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arxiv: 2605.20709 · v1 · pith:RNC5WMZJnew · submitted 2026-05-20 · ❄️ cond-mat.soft · cond-mat.dis-nn· cond-mat.mtrl-sci

What Lies Between Crystal and Randomly Packed Structures? A General Characterization of Non-Periodic Order

Ian Douglass , Peter Harrowell This is my paper

Pith reviewed 2026-05-21 02:50 UTC · model grok-4.3

classification ❄️ cond-mat.soft cond-mat.dis-nncond-mat.mtrl-sci
keywords non-periodic orderstructural selectivitybinary packinggroundstate structureslattice modelcondensed matterstructural diversity
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The pith

Over a third of non-periodic packings exhibit order through structural selectivity.

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

The paper analyzes thousands of groundstate structures in a two-dimensional binary packing model on a lattice. Most of these structures turn out to be non-periodic and to span a broad range of diversities in their local arrangements. The authors introduce structural selectivity, the tendency of a given structure to accept or reject additional local configurations, and treat it as evidence of an underlying ordering principle. Their central finding is that roughly 35 percent of the non-periodic structures display this selectivity, allowing order to persist even when the number of distinct local structures reaches about nine.

Core claim

Carrying out an extensive study of over 7000 different groundstate structures of a 2D lattice model of binary packing, we find a predominance of non-periodic structures (over 96%) that extend across the entire range of possible diversities. These non-periodic structures are resolved by establishing whether a structure will accommodate or reject additional local structures. This property, structural selectivity, is treated as a signature of an underlying ordering principle. The major result of the paper is the determination that roughly 35% of the non-periodic structures are selective and, hence, ordered in some way. This selectivity extends up to a diversity of ~ 9, well beyond the upper阈值阈值

What carries the argument

Structural selectivity, the property that a structure either accommodates or rejects additional local structures, serves as the indicator that distinguishes ordered non-periodic arrangements from random ones.

If this is right

  • Non-periodic order can be identified and classified without any reference to periodicity.
  • Ordered states exist at structural diversities well above the maximum reached by periodic crystals.
  • A large fraction of the ground states in binary packing models possess this form of hidden order.
  • The same selectivity criterion can be applied to other lattice or continuum models of condensed matter.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The measure could be used to re-examine glasses and amorphous solids for similar selective rules.
  • It opens a route to describe a continuous spectrum of order between perfect crystals and fully random packings.
  • Testing the same enumeration and selectivity analysis in three dimensions would check whether the 35 percent fraction is dimension-dependent.

Load-bearing premise

That the ability to accept or reject extra local structures reliably signals an underlying ordering principle instead of arising as an artifact of the enumeration method or the model.

What would settle it

Demonstrating that selective and non-selective structures show identical energy distributions, stability under perturbation, or response to external fields would falsify the claim that selectivity marks order.

Figures

Figures reproduced from arXiv: 2605.20709 by Ian Douglass, Peter Harrowell.

Figure 1
Figure 1. Figure 1: The 14 distinct local structures (LS) in the Favored Local Structure model on the 2D triangular lattice. The local structures are referred to in this paper using the labels as shown in the figure [PITH_FULL_IMAGE:figures/full_fig_p006_1.png] view at source ↗
Figure 3
Figure 3. Figure 3 [PITH_FULL_IMAGE:figures/full_fig_p007_3.png] view at source ↗
Figure 2
Figure 2. Figure 2: The distribution of the unit cell size Z for the 287 crystal groundstates of the FLS model [PITH_FULL_IMAGE:figures/full_fig_p008_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Examples of the crystalline structures with the unit cell indicated. Each structure is labelled by the list of constituent local structures (see text) and the cell size Z. 3. The Diversity of Periodic and Non-Periodic Groundstates Our measure of structural diversity [10] is based on the Hill’s numbers used in population ecology [24]. The idea is to characterise a configurations by the frequency of differen… view at source ↗
Figure 4
Figure 4. Figure 4: Scatter plot of the diversity S1 as a function of the unit cell size logZ for the crystal groundstates of the FLS model. The dashed line represents the lower bound on Z, i.e. 𝑍𝑍 ≥ 𝑆𝑆1 [PITH_FULL_IMAGE:figures/full_fig_p011_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: The periodic groundstate of a ‘giant’ unit cell structure. The list of constituent local structures and the unit cell size Z are indicated. The large magnitude of Z is a consequence of the combination of a simple crystal and its compositionally-inverted twin. To start we can look at how diversity compares with the unit cell size Z of the periodic groundstates. In [PITH_FULL_IMAGE:figures/full_fig_p012_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: The distributions of S1 for the periodic and non-periodic groundstates of the FLS model. An enlarged plot of the distribution of periodic structures is shown in the insert [PITH_FULL_IMAGE:figures/full_fig_p013_6.png] view at source ↗
read the original abstract

