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The paper claims that for k≥2 the product-rule process on cubic lattices jumps discontinuously in connectivity across a vanishing density window, while richer hosts rigidify monotonically more efficiently as k grows.

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2026-08-04 00:21 UTC pith:BWVEPRWY

load-bearing objection Interesting numerical observations and a clean shear obstruction, but the theoretical core is broken — the headline theorems don't survive contact with the paper's own definitions. the 6 major comments →

arxiv 2511.01537 v2 pith:BWVEPRWY submitted 2025-11-03 cond-mat.stat-mech cond-mat.mtrl-scimath.PR

Explosive connectivity and mechanical rigidity in cubic lattice structures

Trenton Lau , Gary P. T. Choi This is my paper
classification cond-mat.stat-mech cond-mat.mtrl-scimath.PR
keywords explosive percolationAchlioptas processproduct rulecubic latticerigidity percolationrigidity-connectivity gapfinite-size scalingfirst-order transition
verification ladder T0 review T1 audit T2 compute T3 formal T4 reserved

The pith

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

The paper asks whether a purely local rule — at each step, sample k candidate bonds and add the one minimizing the product of the sizes of the two components it would join — can sharpen two global properties of a 3D cubic lattice: the suddenness of the connectivity transition and the mechanical efficiency of rigidification. For connectivity it claims that for every k≥2 a 'merger-cascade window' of sublinear width accumulates a linear number of inter-component mergers, so the giant-component fraction jumps by a macroscopic amount across an infinitesimal density interval — a first-order, explosive transition — and that the threshold is monotonically delayed as k grows. For rigidity it claims that on the richly-connected Intra host (axis, face-diagonal, and body-diagonal bonds) increasing k monotonically increases the expected rigidity gain of each added bond, shrinking the rigidity–connectivity gap; a conditional progress function linking local product score to global mechanical utility is offered as the mechanism, while the sparse nearest-neighbor host is shown to be blocked from global rigidity by layered shear flexes. A sympathetic reader would care because these results would convert a cheap local heuristic into a design principle: more choice buys sharper transitions and mechanically cheaper rigidification, with implications for growing or printing stiff lattice structures bond by bond.

Core claim

The central claim (Theorem IV.37): on bounded-degree cubic-lattice hosts, for every k≥2 there is a size-independent δ>0 and a density window of vanishing width across which the expected largest-component fraction rises by at least δ — a genuine first-order jump in the thermodynamic limit, produced by a merger-cascade window (Theorem IV.25) in which Θ(N) inter-component mergers occur in o(N) steps. The companion claim (Theorem IV.51): under two stated assumptions, the expected single-step rigidity gain is non-decreasing in k, explaining the numerically observed shrinkage of the rigidity–connectivity gap on the richly-connected Intra host. The paper also proves the connectivity threshold is mo

What carries the argument

The central object is the product score s(e)=|C(u)|·|C(v)| — the product of the sizes of the two components a candidate bond would join — with the rule adding the sampled bond of minimal score. Three devices carry the argument: a mesoscopic-sparsity lemma asserting the inclusive susceptibility stays uniformly bounded up to the pseudo-threshold pc,α, guaranteeing all components are sublinear just before the transition; a martingale concentration bound on windowed merge counts, plus a counting lemma on inter- versus intra-component missing edges, showing a linear number of merges land in a sublinear step window (the merger cascade); and, for rigidity, the conditional progress function P(s)=E[r

Load-bearing premise

The load-bearing premise is that the inclusive susceptibility — the average component size of a random vertex, giant component included — stays bounded by a constant independent of lattice size right up to the pseudo-threshold pc,α, which cannot hold at pc,α itself since the giant's α-fraction alone contributes at least α²N, and which rests on a lemma (|Cmax|/N ≥ χ_L) that the paper's own supplementary lemma S3.1 proves in the opposite direction (χ_L ≤ |Cmax|).

What would settle it

On the NN lattice with k=2, evaluate the inclusive susceptibility χ_L at the pseudo-threshold pc,α: because the largest component has fraction ≥ α there, χ_L ≥ α²N grows with L, directly contradicting the O(1) bound asserted by Proposition IV.7 and Lemma IV.19 — the premise on which the merger-cascade window and Theorem IV.37 rest; this one measurement, together with an extrapolation of the order-parameter jump across the claimed window (Theorem IV.37 says ≥ δ>0, Corollary IV.44 says 0), settles the matter.

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If this is right

  • For k≥2, the cubic-lattice connectivity transition is claimed to be genuinely first-order: a macroscopic fraction of vertices join the giant component inside a density window whose width tends to zero (Theorem IV.37).
  • More choice monotonically delays percolation: pc,α(k) is non-decreasing in k, proved for every finite lattice by coupling processes with different k on shared randomness (Theorem IV.42).
  • On the Intra host, rigidity becomes monotonically cheaper with k: expected rank gain per added edge rises and expected redundant edges fall, so the rigidity–connectivity gap shrinks — from about 0.42 at k=1 to 0.25 at k=32 at the largest simulated size.
  • On the NN host, no finite density can rigidify the lattice: explicit layered-shear flexes survive any o(N) bond additions, so the local rule cannot overcome the geometric obstruction.
  • Finite-size signatures claimed to accompany the first-order behavior are bimodal order-parameter distributions and susceptibility scaling with exponent γ→1 for k≥2; transition sharpness is non-monotonic in k, peaking at intermediate k before large-k stepwise growth sets in.

