{"id":"81988c93-9534-4887-8f70-4b63b789ef83","arxiv_id":"2412.05031","paper_version":2,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":1,"one_line_summary":"The Jacobian of emergent mechanical features with respect to species-level parameters is full rank in every tested disordered solid, so any set of features can be tuned independently given enough parameters.","lead":"This paper shows that the mechanical properties of disordered solids, such as elastic constants and vibration frequencies, can be tuned independently as long as there are at least as many control parameters. It introduces a Jacobian-based measure of independence and shows that disorder differs sharply from crystals, where independent tuning is limited.","discovery_kind":"new_method","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Universal 'always' claim is not supported: parameterization dependence and admitted possible exceptions (e.g., allosteric features) leave rank(J)=min(ny,nθ) unproven outside tested scenarios.","rationale":"The reader's weakest assumption was that the universal statement is extrapolated from 40 scenarios and lacks a mechanism, with numerical rank tolerance as a secondary issue. My concern sharpens this: the paper itself concedes both parameterization dependence and the possibility of exceptions, directly contradicting the abstract's unqualified 'always' and 'regardless of target features.' The most concrete, non-tautological risk is the allosteric response feature class, which prior work (Rocks et al.) suggests has sublinear tunability and which the paper explicitly flags as a possible exception but does not test. A failure of full rank in that class would falsify the abstract's claim; a success would resolve the tension and support a qualified universal statement. The numerical tolerance issue is real but secondary, because even with an exact arithmetic test on small systems, the universality question would remain. I therefore do not change the reader's CONDITIONAL verdict: the paper should either restrict its claims to the tested feature/parameter classes, provide a mechanism or a broader counterexample search, and report numerical rank tolerances. The proposed test directly addresses the most serious gap between the central claim and the evidence.","tokens_in":8614,"tokens_out":10847,"duration_ms":118244,"concrete_test":"Apply the authors' AD-based Jacobian pipeline to the same disordered networks used in Scenario 1, but take as features y the allosteric responses of Rocks et al. (target-site displacements in response to a source bond) with nθ ≥ ny species diameters as parameters. Compute the singular values of J and declare rank by a stated tolerance, e.g., singular values less than 10^-8 times the largest singular value are zero. If rank(J) < min(ny,nθ) for any configuration, the abstract's 'regardless of target features' is falsified. If full rank is found, the paper's conjecture of universality would be strengthened and the Rocks et al. discrepancy would need another explanation.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central claim, as stated in the abstract, is that the mechanical properties of disordered solids are 'always fully independent regardless of the target features, tunable parameters, and details of particle-particle interactions.' The paper's own body undermines this universal phrasing. In the final paragraphs, the authors 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. (Ref. [8]), where the number of simultaneously tunable allosteric responses grows sublinear with system size. These admissions mean the abstract's 'regardless of target features' and 'regardless of tunable parameters' cannot be literally true. Moreover, no mathematical mechanism is provided that would force rank(J)=min(ny,nθ) for arbitrary feature-parameter pairs in disordered solids; full rank is a generic property of unstructured matrices, but J is structured by the physics (equilibrium, sum rules, and the specific parameterization). The 40 scenarios, while broad, do not include allosteric response features, very low pressures near unjamming, or degenerate parameterizations. The load-bearing assumption is therefore that the tested set is representative of the entire class; the paper itself provides reason to doubt this.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","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.","tokens_in":8864,"tokens_out":3325,"duration_ms":36231,"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":[{"comment":"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.","section":"Abstract and final paragraphs"},{"comment":"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.","section":"Fig. 1 and surrounding text"},{"comment":"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.","section":"Parameterization dependence, final paragraph"}],"minor_comments":[{"comment":"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.","section":"Fig. 3 and related text"},{"comment":"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.","section":"Fig. 1 and 'Primary result' section"},{"comment":"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.","section":"Introductory paragraphs"}],"recommendation":"major_revision","confidential_remarks":"The paper has a strong core: the Jacobian formalism is clean, the AD implementation is a practical contribution, and the evidence across 40 scenarios is broad. However, the abstract's universal claim ('always fully independent ... regardless of ...') is not supported by the paper's own caveats, and a referee must weigh this heavily. The authors are evidently aware of the limitations (parameterization dependence, possible exceptions), so a revision that qualifies the central claim and adds numerical thresholds for rank determination could make this a solid paper. I recommend major revision rather than rejection because the core findings for the tested scenarios are valuable and the required changes are local to the framing and the reporting of numerical rank."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Colleague,\n\nThe short version: this paper gives a clean Jacobian/SVD definition of \"fully independent response\" in disordered solids and backs it with a surprisingly broad numerical campaign (40 scenarios, 2d/3d, multiple potentials, system sizes up to N=4096). The central observation — rank(J)=min(ny,nθ) for all tested disordered systems, but not for crystals — is new, concrete, and genuinely useful for inverse design. The AD-based Jacobian calculation is a real technical asset. Credit where due: the paper formalizes a notion that earlier bond-level response work only gestured at, and the correlation between singular values and maximum linear tunability (ζ≈γ*s) is a practically valuable result, even if partly definitional.