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REVIEW 3 major objections 3 minor 43 references

Fully independent response in disordered solids

T0 review · 3 major / 3 minor · reviewed 2026-08-11 · deepseek-v4-flash

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Pith's one-line read Disordered solids are fully independent: any target features can be tuned independently as long as there are at least as many adjustable parameters.

desk verdict Solid Jacobian-based formalism and broad numerical evidence for full independent response in disordered solids; the universal 'always' claim is an extrapolation beyond the tested scenarios. read the letter →

arxiv 2412.05031 v2 pith:FN6LLKSO submitted 2024-12-06 physics.comp-ph cond-mat.dis-nncond-mat.mtrl-scicond-mat.soft

Mengjie Zu , Aayush Desai , Carl P. Goodrich This is my paper · ORCID
classification physics.comp-phcond-mat.dis-nncond-mat.mtrl-scicond-mat.soft
keywords disorderedsolidsindependentresponseinversedesignelasticmoduliautomaticdifferentiationsingularvaluedecompositionjammedpackingsfull-rankJacobian
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 seeks to establish that the mechanical properties of disordered solids are not merely independently tunable in practice, but fully independent in a formal sense. The central quantity is the Jacobian matrix that records how each property changes when each adjustable parameter changes; the paper claims this matrix always has the maximum possible rank in disordered systems, while in crystals it does not. That means any list of target properties—elastic constants, stress components, vibrational frequencies—can be adjusted independently, provided the number of tuning parameters is at least the number of targets. The authors support the claim with 40 simulation scenarios spanning two and three dimensions, different interactions, pressures, system sizes, and parameter choices, and they show the result enables one-step inverse design and simultaneous multi-feature tuning beyond the linear regime. If right, this removes a presumed fundamental limit on what amorphous materials can be designed to do.

What carries the argument

The load-bearing object is the Jacobian $J_{ij}=\partial y_i/\partial\theta_j$, computed at the prepared state by automatic differentiation through the energy minimization and feature calculation. Its singular value decomposition $J=USW^T$ gives the rank as the number of non-zero singular values; the right singular vectors are called design modes, the left singular vectors are called compatible feature strains, and the singular values are interpreted as susceptibilities of feature combinations to parameter combinations. Full independence is defined as rank$(J)=\min(n_y,n_\theta)$, equivalently all singular values positive. The same SVD machinery also supplies the one-step inverse-design formula $\Delta\theta^*\approx J^+\Delta y^*$, where $J^+$ is the pseudo-inverse, and the susceptibility values are used to define the maximum linear tunability $\zeta\equiv\gamma^* s$, the largest feature change achievable before nonlinear corrections or structural rearrangements break the linear approximation.

What would settle it

Pick any mechanically stable disordered spring network not of the soft-sphere form studied here, for instance a central-force network on a random graph, and compute the Jacobian $J$ for a feature list that includes a large set of allosteric responses. If the numerical rank of $J$ (counting singular values above the noise floor) falls below $\min(n_y,n_\theta)$ for any such system, the paper's universal claim is falsified; the paper predicts no such zero singular value will appear.

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Extended reading notes

Core claim

On its own terms, the paper's discovery is that for athermal disordered solids the Jacobian $J_{ij}=\partial y_i/\partial\theta_j$ relating feature changes to parameter changes has rank $\min(n_y,n_\theta)$ in every scenario tested, including 2D and 3D jammed packings with and without attractions, system sizes from $N=256$ to $N=4096$, multiple pressures, and features drawn from elastic constants, stress-tensor components, and the first vibrational mode frequencies. Because the rank of a matrix cannot exceed the smaller dimension, this is the maximum possible rank, and the paper calls this full independence: all feature strains are compatible with some parameter change when $n_\theta\ge n_y$, and all design modes are relevant when $n_y\ge n_\theta$. The singular values of $J$ are all strictly positive in these disordered cases, though they span orders of magnitude, whereas for a triangular-lattice crystal some singular values vanish, so only partial independence is possible. The smallest singular value stays approximately constant as $N$ grows, which the paper takes as evidence that full independence survives in the thermodynamic limit. The authors verify exact full rank for each of ten independent systems per scenario and describe the result as a fundamental difference between order and disorder.

