/2'; 'the the edge lengths' appears in Section 2.1; 'minium' appears in the proof of Theorem 3.3; 'frameowrk' appears in Section 7.3.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity; the rigidity-order theorem is derived from independent analytic-growth and flex lemmas, not from its own conclusion.

full rationale

The paper's derivation chain is self-contained rather than circular. Theorem 3.13 proves that, for stiff-bar energies, a trajectory is a (j,k)-flex if and only if it is a (j,2k)-energy-flex, using Faà di Bruno's formula and the positivity of the second derivative of each edge energy. Lemma 3.10 then converts an energy-flex into a sometimes-slow growth bound of order (2k+2)/j. Theorem 4.1 combines these with the external analytic-growth theorem of Barone-Netto et al. [4]: every (j,k)-flex forces (k+1)/j ≤ s/2, while a trajectory realizing the tight growth order s has Taylor expansion starting at an even order 2k0+2 and distance from p starting at order j0, so Remark 3.11 gives the two-sided growth order (2k0+2)/j0 and Theorem 3.13 converts this trajectory into a (j0,k0)-flex, attaining the maximum in Equation (8). The right-hand side of (8) is independent of the energy, which proves the energy-independence claim without assuming it. The later rigidity certificates (Theorems 5.1, 5.2, 5.3) are proved by constructing indicative trajectory families and applying the fourth- or higher-order derivative test; they do not invoke the classical theorems they reprove. No fitted parameter is renamed as a prediction. The only self-citations, [18] and [21], are background or are explicitly described as not providing the new proof, so they are not load-bearing. The external theorem [4] supplies the existence and rationality of the tight growth order and the distinguished Puiseux-leading exponent; although Definition 3.5 read alone permits multiple s to satisfy 's-tight', the following discussion in Theorem 3.6 selects the leading exponent of the minimal-value function, so this is a presentation imprecision rather than a circular reduction. Overall the main results are derived from stated hypotheses and external analytic facts, with no step that reduces to its own input by construction.

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

The central claim rests on standard analyticity and rigidity-theory background, not on fitted parameters or newly postulated physical entities. The only non-elementary external input is the Barone-Netto growth theorem for analytic functions, which the paper cites without proof.

assumptions (4)
  • domain assumption Every flexible bar-and-joint framework has an analytic finite flex
    Used in Remark 3.7 to assert that non-rigid frameworks have sometimes-infinity energy growth; cited to Gluck [15] and Milnor [25].
  • domain assumption A real-analytic function with a strict local minimum has a rational tight growth order and a trajectory attaining that order
    This is Theorem 3.6, imported from Barone-Netto et al. [4]. It underpins Definition 4.2 and Theorem 4.1 and is not proved inside the paper.
  • domain assumption Stiff-bar energies are analytic at the pinned configuration, have strict local minima, and have positive curvature at each bar length
    Definition 3.1 restricts the entire rigidity-order theory to this class of edge-based energies.
  • standard math Standard linear-algebra and pinning lemmas for trivial trajectories and isometries
    Appendix A uses folklore pinning results and Connelly-Whiteley lemmas [8, Lemma 4.2.1 etc.] to pass between pinned and unpinned frameworks.

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Cite this review

Pith. "Pith review of Higher Order Rigidity and Energy." pith.science (2026). https://pith.science/paper/OXE3MXQ5

@misc{pith2026250603108,
  author       = {Pith},
  title        = {Pith review of: Higher Order Rigidity and Energy},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/OXE3MXQ5}},
  note         = {Machine review of arXiv:2506.03108}
}
abstract

In this paper, we revisit the notion of higher-order rigidity of a bar-and-joint framework. In particular, we provide a link between the rigidity properties of a framework, and the growth order of an energy function defined on that framework. Using our approach, we propose a general definition for the rigidity order of a framework, and we show that this definition does not depend on the details of the chosen energy function. Then we show how this order can be studied using higher order derivative tests. Doing so, we obtain a new proof that the lack of a second order flex implies rigidity. Our proof relies on our construction of a fourth derivative test, which may be applied to a critical point when the second derivative test fails. We also obtain a new proof that when the dimension of non-trivial first-order flex coefficients $\p'$ equals $1$, then the lack of a $k$th order flex for some $k$ implies a framework is rigid. The higher order derivative tests that we study here may have applications in more general optimization problems.

Figures

Figures reproduced from arXiv: 2506.03108 by the authors.