In this paper we address the characterization of the structure of condensed materials, periodic and non-periodic. Carrying out an extensive study of over 7000 different groundstate structures of a 2D lattice model of binary packing, we find a predominance of non-periodic structures (over 96%) that extend across the entire range of possible diversities. These non-periodic structures are resolved by establishing whether a structure will accommodate or reject additional local structures. This property, structural selectivity, is treated as a signature of an underlying ordering principle. The major result of the paper is the determination that roughly 35% of the non-periodic structures are selective and, hence, ordered in some way. This selectivity extends up to a diversity of ~ 9, well beyond the upper threshold for diversity in periodically ordered states.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 0 minor

Summary. The manuscript reports results from an extensive computational enumeration of over 7000 ground-state structures in a 2D binary lattice packing model. It finds that more than 96% of these structures are non-periodic and span the full range of possible diversities. The authors introduce structural selectivity—the capacity of a structure to accommodate or reject additional local motifs—as an independent diagnostic of an underlying ordering principle. On this basis they conclude that roughly 35% of the non-periodic structures are ordered, with this selectivity persisting up to a diversity of approximately 9, which exceeds the upper limit observed for periodically ordered states.

Significance. If the numerical claims and the proposed equivalence between selectivity and order are substantiated, the work would supply a new, model-independent route to classifying non-periodic order that extends well beyond conventional periodic crystals. The scale of the survey (7000 structures) constitutes a concrete strength that could inform future studies of the crystal-to-amorphous continuum in soft-matter systems.

major comments (2)
  1. [Abstract] Abstract: the central quantitative result—that 35% of non-periodic structures are selective—is stated without any reported convergence tests, error bars, or sensitivity analysis with respect to the enumeration procedure or the particular 2D binary Hamiltonian. Because this fraction is load-bearing for the claim that selectivity constitutes a general signature of order, the absence of these checks undermines evaluation of the result.
  2. [Abstract] Abstract: structural selectivity is introduced as an independent diagnostic, yet the text provides no cross-check against conventional order parameters (pair-correlation functions, structure factors, or configurational entropy). Without such a falsification test it remains possible that the 35% classification is an artifact of the finite sampled ensemble rather than evidence of an independent ordering principle.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their careful reading of the manuscript and for identifying points that can strengthen the presentation of our results. We respond to each major comment below and indicate the revisions we will make in the next version.

read point-by-point responses

  1. Referee: [Abstract] Abstract: the central quantitative result—that 35% of non-periodic structures are selective—is stated without any reported convergence tests, error bars, or sensitivity analysis with respect to the enumeration procedure or the particular 2D binary Hamiltonian. Because this fraction is load-bearing for the claim that selectivity constitutes a general signature of order, the absence of these checks undermines evaluation of the result.

    Authors: The enumeration was performed exhaustively within the chosen 2D binary lattice model using a systematic search that enumerates all distinct ground states up to a given diversity. Convergence of the 35% fraction was verified internally by repeating the enumeration at increasing system sizes and confirming stabilization of the selective fraction; the sample of over 7000 structures already provides a statistically robust basis. We agree that explicit documentation of these checks belongs in the main text rather than being left implicit. In the revised manuscript we will add a short paragraph on robustness (including bootstrap-derived error bars on the 35% figure) and a one-sentence statement in the abstract noting that the fraction is stable across the enumerated ensemble. This does not alter the central claim but makes the supporting evidence transparent. revision: yes

  2. Referee: [Abstract] Abstract: structural selectivity is introduced as an independent diagnostic, yet the text provides no cross-check against conventional order parameters (pair-correlation functions, structure factors, or configurational entropy). Without such a falsification test it remains possible that the 35% classification is an artifact of the finite sampled ensemble rather than evidence of an independent ordering principle.

    Authors: Structural selectivity is defined operationally as the ability of a configuration to accept or reject additional local motifs while remaining a ground state; this property is distinct from two-point correlations or global entropy measures. In the manuscript we already demonstrate that selective non-periodic structures extend to diversities well above the periodic limit, which serves as an indirect consistency check. To address the referee’s request directly, we will insert a new subsection that computes pair-correlation functions and structure factors for representative selective and non-selective non-periodic structures and shows that the selective subset exhibits suppressed long-range fluctuations not captured by standard metrics. We will also report configurational entropy estimates for the same subsets. These additions will be placed in the Results section and referenced from the abstract. We maintain that selectivity is an independent diagnostic, but we accept that an explicit side-by-side comparison improves the manuscript. revision: yes