Where Pith is reading between the lines

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

  • Editorial inference: the paper's own quoted theorem for fixed k (Theorem IV.43) and its Corollary IV.44 — the transition is continuous and the jump satisfies ∆(k)=0 in the thermodynamic limit — sit in tension with Theorem IV.37's positive-jump claim; read sympathetically, the durable content is the finite-size first-order crossover at realistic lattice sizes rather than a literal infinite-size dis
  • Editorial inference: a discriminating test the authors did not report is a per-step growth-rate histogram of the largest component; if the merger cascade is real, the increments should concentrate into a heavy-tailed spike inside the claimed sublinear window, which would remain visible even if the infinite-size jump does not survive.
  • Editorial inference: the monotonic-efficiency mechanism, if correct, should generalize to any sufficiently dense non-bipartite host whose bond directions span 3D (the property that makes the Intra host special) and should fail on planar or bipartite hosts — a host-by-host prediction checkable in 2D analogs.
  • Editorial inference: the intermediate-k sharpening implies an optimal choice parameter k_opt(L) that should grow with system size; mapping k_opt(L) across hosts would convert the theoretical picture into a practical fabrication heuristic — add just enough local choice to buy the stiffness gain, since sharpness degrades beyond k_opt.

Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

6 major / 6 minor

Summary. This paper studies percolation under the k-choice Achlioptas product rule on two three-dimensional cubic-lattice hosts, the nearest-neighbor (NN) model and the Intra model with face and body diagonals. The authors report numerical simulations (Section III) showing a sharpening connectivity transition for k>=2, susceptibility peaks scaling with exponent near 1, and a rigidity–connectivity gap in the Intra host that shrinks as k increases. The theoretical part claims: (i) a rigorous proof of sublinear merger-cascade windows for k>=2 leading to a first-order connectivity transition (Theorems IV.25, IV.28, IV.37); (ii) monotone delay of the threshold in k (Theorem IV.42); and (iii) a conditional progress-function model for monotonic rigidification efficiency (Theorem IV.51), said to rest on two physically motivated assumptions. The SI provides supporting material including a proxy-model rigidity proof for Erdős–Rényi Intra subgraphs.

Significance. Were the connectivity theorems valid, the paper would be significant: a rigorous first-order jump for fixed k on a fixed-dimensional lattice would sit in tension with established continuity results for Achlioptas processes and would reframe the interpretation of explosive-percolation simulations. The numerical observation of a host-dependent, k-dependent rigidity–connectivity gap is potentially interesting, and the manuscript is transparent in identifying which rigidity results are conditional. Credit is due for attempting a self-contained framework with SI proofs. However, the central proofs contain a factor-N error in the susceptibility bound, an arithmetic impossibility in the merger-cascade window, and a direct contradiction between Theorem IV.37 and Corollary IV.44; the rigidity theorem is explicitly conditional on an unproved assumption. As it stands, the theoretical framework does not support the headline claims, and the numerical evidence alone (L<=10 in the body) is too limited to carry them.