Load-bearing premise

The universal claim rests on the assumption that the 40 simulation scenarios are representative of all disordered solids; the paper itself concedes exceptions may exist and provides no mathematical proof that full rank is forced by disorder.

Editorial extensions

If this is right

  • Any set of $n_y$ mechanical features can be tuned independently in a disordered solid whenever $n_\theta\ge n_y$, so there is no combinatorial cap of the kind found in crystals.
  • For sufficiently small targets, inverse design reduces to a single matrix multiplication $\Delta\theta^*=J^+\Delta y^*$, requiring no iterative optimization.
  • The singular-value spectrum predicts how easy each feature combination is to tune: susceptibility $s$ is correlated with the maximum linear tunability $\zeta=\gamma^*s$, so high-susceptibility directions support larger precise feature changes.
  • Multi-feature design succeeds beyond the linear regime: all six elastic constants in a 2D packing were simultaneously moved to targets 10% away using gradient descent plus one linear correction, even through a structural rearrangement.
  • Full independence persists with system size: the smallest susceptibility is roughly constant from $N=256$ to $N=4096$, so even anisotropic elastic directions that vanish in the thermodynamic limit remain tunable.

Reading between the lines

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

  • If full rank is generic rather than an accident of these packings, the order-to-disorder transition should be visible as rank($J$) rising from the crystal's plateau to $\min(n_y,n_\theta)$ as defects are introduced; computing $J$ for polycrystals or lightly defected crystals would map that transition directly.
  • The documented correlation between susceptibility and maximum linear tunability suggests a design heuristic the paper does not state: when many features must be controlled simultaneously, assign the most stringent targets to high-susceptibility modes and leave low-susceptibility modes for loose constraints, since those modes tolerate smaller parameter excursions.
  • Because the Jacobian method only requires a differentiable forward model, the same full-rank test can be applied to non-mechanical features such as thermal transport or relaxation spectra of model glasses; the paper claims broad applicability but demonstrates it only for mechanical features of athermal packings.
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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

3 major / 3 minor

Summary. The manuscript formalizes the notion of independent response in disordered solids by linearizing the relationship between emergent features (elastic constants, stress-tensor elements, vibrational frequencies) and species-level interaction parameters through a Jacobian matrix J (Eq. 2). Independence is quantified by the rank of J, with full independence defined as rank(J)=min(n_y,n_theta). The central claim is that disordered solids are 'always fully independent' across a wide range of scenarios, in sharp contrast to crystals, whose Jacobian rank saturates below min(n_y,n_theta). The paper also examines the singular values of J as susceptibilities, defines a 'maximum linear tunability' zeta = gamma* s, and demonstrates simultaneous nonlinear tuning of all six elastic constants in a 2D example. The evidence consists of rank measurements and susceptibility spectra for 40 scenarios covering different dimensions, potentials, pressures, system sizes, and parameter types, with careful use of automatic differentiation to compute J.

Significance. If the central claim holds, this is a notable conceptual advance: it transforms the empirical principle of independent bond-level response into a quantitative, Jacobian-based framework that can guide inverse design in disordered solids. The use of automatic differentiation to propagate derivatives through energy minimization is a practical and general contribution. The breadth of tested scenarios (2D/3D, repulsive/attractive potentials, varying pressure and system size) is a genuine strength, and the comparison with crystals sharpens the physical message. However, the universal wording of the main claim goes beyond what the evidence and the paper's own caveats support, and the numerical determination of rank is not fully specified. These issues are load-bearing because the abstract and title assert a universal property.