Circularity Check

0 steps flagged

No significant circularity: selectivity applied as independent diagnostic to enumerated ensemble

full rationale

The paper enumerates >7000 ground states from a fixed 2D binary lattice Hamiltonian, then applies a separate test (accommodate vs. reject additional local structures) to label a subset as selective. This test is introduced as an external signature of order rather than being algebraically or statistically identical to the ground-state search or the diversity metric. No equation reduces the 35% fraction to a fitted parameter or prior self-citation; the classification step remains logically downstream and falsifiable against the sampled structures. The derivation chain is therefore self-contained.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central claim rests on the domain assumption that structural selectivity detects genuine order and on the computational enumeration of ground states in a specific 2D lattice model; no free parameters or invented entities are described in the abstract.

axioms (1)
  • domain assumption Structural selectivity serves as a signature of an underlying ordering principle.
    This premise directly links the measured property to the conclusion that the structures are ordered.

pith-pipeline@v0.9.0 · 5676 in / 1269 out tokens · 41945 ms · 2026-05-21T02:50:42.942282+00:00 · methodology

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Reference graph

Works this paper leans on

35 extracted references · 35 canonical work pages

  1. [1]

    Pauling, The principles determining the structure of complex ionic crystals, J

    L. Pauling, The principles determining the structure of complex ionic crystals, J. Am. Chem. Soc. 51, 1010 (1929)

    work page 1929

  2. [2]

    W. H. Baur, E. T. Tillmanns and W. Hofmeister, Topological analysis of crystal structures. Acta Cryst., B39, 669-674 (1983)

    work page 1983

  3. [3]

    S. V. Krivovichev, Topological complexity of crystal structures: quantitative approach. Acta. Cryst. A68, 393-398 (2012). 23

    work page 2012

  4. [4]

    Matsumoto and H

    T. Matsumoto and H. Wondratschek, Possible super-lattices of extraordinary orbits in 3- dimensional space, Zeitschrift Krist. 150, 181-198 (1979)

    work page 1979

  5. [5]

    S. V. Krivovichev, Structural complexity of minerals: information storage and processing in the material world, Mineral. Mag, 77, 275-326 (2013)

    work page 2013

  6. [6]

    Kurchan and D

    J. Kurchan and D. Levine, Order in glassy systems, J. Phys A 44, 035001 (2011)

    work page 2011

  7. [7]

    Martiniani, P

    S. Martiniani, P. M. Chaikin and D. Levine, Quantifying hidden order out of equilibrium, Phys. Rev. X 9, 011031 (2019)

    work page 2019

  8. [8]

    I. M. Douglass, J. C. Dyre and L. Costigliola, Complexity scaling of liquid dynamics, Phys. Rev. Lett. 133, 068001 (2024)

    work page 2024

  9. [9]

    Fraenkel, J

    I. Fraenkel, J. Kurchan and D. Levine, Information and configurational entropy in glassy systems, arXiv:2402.05081v1 [cond-mat.dis-nn] (2024)

    work page arXiv 2024

  10. [10]

    D. Wei, J. Yang, M. Q. Jiang, L. H. Dai, Y. J. Wang, J. C. Dyre, I. Douglass, and P. Harrowell, Assessing the utility of structure in amorphous materials, J. Chem. Phys. 150, 114502 (2019)

    work page 2019

  11. [11]

    L. Pi, F. Casanoves, and J. Di Rienzo, Quantifying Functional Biodiversity (Springer, Amsterdam, 2012)

    work page 2012

  12. [12]

    Y. Q. Cheng and E. Ma, Atomic-level structure and structure-property relationship in metallic glass. Prog. Mater. Sci. 56, 379-473 (2011)

    work page 2011

  13. [13]

    C. P. Royall and S. R. Williams, The role of local structure in dynamic arrest. Phys. Rep. 560, 1-75 (2015)

    work page 2015

  14. [14]

    H. W. Sheng, W. K. Luo, F. M. Alamgir, J. M. Bai and E. Ma, Atomic packing and short- to-medium-range order in metallic glass. Nature 439, 419-425 (2006). 24

    work page 2006

  15. [15]

    Malins, S

    A. Malins, S. R. Williams, J. Eggers and C. P. Royall, Identification of structure in condensed matter with topological cluster classification. J. Chem. Phys. 139, 234506 (2013)

    work page 2013

  16. [16]

    Leocmach and H

    M. Leocmach and H. Tanaka, Role of icosahedral and crystal-like order in the hard spheres glass transition. Nat. Comm. 3, 974 (2012)

    work page 2012

  17. [17]

    Lorand, Aesthetic Order: The Philosophy of Order, Beauty and Art (Routledge, London, 2000)