major comments (6)
  1. [§II.D/§IV.B, Lemma IV.4, Prop IV.7, Lemma IV.19; SI Lemma S3.1] The uniform mesoscopic-sparsity bound chi_L(p) <= eta for all p <= pc,alpha (Prop IV.7, Lemma IV.19) is false. Lemma IV.4's inequality chi_L <= |Cmax|/N is a factor-N error: from sum a_i^2 <= a_max sum a_i = a_max N one obtains chi_L <= |Cmax|, not |Cmax|/N; SI Lemma S3.1 repeats the same algebra error (concluding a_max N/N = |Cmax|/N). Moreover, at the pseudo-threshold E[|Cmax|] = alpha N and chi_L >= S_max^2/N, so E[chi(pc,alpha)] >= alpha^2 N, which is unbounded; Lemma IV.19's own display E[chi] <= N*P_N <= alpha N gives a divergent bound, not a constant K. Theorems IV.21, IV.24, and IV.31 all inherit this false input.
  2. [§IV.C, Theorem IV.25; Proposition IV.33; Proposition IV.35] Theorem IV.25 asserts X_{t-,t+} >= c*N merges in a window of width w=o(N). Since the merge count X is a sum of w indicator variables (Def IV.8), X <= w deterministically; a sublinear window cannot contain a linear number of merges. The proof first fixes w=N^gamma then switches to 'w=c0N ... while still w=o(N)', which is a contradiction in terms: c0N is linear. Proposition IV.33 concatenates N^{1/3} subwindows of width N^{2/3}, total length Theta(N), not N^{2/3} as its statement claims. Proposition IV.35 states 'total width o(N)' but proves W=Bw=Theta(N). Thus Theorem IV.28's jump criterion has no valid sublinear-window input, and Theorem IV.37's o(1)-density jump is unsupported.
  3. [§IV.C–D, Theorem IV.37 vs. Corollary IV.44] Theorem IV.37 concludes a genuine thermodynamic discontinuity: liminf_L [P_N(p+(L))-P_N(p-(L))] >= delta>0 across density intervals of width ->0, and the text calls this 'a hallmark of a first-order (explosive) transition.' This directly contradicts Corollary IV.44 (Delta(k)=0 for every fixed k) and the abstract's statement that the transition 'is continuous in the infinite thermodynamic limit.' The manuscript never reconciles these claims; the finite-size 'signature' language of Section IV.D and the thermodynamic jump statement of Theorem IV.37 are different assertions, and the proof of the theorem does not bridge them.
  4. [§IV.C, Lemma IV.22, Lemma IV.31, Proposition IV.23] Lemma IV.31's proof counts Theta(N^2) inter-component vertex pairs and concludes 'a Theta(N) pool of inter-component host edges,' conflating vertex pairs with bounded-degree host edges; the inter-component edge count is O(Delta N) regardless of pair counts, and a linear lower bound additionally requires average component size O(1), which Proposition IV.23 does not provide (it guarantees only o(N) component sizes; with size N^{2/3}, inter-component edges can be sublinear). Lemma IV.22 similarly asserts linearity under hypotheses that yield only N/S. Proposition IV.23 and Lemma IV.29 define tc,alpha through the expectation P_N but treat it as an almost-sure hitting time; the stated contradiction with 'the definition of tc,alpha' is therefore invalid.
  5. [§IV.G, Theorem IV.51; Appendix S5; Theorem IV.42] Theorem IV.51 is stated unconditionally, but its proof begins 'Conditional on the validity of the Monotonic Density assumption (Theorem IV.48)' — Assumption IV.48 is an unproved assumption, not a theorem. Lemma IV.47, the non-increasing property of P(s), is deferred to Appendix S5 and proved there only under Assumption S5.3, the same monotone-density assumption. Lemma IV.49 / Hypothesis IV.50 is established only for an Erdős–Rényi proxy (Theorem S4.3) and then explicitly hypothesized to carry over to the Achlioptas process. Moreover, the proof invokes the coupling of Theorem IV.42, whose candidate sets are sampled from edges unused in either process and hence do not reproduce either process's marginal distribution. The monotone-efficiency claim is therefore unsupported at every load-bearing step; the theorem statement should be made explicitly conditional.
  6. [§IV.C–B, Theorem IV.21; Lemmas IV.24, IV.32] Several steps in the cascade are asserted rather than proved. Theorem IV.21's proof contains 'Details are routine', 'A double counting argument ... shows', and 'This handles the small-s case', and its union bound over small sets is admitted to be 'superpolynomially large if taken literally'; no valid bound is supplied. Lemma IV.24 and Lemma IV.32 infer that the product rule selects an inter-component edge whenever any sampled candidate is inter-component, but the rule selects the minimum product; an intra-component candidate inside a small component can have a smaller product, so the claimed per-step merge probability p* is not established. These gaps are inherited by Theorem IV.25 and Proposition IV.33.
minor comments (6)
  1. [Abstract vs. Section III] The abstract's 'massive-scale simulations up to L=192 (N approx 7e6) with 20,000 independent realizations' do not appear in the main text or SI; Section III reports L=1,...,10 with 1,000 realizations and Tables S1/S2 cover L<=10. The headline finite-size-scaling claims should be tied to data actually presented.
  2. [Abstract vs. Tables S1/S2] The abstract states that for k>=8 the peak-susceptibility exponent is gamma=1.000; Tables S1 and S2 report, e.g., k=8: gamma=1.118 for both hosts, and Intra k=17-32: gamma between 0.978 and 0.998. The stated exact value 1.000 is not what the tables show.
  3. [SI Lemma S3.1] The final step of SI Lemma S3.1, 'a_max N/N = |Cmax(t)|/N', is a factor-N algebra error; the correct consequence is |Cmax(t)| (see Major Comment 1). The SI proof should be corrected in any revision.
  4. [Theorem IV.51 proof; Assumption IV.48; Hypothesis IV.50] The proof of Theorem IV.51 refers to 'Theorem IV.48' for the Monotonic Density assumption, and Lemma IV.49's justification is labelled 'Justification' based on a proxy model. Assumptions and hypotheses should be labelled consistently so the reader can track what is proved vs. assumed.
  5. [Lemma IV.14/Theorem IV.15] Lemma IV.14 claims |M_s-M_{s-1}|<=1 for the Doob martingale, but its proof abandons M and proves bounded differences for the partial-sum martingale S_u instead. Theorem IV.15's inequality is correct via S_u; the lemma statement should be revised or removed.
  6. [Section III.A and Fig. 4 caption] The claim that gamma 'rapidly approaches 1' for k>=2 is weakened by the values in Tables S1/S2 for k=2 (gamma ~ 1.18) and k=17-32 (gamma slightly below 1); the text should discuss deviations from gamma=1 rather than assert approach. Also, the figure captions label susceptibility as chi_max for the Intra model while the text defines chi'_max; please make notation consistent.

Circularity Check

2 steps flagged

Partially circular: Theorem IV.42 assumes the monotone-coupling conclusion it purports to prove, and the merger-cascade window contradicts its own o(N) definition; the rigidity theorem is explicitly conditional.

specific steps
  1. other [Section IV.F, Theorem IV.42 (Monotone delay under increasing choice), proof step (c)]
    "Adding edges never decreases component sizes; thus for any e, S^{G(k2)}_{t-1}(e) ≤ S^{G(k1)}_{t-1}(e), because G(k2)_{t-1} has had at least as much 'small-merge filtering' as G(k1)_{t-1} (we formalize this by induction below, but at this point it suffices to use the simple fact that component sizes are nondecreasing as edges are added, and the k2 process picks no larger product than the k1 process, step-by-step)."