\n\nSoft spots. The abstract says \"always fully independent... regardless of target features, tunable parameters, and details of interactions.\" The body is more careful: the authors admit parameterization dependence and say exceptions might exist, explicitly citing Rocks et al. where allosteric multifunctionality grows sublinearly. That tension is real. The 40 scenarios are broad but don't include allosteric target features, very low pressures near unjamming, or degenerate parameterizations. So the universal phrasing overreaches. Also, exact rank is asserted without stated numerical tolerances; \"verified through 10 independent measurements\" is fine but rank counting depends on a cutoff. No code or data are shipped, which matters for a claim of this scope. None of this sinks the paper. The core result — that full rank holds across a wide, systematic scenario matrix in disordered athermal packings — is well supported, and the authors are transparent about the extrapolation. The finite-size analysis (smallest singular value roughly constant in N) is a good check.\n\nWho this is for: people doing inverse design in soft matter and metamaterials, and anyone working on the order/disorder dichotomy in elastic response. It deserves a serious referee. The main fix is to align the abstract with the actual scope, release the code/data or at least the scenario generation details, and pick a rank threshold honestly.\n\nRecommendation: engage, with revision. This is a credible, reproducible-in-principle study with a clear framework. Send it to review.","headline":"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.","tokens_in":9333,"tokens_out":1453,"would_cite":true,"duration_ms":207679,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":[],"pacs":[],"model":"deepseek-v4-flash","headline":"Disordered solids are fully independent: any target features can be tuned independently as long as there are at least as many adjustable parameters.","keywords":["disordered solids","independent response","inverse design","elastic moduli","automatic differentiation","singular value decomposition","jammed packings","full-rank Jacobian"],"falsifier":"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.","tokens_in":8435,"feed_emoji":"🎛️","tokens_out":13673,"duration_ms":127344,"temperature":0.7,"pith_summary":"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.","feed_headline":"Disorder makes every mechanical property independently tunable","feed_subtitle":"Full-rank response in 40 scenarios means moduli, stress, and vibrations can be set independently.","key_machinery":"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.","core_discovery":"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.","pith_inferences":["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."],"forward_implications":["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."],"supporting_citations":[{"why":"Defines the principle of independent bond-level response that this paper formalizes and generalizes.","marker":"[1]"},{"why":"Demonstrates bond-level local responses that let elastic properties be manipulated separately, motivating the Jacobian formulation.","marker":"[2]"},{"why":"Provides the automatic-differentiation nonlinear optimization method reused to tune elastic constants beyond the linear regime.","marker":"[16]"},{"why":"Supplies the FIRE relaxation algorithm used to prepare the athermal jammed packings tested.","marker":"[17]"},{"why":"Automatic differentiation survey; the technique used to compute the Jacobian exactly through energy minimization.","marker":"[20]"},{"why":"SVD of the equilibrium matrix, the basis for interpreting left and right singular vectors as feature strains and design modes.","marker":"[23]"},{"why":"Argues single rearrangements have vanishing effect with system size, supporting a well-defined linear regime in the thermodynamic limit.","marker":"[25]"}],"fun_headline_variants":["Disorder grants full independent tunability of mechanical properties","Full-rank response: disordered solids let you tune every property at once","In disordered solids, every mechanical feature is independently adjustable","Disorder wins: all mechanical properties become independently tuneable","Maximal independence: disordered solids allow full control of each property"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"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.","fun_headline_variants_meta":{"raw":{"variants":["Disorder grants full independent tunability of mechanical properties","Full-rank response: disordered solids let you tune every property at once","In disordered solids, every mechanical feature is independently adjustable","Disorder wins: all mechanical properties become independently tuneable","Maximal independence: disordered solids allow full control of each property"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.00021,"raw_usage":{"total_tokens":1443,"prompt_tokens":1011,"completion_tokens":432,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":627,"completion_tokens_details":{"reasoning_tokens":350}},"tokens_in":627,"tokens_out":432,"duration_ms":5132,"temperature":1.0,"reasoning_tokens":350,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-11T20:58:21.157402+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"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.","supporting_citations":[{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Defines the principle of independent bond-level response that this paper formalizes and generalizes."},{"cited_title":"Hexner, A","cited_arxiv_id":null,"evidence_quote":"Demonstrates bond-level local responses that let elastic properties be manipulated separately, motivating the Jacobian formulation."},{"cited_title":"Zu and C","cited_arxiv_id":null,"evidence_quote":"Provides the automatic-differentiation nonlinear optimization method reused to tune elastic constants beyond the linear regime."},{"cited_title":"Pellegrino, Structural computations with the singular value decomposition of the equilibrium matrix, Interna- tional Journal of Solids and Structures 30, 3025 (1993)","cited_arxiv_id":null,"evidence_quote":"SVD of the equilibrium matrix, the basis for interpreting left and right singular vectors as feature strains and design modes."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Argues single rearrangements have vanishing effect with system size, supporting a well-defined linear regime in the thermodynamic limit."}],"review_version":1}