major comments (3)
  1. [Abstract and final paragraphs] The abstract claims that mechanical properties of disordered solids are 'always fully independent ... regardless of the target features, tunable parameters, and details of particle-particle interactions.' This universal statement is not supported by the evidence presented and is contradicted by the paper's own caveats: the final paragraphs state that 'our definition of independent response necessarily depends on the parameterization' and that 'it is certainly possible that exceptions do exist,' explicitly citing Rocks et al. [8], where the number of simultaneously tunable allosteric responses grows sublinearly with system size. To make the central claim load-bearing, the authors must either provide a mathematical mechanism forcing full rank under generic conditions or restate the claim as a property of the tested class of disordered solids and parameterizations, with a precise characterization of that class.
  2. [Fig. 1 and surrounding text] The paper states that 'rank(J) = min(n_y,n_theta) exactly' and that for disordered systems the susceptibilities are 'strictly positive.' Singular values from a numerical SVD are never exactly zero in floating-point arithmetic, so a rank tolerance must be specified. Without reporting the threshold used to count non-zero singular values, the claim of exact rank is not falsifiable from the data. This concern is concrete: Scenario 12 in Fig. 2 shows a band of small but non-zero susceptibilities whose separation from numerical noise is unclear. Please report the tolerance, the smallest singular values for representative systems, and ideally a comparison with a randomized null ensemble to demonstrate that the small singular values are physically meaningful.
  3. [Parameterization dependence, final paragraph] The paper acknowledges that 'independent response necessarily depends on the parameterization,' yet the abstract and the central result claim independence 'regardless of ... tunable parameters.' The species-reassignment procedure used to generate parameter spaces is a specific construction (randomly assigning particles to species while preserving particle sizes), and it is not shown to be unbiased or representative of all possible parameterizations. For instance, a parameterization that scales all diameters by a common factor or one that only varies a single species-level energy scale would have rank one. The manuscript should state the conditions on the parameterization under which the full-rank result is expected, or modify the universal claim to apply only to non-degenerate parameterizations of the type tested.
minor comments (3)
  1. [Fig. 3 and related text] The definition of the linear-regime boundary gamma* uses a 10% deviation criterion, and gamma_tilde is defined by the quadratic term reaching 10% of the linear term. The robustness of the zeta-versus-s correlation to this choice of threshold is not discussed; a brief sensitivity check would be helpful.
  2. [Fig. 1 and 'Primary result' section] The paper says 'verified through 10 independent measurements' for each rank data point. Please clarify whether these are 10 different initial configurations, 10 different species reassignments, or both, and whether rank is computed for each system individually or for an averaged Jacobian.
  3. [Introductory paragraphs] The text mentions 'order-N modes of vibration' in the introduction but the feature vector uses only the 'first 10 nonzero vibrational mode frequencies' in Fig. 1 and elsewhere. This discrepancy should be reconciled so the reader knows whether the full spectrum was tested.

Circularity Check

1 steps flagged · score 1.0 of 10

No significant circularity in the central rank claim; one minor definitional correlation in ζ ≡ γ*s.

  1. self definitional [Section 'Maximum linear tunability and nonlinear optimization', Fig. 4a; definition of ζ]
    "we can use our measurements of γ∗ in directions of relevant design modes with susceptibility s to define the 'maximum linear tunability' ζ ≡ γ∗s. This can be thought of as γ∗ in units of features, rather than parameters, and quantifies how far the features can be precisely tuned without resorting to nonlinear optimization techniques. As γ∗ is at best weakly correlated with s, Fig. 4a shows that ζ is correlated with susceptibility."

    ζ is defined as the product γ*s, and γ* is measured separately. The claimed correlation between ζ and s therefore reduces to the near-independence of γ* from s: if γ* were exactly constant, ζ = const·s would be perfectly correlated by definition. The paper itself states that γ* is at best weakly correlated with s, so the Fig. 4a correlation is largely inherited from the definition rather than an independent empirical test of the predictive value of the susceptibility. This is a peripheral, secondary observation; the central rank(J)=min(ny,nθ) result does not depend on it.