    The displayed inequality compares component sizes of the two coupled processes at the same time, which is exactly the monotonicity the theorem must prove. The justification invokes 'the k2 process picks no larger product than the k1 process', i.e., the theorem's own conclusion. The promised induction is never supplied, and the later largest-component comparison repeats the same asserted inequality. Thus Theorem IV.42 does not derive monotone delay from the coupling; it assumes it. Theorem IV.51 uses 'the synchronous coupling from Theorem IV.42' to obtain stochastic ordering of scores, so the monotone-efficiency claim inherits this circularity.

  2. other [Section IV.C, Theorem IV.25 (Merger-cascade window for k≥2), proof]
    "or simply choose γ close to 1 so that w=c_0N for small constant c_0>0 while still w=o(N), if one prefers a single window argument; both viewpoints lead to the same conclusion since we only need linear-in-N total merges over an o(N) span"

    By Definition IV.8, X_{t−,t+} is a sum of w merge indicators, so X_{t−,t+} ≤ w deterministically. The theorem asserts X_{t−,t+} ≥ c*N with w=o(N), which is impossible. The proof's escape 'w=c_0N while still w=o(N)' contradicts the definition of o(N): c_0N is linear. Proposition IV.33 likewise concatenates N^{1/3} subwindows of width N^{2/3} into total length Θ(N). Hence the 'sublinear merger-cascade window' that feeds Theorem IV.28 and Theorem IV.37 is asserted by contradicting the process definitions rather than derived from them.

full rationale

The numerical portions are self-contained and externally benchmarked (k=1 matches the known cubic-lattice threshold p_c≈0.24881), so they do not introduce circularity. The genuine circular reduction is Theorem IV.42: its proof needs the component-size comparison it is trying to establish, and justifies it by the theorem's conclusion. This infects the rigidity half, since Theorem IV.51 explicitly relies on the synchronous coupling from Theorem IV.42. Separately, the first-order connectivity chain has a definitional breakdown: the claimed sublinear window with a linear number of merges contradicts X≤w, and the proof's w=c_0N is not o(N); this is an internal inconsistency, but it is severe because the deterministic jump criterion (Theorem IV.28) has no valid sublinear input. The rigidity theorem is honestly labeled conditional: Assumption IV.48 states 'a formal proof is beyond the scope of this paper', Lemma IV.49 relies on Hypothesis IV.50, and the proof begins 'Conditional on the validity of the Monotonic Density assumption'. These are limitations rather than concealed circularity, but they mean that half of the headline results is not a first-principles derivation. No load-bearing self-citation chain or imported uniqueness theorem was found; the self-citations to prior origami/kirigami work are motivational, and the continuity citations are external.

Axiom & Free-Parameter Ledger

1 free parameters · 5 axioms · 3 invented entities

The theory adds no fitted physical constants; its numerical claims (γ, pc, Δpc) are measurements over L=1..10. The 'proofs' rest on: the false uniform susceptibility bound (Prop IV.7 — χ_L includes the giant component and is Θ(N) at p_{c,α}); an unproved monotone-density assumption (IV.48) that carries the entire rigidity-efficiency theorem; a proxy-model result (S4) transported by hypothesis to the Achlioptas process; generic-placement assumptions; and the bounded-degree/M=Θ(N) facts. The free-parameter entry records the abstract's γ=1.000 claim versus the fitted values.

free parameters (1)
  • FSS exponent γ (abstract claims 1.000 for k≥8) = 0.978–1.232 across fits; 1.118 at k=8; 1.012 at k=32 (Tables S1–S2)
    The central finite-size-scaling claim rests on γ fitted by linear regression over L=1..10; the fitted values do not reproduce the abstract's stated 1.000, and no uncertainties are reported.
axioms (5)
  • ad hoc to paper Uniform mesoscopic sparsity up to the pseudo-threshold: χ_L(p) ≤ η(α,∆,k) for all p ≤ pc,α (Prop IV.7, Lemma IV.19).
    Load-bearing premise of the cascade proof; false as stated because the inclusive susceptibility includes the giant component (χ_L(pc,α) ≥ α²N), and the proof's Lemma IV.4 misstates the bound by a factor of N.
  • ad hoc to paper Assumption IV.48 (Monotonic Average Density): average internal edge density of product-rule components is non-decreasing in size.
    Unproved ('a formal proof is beyond the scope of this paper'); it is the input to Lemma IV.47 and hence Theorem IV.51, and it encodes the qualitative content of the monotone-efficiency conclusion.
  • ad hoc to paper Hypothesis IV.50 / Lemma IV.49 (SDA): large dense components of the Intra host are rigid; proven only for an ER proxy model (Appendix S4) and then transported to the history-dependent Achlioptas process.
    The rigorous proof is for a different process; the transfer is hypothesized (authors: 'We hypothesize that this property... carries over').
  • standard math Generic placements (Assumption II.8 / S1.24): all rigidity statements hold for measure-one vertex placements.
    Standard in rigidity theory; avoids coordinate degeneracies.
  • domain assumption Bounded maximum degree of host graphs and M=Θ(N) (Lemma IV.36).
    True for the NN/Intra lattices; used throughout the cascade proofs and the density-window conversion.
invented entities (3)
  • Conditional progress function P(s)=E[rgain(e)|s(e)=s] no independent evidence
    purpose: Bridge local product-rule score to global mechanical utility; the basis of Theorem IV.51.
    A definition, not an observable; no falsifiable prediction outside the framework; its monotonicity is assumed/proxy-derived rather than established for the Achlioptas process.
  • Effective coordination deff(k)=2M p^conn_c(k)/N no independent evidence
    purpose: Unifying parameter for the rigidity-gap hypothesis (Hypothesis IV.2).
    Defined from the measured threshold in the same simulations; the mean-field gap formula is used to 'predict' a trend that is then verified on the same data that defines deff.
  • Merger-cascade window no independent evidence
    purpose: Postulated sublinear-width window with linear merges, driving the claimed first-order jump.
    No direct measurement of the window is reported; and the construction is arithmetically impossible as stated (X≤w with w=o(N)).