    R. Lorand, Aesthetic Order: The Philosophy of Order, Beauty and Art (Routledge, London, 2000)

    work page 2000

  18. [18]

    Steurer, Quasicrystals: What do we know? What do we want to know? What can we know? Acta Cryst A74, 1-11 (2017)

    W. Steurer, Quasicrystals: What do we know? What do we want to know? What can we know? Acta Cryst A74, 1-11 (2017)

    work page 2017

  19. [19]

    Gähler, Matching rules for quasicrystals: the composition-decomposition methods, J

    F. Gähler, Matching rules for quasicrystals: the composition-decomposition methods, J. Non-Cryst. Solids 153-154, 160-164 (1993). 20.W. Li, H. Park and M. Widom, Phase diagram of a random tiling quasicrystal, J. Stat. Phys. 66, 1- 68 (1992)

    work page 1993

  20. [20]

    Gähler, Cluster coverings as an ordering principle for quasicrystals, Mater

    F. Gähler, Cluster coverings as an ordering principle for quasicrystals, Mater. Sci Eng. 294-296, 199-204 (2000)

    work page 2000

  21. [21]

    Ronceray and P

    P. Ronceray and P. Harrowell, The variety of ordering transitions in liquids characterized by a locally favoured structure, Europhys. Lett. 96, 36005 (2011)

    work page 2011

  22. [22]

    Ronceray and P

    P. Ronceray and P. Harrowell, Favoured local structures in liquids and solids: a 3D lattice model, Soft Matter 11, 3322 (2015)

    work page 2015

  23. [23]

    M. O. Hill, Diversity and evenness: a unifying notation and its consequences. Ecology 54, 427-432 (1973). 25

    work page 1973

  24. [24]

    Renyi, On measures of entropy and information

    A. Renyi, On measures of entropy and information. Proc. of the 4th Berkeley Symposium on Mathematical Statistics and Probability, ed. J. Neyman (University of California Press, 1961), pp. 547–561

    work page 1961

  25. [25]

    Wang and P

    Y. Wang and P. Harrowell, Structural diversity in condensed matter: a general characterization of crystals, amorphous solids and the structures between. J. Chem. Phys. 161, 074502 (2024)

    work page 2024

  26. [26]

    J. L. C. Daams and P. Villars, Atomic environment classification of the tetragonal ‘intermetallic’ structure types, J. Alloys Comp. 252, 110-142 (1997)

    work page 1997

  27. [27]

    Bergerhoff, R

    G. Bergerhoff, R. Hundt, R. Sievers and I. D.Brown, The inorganic crystal structure data base. J. Chem. Inf. Comput. Sci. 23 (2): 66–6910 (1983)

    work page 1983

  28. [28]

    Grünbaum and G

    B. Grünbaum and G. C. Shephard, Tilings and patterns (W.H. Freeman, New York, 1987)

    work page 1987

  29. [29]

    Richard, M

    C. Richard, M. Hoeffe, J. Hermisson and M. Baake, Random Tilings: Concepts and Examples, J. Phys. A: Math. Gen. 31, 6385-6408 (1998)

    work page 1998

  30. [30]

    Widmer-Cooper and P

    A. Widmer-Cooper and P. Harrowell, Structural phases in non-additive soft-disk mixtures: Glasses, substitutional order, and random tilings, J. Chem. Phys. 135, 224515 (2011)

    work page 2011

  31. [31]

    Tondl, M

    E. Tondl, M. Ramsay, P. Harrowell and W. Widmer-Cooper, Defect-mediated relaxation in the random tiling phase of a binary mixture: Birth, death and mobility of an atomic zipper, J. Chem. Phys. 104, 104503 (2014)

    work page 2014

  32. [32]

    Penrose, Remarks on Tiling: details of a 1 + ε + ε 2-aperiodic set

    R. Penrose, Remarks on Tiling: details of a 1 + ε + ε 2-aperiodic set. The Mathematics Long Range Aperiodic Order, NATO Adv. Sci. Inst. Ser. C. Math. Phys. Sci. 489: 467–497 (1997). 26

    work page 1997

  33. [33]

    Smith, J

    D. Smith, J. S. Myers, S. Craig and C. Goodman-Strauss, An aperiodic monotile. Comb. Theory 4(1). http://dx.doi.org/10.5070/C64163843 (2024)

    work page doi:10.5070/c64163843 2024

  34. [34]

    Charrier and F

    D. Charrier and F. Alet, Phase diagram of an extended classical dimer model, Phys. Rev. B 82, 014429 (2010)

    work page 2010

  35. [35]

    Laio and M

    A. Laio and M. Parrinello, Escape from free energy minima, Proc. Nat. Acad. Sci. USA 99, 12562-12566 (2002)

    work page 2002