full rationale

The central claim—that rank(J)=min(ny,nθ) for disordered solids—is a direct numerical measurement and is not circular. The Jacobian J is computed by automatic differentiation through the complete feature pipeline, including energy minimization, from a specified pair potential; no parameter is fitted to the target features, and the contrast with crystals is an external comparison. The 40-scenario sweep is extrapolated to 'always', and the paper itself concedes possible exceptions (e.g., Rocks et al. allostery limits) and the parameterization dependence of the definition, but those are scope and correctness caveats rather than circular steps. The only definitional element is the correlation of ζ ≡ γ*s with susceptibility, which is partly by construction and explicitly framed as a definition. Self-citations (Refs. [1,16,24,25]) provide background or numerical procedures, not a uniqueness theorem or the rank result. Overall, the derivation chain is self-contained apart from one minor secondary definitional observation.

Assumptions & free parameters 1 free parameters · 5 assumptions · 0 invented entities

No parameters were fitted to data in the central claim; simulation inputs such as pressure and system size are stated and varied. No new physical entities are introduced. The claim rests on numerical evidence and standard linear algebra.

free parameters (1)
  • linear-regime threshold = 10% deviation
    Used to define γ* and maximum linear tunability ζ; a convention rather than a fitted parameter, but it shapes the correlation reported in Fig. 4a.
assumptions (5)
  • standard math Rank of a matrix equals the number of nonzero singular values; full rank is generic for random matrices.
    Used in Eq. (3) to translate SVD into independence; makes full rank plausible but does not guarantee it for structured Jacobians.
  • domain assumption Zero-temperature FIRE-minimized packings are representative of disordered solids.
    All results use energy-minimized athermal configurations; thermal, glassy, or driven systems are not tested.
  • ad hoc to paper Randomly reassigning particles into species creates an unbiased parameter space.
    The parameter set is constructed by partitioning big and small particles into new species; full-rank behavior may depend on this construction.
  • domain assumption Automatic differentiation through energy minimization gives the true Jacobian of the feature calculation.
    J is computed with AD; this assumes differentiability of the minimized configuration with respect to parameters, which holds away from contact changes.
  • standard math The linear expansion in Eq. (2) is valid for sufficiently small parameter changes.
    Used to define the linear regime and pseudo-inverse design; tested via γ* analysis.

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Pith. "Pith review of Fully independent response in disordered solids." pith.science (2026). https://pith.science/paper/FN6LLKSO

@misc{pith2026241205031,
  author       = {Pith},
  title        = {Pith review of: Fully independent response in disordered solids},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/FN6LLKSO}},
  note         = {Machine review of arXiv:2412.05031}
}
read the original abstract

Unlike in crystals, it is difficult to trace emergent material properties of amorphous solids to their underlying structure. Nevertheless, one can tune features of a disordered spring network, ranging from bulk elastic constants to specific allosteric responses, through highly precise alterations of the structure. This has been understood through the notion of independent bond-level response -- the observation that in many cases, different springs have different effects on different properties. While this idea has motivated inverse design in numerous contexts, it has not been formalized and quantified in a general context that not just informs but enables and predicts inverse design. Here, we show how to quantify independent response by linearizing the simultaneous change in multiple emergent features, and introduce the much stronger notion of fully independent response. Remarkably, we find that the mechanical properties of disordered solids are always fully independent across a wide array of scenarios, regardless of the target features, tunable parameters, and details of particle-particle interactions. Furthermore, our formulation quantifies the susceptibility of feature changes to parameter changes, which we find to be correlated with the maximum linear tunability. These results formalize our understanding of a key fundamental difference between ordered and disordered solids while also creating a practical tool to both understand and perform inverse design.

Figures

Figures reproduced from arXiv: 2412.05031 by the authors.

Figure 1
Figure 1. FIG. 1. Fully independent response is revealed by rank( [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗
Figure 2
Figure 2. FIG. 2. Susceptibilities, as measured by the singular values of [PITH_FULL_IMAGE:figures/full_fig_p003_2.png] view at source ↗
Figure 4
Figure 4. FIG. 4. Maximum linear tunability and nonlinear optimiza [PITH_FULL_IMAGE:figures/full_fig_p004_4.png] view at source ↗

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