pith-pipeline@v1.3.0-alltime-deepseek · 53909 in / 30805 out tokens · 315105 ms · 2026-08-04T00:21:19.198764+00:00 · methodology

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Cite this review

Pith. "Pith review of Explosive connectivity and mechanical rigidity in cubic lattice structures." pith.science (2026). https://pith.science/paper/BWVEPRWY

@misc{pith2026251101537,
  author       = {Pith},
  title        = {Pith review of: Explosive connectivity and mechanical rigidity in cubic lattice structures},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/BWVEPRWY}},
  note         = {Machine review of arXiv:2511.01537}
}
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read the original abstract

We study explosive connectivity and mechanical rigidity in three-dimensional cubic lattice structures under Achlioptas-type product-rule dynamics. Our work combines extensive numerical simulation with a theoretical framework based on rigorous finite-size scaling. Using massive-scale simulations up to $L=192$ ($N \approx 7 \times 10^6$) with 20,000 independent realizations, we demonstrate that for $k \ge 8$, the peak susceptibility scales with an exponent of $\gamma = 1.000$, and the maximum single-step jump stabilizes at a macroscopic fraction. This confirms that while the transition is continuous in the infinite thermodynamic limit, it exhibits the exact finite-size scaling signatures of a first-order discontinuity in finite physical systems. For rigidity, we discover numerically that for richly-connected hosts, increasing the number of choices $k$ optimally enhances the efficiency of rigidification. To explain this phenomenon, we propose a theoretical model centered on a conditional progress function that links an edge's local product-rule score to its global mechanical utility. We show that while local rigidification efficiency monotonically increases, the global rigidity gap exhibits a ``Goldilocks'' minimum at intermediate $k$ due to the emergence of maximally floppy, tree-like components at large $k$. Altogether, our work provides new insights into the relationship between local dynamics and global connectivity and rigidity in cubic lattice structures via both theory and computation.

Figures

Figures reproduced from arXiv: 2511.01537 by Gary P. T. Choi, Trenton Lau.

Figure 2
Figure 2. Figure 2: FIG. 2 [PITH_FULL_IMAGE:figures/full_fig_p002_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3 [PITH_FULL_IMAGE:figures/full_fig_p005_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: FIG. 4 [PITH_FULL_IMAGE:figures/full_fig_p006_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: FIG. 5 [PITH_FULL_IMAGE:figures/full_fig_p007_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: FIG. 6 [PITH_FULL_IMAGE:figures/full_fig_p008_6.png] view at source ↗

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Reference graph

Works this paper leans on

88 extracted references · 4 canonical work pages · 4 internal anchors

  1. [1]

    Let e={u,v} be a candidate edge not in Gt−1

    The Monotonic Relationship between Score and Redundancy Let (Ft )t≥0 be the natural filtration generated by the k- choice process. Let e={u,v} be a candidate edge not in Gt−1. Its product score is s(e) =|Ct−1 (u)| · |Ct−1 (v)|, and its rank gain is rgain(e)∈ {0,1}. Definition IV .46(Conditional Progress Function).For any score value s>0 , we define thecon...

  2. [2]

    force balance constraints

    Main Theorem: Monotonic Efficiency of Rigidification With these two lemmas rigorously established, we can now prove the main theorem. Theorem IV .51(Monotonic Efficiency with Choice).For the k-choice product-rule process, let E[∆Φ(k) t ] be the expected single-step progress towards rigidity at step t. For any 1≤ k1 <k 2, we have: E[∆Φ(k2) t ]≥E[∆Φ (k1) t ...

  3. [3]

    Stauffer and A

    D. Stauffer and A. Aharony,Introduction to Percolation Theory (Taylor & Francis, 2018)

    2018

  4. [4]

    Christensen and N

    K. Christensen and N. R. Moloney,Complexity and criticality, V ol. 1 (Imperial College Press, 2005)

    2005

  5. [5]

    Majewski and K

    M. Majewski and K. Malarz, Acta Phys. Pol. B38, 2191 (2007)

    2007

  6. [6]

    N. A. M. Araújo, P. Grassberger, B. Kahng, K. J. Schrenk, and R. M. Ziff, Eur. Phys. J. Spec. Top.223, 2307 (2014)

    2014

  7. [7]

    Xun and R

    Z. Xun and R. M. Ziff, Phys. Rev. E102, 012102 (2020)

    2020

  8. [8]

    Meloni, M

    M. Meloni, M. J. Large, J. M. González Domínguez, S. Víctor- Román, G. Fratta, E. Istif, O. Tomes, J. P. Salvage, C. P. Ewels, M. Peláez-Fernández,et al., Nat. Commun.13, 6872 (2022)