Figure 1
Figure 1. (Left) The half flat prism has rigidity order 4. The middle left vertex has been shifted for visualization purposes. (Right) The Leonardo-3 framework [38] has a rigidity order of 8. The upper middle vertex has been shifted for visualization purposes. The green edges are used so that no pins are needed in the framework. The rigidity orders of these examples persist under projective transformations. In Figures 1, 2 an… view at source ↗
Figure 2
Figure 2. (Left) This symmetric flipped triangular prism has a rigidity order of 4. (Middle) This non-symmetric flipped prism has a rigidity order of 3. (Right) This framework of K3,3 has a rigidity order of 3. (The bipartite partitions are visualized in red and green.) All three become second-order rigid under a generic affine transform [PITH_FULL_IMAGE:figures/full_fig_p034_2.png] view at source ↗
Figure 3
Figure 3. (Left, Middle) These frameworks arise from packings of 14 identical spheres. They were found in [19] but were discarded from the data presented in that paper because they did not pass the test for prestress stability. Both have a rigidity order of 3. (Right) A framework of the coned triangular prism. It has a rigidity order of 4. All three become second-order rigid under a generic affine transform. in the system’s r… view at source ↗

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Cited by 2 Pith papers

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Reference graph

Works this paper leans on

39 extracted references · 37 canonical work pages · cited by 2 Pith papers

  1. [1]

    T. G. Abbott. Generalizations of Kempe’s universality theorem. Masters dissertation, Mas- sachusetts Institute of Technology, 2008

    work page 2008

  2. [2]

    Alexandrov

    V. Alexandrov. Implicit function theorem for systems of polynomial equations with vanishing jacobian and its application to flexible polyhedra and frameworks. Monatshefte f¨ ur Mathe- matik, 132:269–288, 2001

    work page 2001

  3. [3]

    Asimow and B

    L. Asimow and B. Roth. The rigidity of graphs. Trans. Amer. Math. Soc., 245:279–289, 1978

    work page 1978

  4. [4]

    Barone-Netto, G

    A. Barone-Netto, G. Gorni, and G. Zampieri. Local extrema of analytic functions. Nonlinear Differential Equations and Applications NoDEA , 3(3):287–303, 1996

    work page 1996

  5. [5]

    Blekherman

    G. Blekherman. Personal Communication, 2025

    work page 2025

  6. [6]

    Connelly

    R. Connelly. The rigidity of certain cabled frameworks and the second-order rigidity of arbi- trarily triangulated convex surfaces. Adv. in Math. , 37(3):272–299, 1980

    work page 1980

  7. [7]

    Connelly and H

    R. Connelly and H. Servatius. Higher-order rigidity—what is the proper definition? Discrete Comput. Geom., 11(2):193–200, 1994

    work page 1994

  8. [8]

    Connelly and W

    R. Connelly and W. Whiteley. Second-order rigidity and prestress stability for tensegrity frameworks. SIAM J. Discrete Math. , 9(3):453–491, 1996. ISSN 0895-4801

    work page 1996

Show all 39 references
  1. [9]

    A. D. Craik. Prehistory of fa` a di bruno’s formula. The American Mathematical Monthly , 112 (2):119–130, 2005

    2005

  2. [10]

    J. Cushing. Extremal tests for scalar functions of several real variables at degenerate critical points. aequationes mathematicae, 13:89–96, 1975. 35

    1975

  3. [11]

    O. K. Damavandi, V. F. Hagh, C. D. Santangelo, and M. L. Manning. Energetic rigidity. i. a unifying theory of mechanical stability. Physical Review E , 105(2):025003, 2022

    2022

  4. [12]

    Donev, R

    A. Donev, R. Connelly, F. H. Stillinger, and S. Torquato. Underconstrained jammed packings of nonspherical hard particles: Ellipses and ellipsoids. Physical Review E—Statistical, Nonlinear, and Soft Matter Physics , 75(5):051304, 2007

    2007

  5. [13]

    Garcea, G

    G. Garcea, G. Formica, and R. Casciaro. A numerical analysis of infinitesimal mechanisms. International journal for numerical methods in engineering , 62(8):979–1012, 2005

    2005

  6. [14]

    G´ asp´ ar and T

    Z. G´ asp´ ar and T. Tarnai. Finite mechanisms have no higher-order rigidity. Acta Technica- academiae Scientiarum Hungaricae , 106:119–126, 1994

    1994

  7. [15]

    H. Gluck. Almost all simply connected closed surfaces are rigid. In Geometric topology (Proc. Conf., Park City, Utah, 1974) , pages 225–239. Lecture Notes in Math., Vol. 438. Springer, Berlin, 1975

    1974

  8. [16]

    S. Guest. The stiffness of prestressed frameworks: a unifying approach. International Journal of Solids and Structures , 43(3-4):842–854, 2006

    2006

  9. [17]