    2022

  9. [9]

    Sahimi,Applications of Percolation Theory, Applied Mathe- matical Sciences, V ol

    M. Sahimi,Applications of Percolation Theory, Applied Mathe- matical Sciences, V ol. 213 (Springer Cham, 2023)

    2023

  10. [10]

    Achlioptas, R

    D. Achlioptas, R. M. D’Souza, and J. Spencer, Science323, 1453 (2009)

    2009

  11. [11]

    R. A. da Costa, S. N. Dorogovtsev, A. V . Goltsev, and J. F. F. Mendes, Phys. Rev. Lett.105, 255701 (2010)

    2010

  12. [12]

    Radicchi and S

    F. Radicchi and S. Fortunato, Phys. Rev. E81, 036110 (2010)

    2010

  13. [13]

    Riordan and L

    O. Riordan and L. Warnke, Science333, 322 (2011)

    2011

  14. [14]

    H. K. Lee, B. J. Kim, and H. Park, Phys. Rev. E84, 020101 (2011)

    2011

  15. [15]

    R. M. D’Souza and J. Nagler, Nat. Phys.11, 531 (2015)

    2015

  16. [16]

    R. M. D’Souza, J. Gómez-Gardeñes, J. Nagler, and A. Arenas, Adv. Phys.68, 123 (2019)

    2019

  17. [17]

    Asimow and B

    L. Asimow and B. Roth, Trans. Am. Math. Soc.245, 279 (1978)

    1978

  18. [18]

    Moukarzel, J

    C. Moukarzel, J. Phys. A: Math. Gen.29, 8079 (1996)

    1996

  19. [19]

    Cheng and M

    J. Cheng and M. Sitharam, J. Comb. Theory, Ser. B105, 26 (2014)

    2014

  20. [20]

    D. J. Jacobs and B. Hendrickson, J. Comput. Phys.137, 346 (1997)

    1997

  21. [21]

    Lee and I

    A. Lee and I. Streinu, Discrete Math.308, 1425 (2008)

    2008

  22. [22]

    Li and G

    R. Li and G. P. T. Choi, Proc. R. Soc. A481, 20240826 (2025)

    2025

  23. [23]

    Rigidity control of general origami structures

    R. Li and G. P. T. Choi, arXiv preprint arXiv:2507.16934 10.48550/arXiv.2507.16934 (2025)

    work page internal anchor Pith review Pith/arXiv arXiv doi:10.48550/arxiv.2507.16934 2025

  24. [24]

    G. P. T. Choi, L. Liu, and L. Mahadevan, Proc. R. Soc. A479, 20220798 (2023)

    2023

  25. [25]

    M. Li, J. Wang, and Y . Deng, Phys. Rev. Lett.130, 147101 (2023)

    2023

  26. [26]

    M. Li, J. Wang, and Y . Deng, Phys. Rev. Res.6, 033319 (2024)

    2024

  27. [27]

    G. P. T. Choi, S. Chen, and L. Mahadevan, Proc. R. Soc. A476, 20200485 (2020)

    2020

  28. [28]

    J. E. Graver, B. Servatius, and H. Servatius,Combinatorial Rigidity, Graduate Studies in Mathematics, V ol. 2 (American Mathematical Society, Providence, RI, 1993)

    1993

  29. [29]

    Garamvölgyi, S

    D. Garamvölgyi, S. J. Gortler, and T. Jordán, Forum Math. Sigma 10, e44 (2022)

    2022

  30. [30]

    Laman, J

    G. Laman, J. Eng. Math.4, 331 (1970)

    1970

  31. [31]

    Cruickshank, B

    J. Cruickshank, B. Jackson, T. Jordán, and S.-i. Tanigawa, arXiv preprint arXiv:2508.11636 10.48550/arXiv.2508.11636 (2025)

    Pith/arXiv arXiv doi:10.48550/arxiv.2508.11636 2025

  32. [32]

    S. M. Cioab˘a, S. Dewar, and X. Gu, Discrete Math.344, 112527 (2021)

    2021

  33. [33]

    Minimum degree conditions for graph rigidity

    M. Krivelevich, A. Lew, and P. Michaeli, arXiv preprint arXiv:2412.14364 10.48550/arXiv.2412.14364 (2024)

    work page internal anchor Pith review Pith/arXiv arXiv doi:10.48550/arxiv.2412.14364 2024

  34. [34]

    Binder, Z

    K. Binder, Z. Phys. B43, 119 (1981)

    1981

  35. [35]

    Efron, inBreakthroughs in Statistics, Springer Series in Statis- tics, edited by S

    B. Efron, inBreakthroughs in Statistics, Springer Series in Statis- tics, edited by S. Kotz and N. L. Johnson (Springer, New York, NY , 1992) pp. 569–593

    1992

  36. [36]

    R. M. Ziff, Phys. Rev. Lett.103, 045701 (2009)

    2009

  37. [37]

    M. E. J. Newman and R. M. Ziff, Phys. Rev. E64, 016706 (2001)

    2001

  38. [38]

    Goerdt, Theor

    A. Goerdt, Theor. Comput. Sci.259, 307 (2001)

    2001

  39. [39]

    Lichev, D

    L. Lichev, D. Mitsche, and G. Perarnau, J. Comb. Theory, Ser. B 167, 250 (2024)

    2024

  40. [40]

    J. A. Hartigan and P. M. Hartigan, Ann. Stat.13, 70 (1985)

    1985

  41. [41]

    Gómez-Gardeñes, S

    J. Gómez-Gardeñes, S. Gómez, A. Arenas, and Y . Moreno, Phys. Rev. Lett.106, 128701 (2011)

    2011

  42. [42]

    Selke, J

    W. Selke, J. Stat. Mech.2007, P04008 (2007)

    2007

  43. [43]

    A. Lew, E. Nevo, Y . Peled, and O. E. Raz, Combinatorica45, 24 (2025)

    2025

  44. [44]

    Frieze and M

    A. Frieze and M. Karo´nski,Introduction to random graphs(Cam- bridge University Press, 2015)

    2015

  45. [45]

    Duminil-Copin, Jpn

    H. Duminil-Copin, Jpn. J. Math.14, 1 (2019)

    2019

  46. [46]

    Perkins, Bull

    W. Perkins, Bull. Am. Math. Soc.62, 113 (2025)

    2025

  47. [47]

    Xing and K

    Y . Xing and K. H. Johansson, SIAM J. Control Optim.62, 1521 (2024)

    2024

  48. [48]

    Durrett,Probability: Theory and Examples, 5th ed., Cam- bridge Series in Statistical and Probabilistic Mathematics, V ol

    R. Durrett,Probability: Theory and Examples, 5th ed., Cam- bridge Series in Statistical and Probabilistic Mathematics, V ol. 49 (Cambridge University Press, 2019)

    2019

  49. [49]

    Binder and D

    K. Binder and D. P. Landau, Phys. Rev. B30, 1477 (1984)

    1984

  50. [50]

    Chae, S.-H

    H. Chae, S.-H. Yook, and Y . Kim, Phys. Rev. E85, 051118 (2012)

    2012

  51. [51]

    Y . S. Cho, Eur. Phys. J. B97, 58 (2024)

    2024

  52. [52]

    Olvera-Cravioto, J

    M. Olvera-Cravioto, J. Appl. Probab.59, 1261 (2022)

    2022

  53. [53]

    Grassberger, C

    P. Grassberger, C. Christensen, G. Bizhani, S.-W. Son, and M. Paczuski, Phys. Rev. Lett.106, 225701 (2011)

    2011

  54. [54]

    G. Lois, J. Blawzdziewicz, and C. S. O’Hern, Phys. Rev. Lett. 100, 028001 (2008)

    2008

  55. [55]

    P. S. Noerr, J. E. Zamora Alvarado, F. Golnaraghi, K. E. Mc- Closkey, A. Gopinathan, and K. Dasbiswas, Proc. Natl. Acad. Sci. U.S.A.120, e2301555120 (2023)

    2023

  56. [56]

    Geometrically Frustrated Assembly at Finite Temperature: Phase Transitions from Self-Limiting to Bulk States

    N. Hackney and G. Grason, arXiv preprint arXiv:2508.21688 10.48550/arXiv.2508.21688 (2025)

    work page internal anchor Pith review Pith/arXiv arXiv doi:10.48550/arxiv.2508.21688 2025

  57. [57]

    J. U. Surjadi, B. F. G. Aymon, M. Carton, and C. M. Portela, arXiv preprint arXiv:2409.01533 10.48550/arXiv.2409.01533 (2024)

    work page internal anchor Pith review Pith/arXiv arXiv doi:10.48550/arxiv.2409.01533 2024

  58. [58]

    Mehta, M

    P. Mehta, M. Bukov, C.-H. Wang, A. G. Day, C. Richardson, C. K. Fisher, and D. J. Schwab, Phys. Rep.810, 1 (2019)

    2019

  59. [59]

    Ghrist, Bull

    R. Ghrist, Bull. Am. Math. Soc.45, 61 (2008)

    2008

  60. [60]

    J. A. Tropp, Found. Trends Mach. Learn.8, 1 (2015)

    2015

  61. [61]

    Friedgut, J

    E. Friedgut, J. Am. Math. Soc.12, 1017 (1999)

    1999

  62. [62]

    A. Lew, E. Nevo, Y . Peled, and O. E. Raz, Bull. London Math. Soc.55, 490 (2023)

    2023

  63. [63]

    Krivelevich, A

    M. Krivelevich, A. Lew, and P. Michaeli, J. Comb. Theory, Ser. B175, 126 (2025)

    2025

  64. [64]

    Peled and N

    Y . Peled and N. Peleg, arXiv preprint arXiv:2412.13127 10.48550/arXiv.2412.13127 (2024)

    Pith/arXiv arXiv doi:10.48550/arxiv.2412.13127 2024

  65. [65]

    Molloy and B

    M. Molloy and B. Reed, Random Struct. Algorithms6, 161 (1995)

    1995

  66. [66]

    Molloy and B

    M. Molloy and B. Reed, Combin. Probab. Comput.7, 295 (1998)

    1998

  67. [67]

    M. F. Thorpe, P. M. Duxbury, D. J. Jacobs, and M. V . Chubynsky, inRigidity Theory and Applications, edited by M. F. Thorpe and P. M. Duxbury (Kluwer Academic/Plenum Publishers, New York, NY , 1999) pp. 1–15

    1999

  68. [68]

    Bollobás, inModern Graph Theory, Graduate Texts in Math- ematics, V ol

    B. Bollobás, inModern Graph Theory, Graduate Texts in Math- ematics, V ol. 184 (Springer, New York, NY , 1998) pp. 215–252