    Guo and T.-S

    F. Guo and T.-S. Pham. On types of degenerate critical points of real polynomial functions. Journal of Symbolic Computation , 99:108–126, 2020

    2020

  10. [18]

    Holmes-Cerfon, L

    M. Holmes-Cerfon, L. Theran, and S. J. Gortler. Almost-rigidity of frameworks. Communica- tions on Pure and Applied Mathematics , 74(10):2185–2247, 2021

    2021

  11. [19]

    M. C. Holmes-Cerfon. Enumerating rigid sphere packings. SIAM Review, 58(2):229–244, 2016

    2016

  12. [20]

    J. E. Jones. On the determination of molecular fields.—i. from the variation of the viscosity of a gas with temperature. Proceedings of the Royal Society of London. Series A, Containing Papers of a Mathematical and Physical Character , 106(738):441–462, 1924

    1924

  13. [21]

    Kallus and M

    Y. Kallus and M. Holmes-Cerfon. Free energy of singular sticky-sphere clusters. Physical Review E, 95(2):022130, 2017

    2017

  14. [22]

    W. T. Koiter. On the stability of elastic equilibrium . National Aeronautics and Space Admin- istration, 1967

    1967

  15. [23]

    Malins, S

    A. Malins, S. R. Williams, J. Eggers, H. Tanaka, and C. P. Royall. Geometric frustration in small colloidal clusters. Journal of Physics: Condensed Matter , 21(42):425103, 09 2009

    2009

  16. [24]

    M. L. Manning. Rigidity in mechanical biological networks. Current Biology, 34(20):R1024– R1030, 2024

    2024

  17. [25]

    J. Milnor. Singular points of complex hypersurfaces , volume No. 61 of Annals of Mathematics Studies. Princeton University Press, Princeton, NJ; University of Tokyo Press, Tokyo, 1968

    1968

  18. [26]

    P. M. Morse. Diatomic molecules according to the wave mechanics. ii. vibrational levels. Physical review, 34(1):57, 1929

    1929

  19. [27]

    Nawratil

    G. Nawratil. A global approach for the redefinition of higher-order flexibility and rigidity. Mechanism and Machine Theory , 205:105853, 2025. 36

    2025

  20. [28]

    Nawratil

    G. Nawratil. On flexes associated with higher-order flexible bar-joint frameworks. arXiv preprint arXiv:2502.01124, 2025

    2025 arXiv

  21. [29]

    Nocedal and S

    J. Nocedal and S. Wright. Numerical optimization. Springer. Springer, 2nd edition, 00 2006

    2006

  22. [30]

    J. C. Roback, A. Nagrath, S. Kristipati, C. D. Santangelo, and R. C. Hayward. Tuning stiffness of mechanical metamaterial unit cells via transitions to second-order rigid and pre-stressed states. Soft Matter , 2025

    2025

  23. [31]

    S. Roman. The formula of faa di bruno. The American Mathematical Monthly, 87(10):805–809, 1980

    1980

  24. [32]

    I. K. Sabitov. Local theory of bendings of surfaces. In Geometry III: Theory of Surfaces, pages 179–250. Springer, 1992

    1992

  25. [33]

    G. Salerno. How to recognize the order of infinitesimal mechanisms: a numerical approach. International journal for numerical methods in engineering , 35(7):1351–1395, 1992

    1992

  26. [34]

    Schwerdtfeger and D

    P. Schwerdtfeger and D. J. Wales. 100 years of the lennard-jones potential.Journal of Chemical Theory and Computation , 20(9):3379–3405, 2024

    2024

  27. [35]

    H. Stachel. A proposal for a proper definition of higher-order rigidity. In Tensegrity Workshop, volume 2007, 2007

    2007

  28. [36]

    H. Stachel. A (3, 8)-flexible bar-and-joint framework? In Presentation Slides, AIM Workshop on Rigidity and Polyhedral Combinatorics, Palo Alto/CA , volume 2007, 2007

    2007

  29. [37]

    T. Tachi. A proper definition of higher order rigidity. arXiv preprint arXiv:2410.15541 , 2024

    2024 arXiv

  30. [38]

    Tarnai and A

    T. Tarnai and A. Lengyel. A remarkable structure of leonardo and a higher-order infinitesimal mechanism. Journal of Mechanics of Materials and Structures , 6(1):591–604, 2011

    2011

  31. [39]

    Vassart, R

    N. Vassart, R. Laporte, and R. Motro. Determination of mechanism’s order for kinematically and statically indetermined systems. International Journal of Solids and Structures , 37(28): 3807–3839, 2000. A Rigidity and pinning Proof of Proposition 2.4 Suppose that ( G, p) is pin...

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