    1998

  69. [69]

    Waagen and R

    A. Waagen and R. M. D’Souza, Eur. Phys. J. B87, 267 (2014). 25 Supplementary Information Appendix S1: Theoretical Preliminaries For completeness and self-containedness, in this supple- mentary section, we provide the detailed descriptions of the concepts and preliminaries in graph theory and rigidity theory relevant to our work

    2014

  70. [70]

    Definition S1.1(Graph).A (simple) graph G= (V,E) consists of a finite set V ofverticesand a set E⊆ {{u,v}:u,v∈V,u̸= v}ofedges

    Host Families, Processes, and Order Parameters We consider finite host graphs GL = (VL,E L) with bounded maximum degree ∆ on 3D cubic lattice structures, where N= |VL|andM=|E L|. Definition S1.1(Graph).A (simple) graph G= (V,E) consists of a finite set V ofverticesand a set E⊆ {{u,v}:u,v∈V,u̸= v}ofedges. We sayuandvareadjacentif{u,v} ∈E. Definition S1.2(H...

  71. [71]

    Call thisk 2-setS t ={e t,1 ,

    From the master permutation π, scan forward and collect the first k2 edges that are not yet present ineitherprocess at stept−1. Call thisk 2-setS t ={e t,1 , . . . ,et,k2}

  72. [72]

    ,et,k1} ⊂St, and define thek 2-set for the larger-kprocess asS (k2) t =S t

    Define the k1-set for the smaller-k process as the first k1 edges within St, i.e., S(k1) t ={e t,1 , . . . ,et,k1} ⊂St, and define thek 2-set for the larger-kprocess asS (k2) t =S t

  73. [73]

    Compute product scores with respect to thecurrent graphs: s(ki) t (e) = CGt−1 (ki) (u) · CGt−1 (ki) (v) fore={u,v} ∈S (ki) t ,i∈ {1,2}. Then choose e∗ t,(ki) ∈S (ki) t minimizing s(ki) t (e) (break ties uniformly at randomusing the same tie-breaking ran- domness for both processes restricted to their own can- didate sets)

  74. [74]

    the largest component has size at least m by time t

    UpdateG t (ki) by addinge ∗ t,(ki) toG t−1 (ki). Lemma S1.6(Suppressive k-coupling).With the coupled con- struction above, for every step t≥1we have min e∈S(k2) t s(k2) t (e)≤min e∈S(k1) t s(k1) t (e). In words: the product score of the edge actually chosen by the k2-choice process at step t is at most the product score of the edge chosen by the k1-choice...

  75. [75]

    floppy” or “rigid

    Graphs, Frameworks, and Rigidity Definition S1.9(Embedding / Framework in Rd).Fix a dimen- sion d≥1 . Abar-joint framework(or simplyframework) in Rd is a pair (G,P) where G= (V,E) is a graph and p:V→R d assigns to each vertex v∈V aposition p(v)∈R d. We interpret each edge {u,v} ∈Eas a rigid bar of fixed length between the pointsp(u)andp(v). Definition S1....

  76. [76]

    Therigidification cost ∆t(G,P) is the minimal number of edges one must add to G to obtain a graph H⊇G such that (H,P) has f(H,P) =0 (i.e., is infinitesimally rigid)

    Rank Gain, Redundancy, and Rigidification Cost Definition S1.18(Rigidification cost).Let (G,P) be given. Therigidification cost ∆t(G,P) is the minimal number of edges one must add to G to obtain a graph H⊇G such that (H,P) has f(H,P) =0 (i.e., is infinitesimally rigid). If no such H exists, set∆t(G,P) = +∞. Lemma S1.19(Each independent flex needs at least...

  77. [77]

    layered shear

    Rigidity of Cubic Unit Cells Lemma S1.21(A single NN cell is not rigid).A framework whose graph is a 2×2×2 NN cubic cell is not generically rigid. Proof. The proof is a direct application of Maxwell’s condition. A standard NN cell has |V|=8 vertices and |E|=12 edges. The necessary number of edges for generic rigidity in 3D is 3|V| −6=3(8)−6=18 . Since the...

  78. [78]

    This means ∆(k) =0 for all k≥1

    Convergence of the jump ∆(k):From Theorem IV .43 and its corollary, we know that for any fixed k, the transition is continuous in the thermodynamic limit. This means ∆(k) =0 for all k≥1 . The sequence {∆(k)}k≥1 is therefore {0,0,0, . . .}. The limit of this sequence is trivially ∆(∞) =0. A non-zero jump in the deterministic limit could only be achieved if...

  79. [79]

    A critical part of this argument is that for a large, dense component, adding another internal edge is almost certainly redundant

    Introduction and Formal Statement of the Problem The central claim of monotonic rigidification efficiency in the main paper hinges on the connection between a local se- lection rule and a global mechanical property. A critical part of this argument is that for a large, dense component, adding another internal edge is almost certainly redundant. This idea ...

  80. [80]

    unbiased

    Strategy: Proof on a Tractable Model Our strategy is to prove the SDA property for the giant component of anErd˝ os-Rényi random subgraph of the Intra- host graph. We consider the graph GIntra(N,p) where each of the M potential edges of the full Intra-host graph on N= (L+1) 3 vertices is included independently with probability p. If the property holds for...

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