{"id":"d45eac3c-10c6-4c58-8d2e-a669d8c9bf43","arxiv_id":"1908.09435","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":4,"one_line_summary":"The ensemble shear modulus of jammed spheres rises with pressure because upward jumps at contact rearrangements overcome the linear softening each packing shows inside a fixed contact network.","lead":"Using computer simulations of squishy spherical particles, this paper shows that the average stiffness of a jammed packing rises with pressure because sudden jumps at particle rearrangements outweigh a steady softening within each undisturbed configuration. The work explains a long-standing discrepancy between single-packing and averaged measurements, and reports that compressing a jammed packing can sometimes make it unjam.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Gi is measured with a frozen-contact, double-sided-spring probe; the within-family linear decrease and the upward-jump compensation could be artifacts of this unphysical contact treatment.","rationale":"Agreement with reader: the reader's weakest assumption points to the same protocol. I sharpen it: the double-sided spring is not just a benign linear-response device; it allows contacts in tension, which the repulsive model forbids, and it freezes out the one physical channel (new contacts) that could first-order stiffen the response. The negative Gi values reported in Fig. 2(a) are direct evidence that this probe measures something different from the physical modulus of a stable repulsive packing; such states would shear spontaneously in a true quasistatic measurement. Because the entire explanation is an accounting identity (within-family change + jumps), all physical content sits in the sign and magnitude of the two terms; a probe bias can flip both. This is therefore more load-bearing than the descriptive scaling-function fits (which would be a concern only if the two-term decomposition were absent) and the compression-unjamming side claim (deferred to the Supplemental Material and not needed for the shear-modulus result). Independent support exists: Eq. (5) follows from energy conservation for a fixed contact network, and the ensemble fits recover the known α≈1 and β≈0.5 exponents, so I do not recommend rejection. The appropriate status is conditional acceptance pending the protocol check, which is what the reader already concluded. Hence verdict unchanged.","tokens_in":7375,"tokens_out":23870,"duration_ms":255913,"concrete_test":"Re-measure Gi for a subset of the same packings (N = 64, 128, 256 disks) with the actual single-sided potential of Eq. (6) and full energy minimization after each strain step, allowing contacts to form and break, at step sizes from Δγ = 5×10^-9 down to 5×10^-13. Extract the zero-strain derivative of the virial stress and compare within-family slopes λ_i, the sign and magnitude of jumps at rearrangements, and the ensemble averages ⟨Gf+Gs⟩ and ⟨Gr⟩. If the full single-sided measurement reproduces the linear decrease and upward jumps, the protocol is validated; if not, the compensation mechanism in Fig. 3(b) fails. Also repeat with negative shear to check for directional bias.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central mechanism — within-family softening plus upward jumps — rests on the specific definition of Gi. On p. 3 the authors measure shear response by assuming a double-sided linear spring for existing contacts and excluding new contacts that form during the applied strain. For a purely repulsive system this lets contacts support tension and removes a first-order stiffening channel; the resulting frozen-contact modulus is not the standard quasistatic shear modulus used to establish the p^α/p^β scaling cited in Eq. (2). The paper itself notes that 'some of the packings are unstable with Gi < 0' (Fig. 2a), showing the probe can disagree qualitatively with the positive-definite modulus of a stable repulsive packing. If about-to-break contacts are artificially kept by the double-sided spring, the fitted λ_i and the jump statistics at rearrangements are biased, and the claim that jumps are on average upward and compensate the linear decrease may be an artifact of the measurement protocol.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"This Letter studies the pressure dependence of the static shear modulus of jammed packings of frictionless, purely repulsive disks and spheres under athermal quasistatic shear. The authors report that for an individual packing, the shear modulus decreases linearly with pressure along a geometrical family of fixed contact network, following G^i = G^i_0 - lambda_i p, while discontinuous upward jumps in G^i occur at rearrangements between families. They partition the ensemble-averaged modulus into first-family, change-in-family, and rearrangement contributions, and show that the sum is described by a scaling function that crosses over from a low-pressure power law with exponent alpha ~ 1 to a high-pressure power law with exponent beta ~ 0.5, matching earlier results. They also report the phenomenon of compression unjamming, in which a jammed packing becomes unjammed upon isotropic compression.","tokens_in":7615,"tokens_out":13128,"duration_ms":143825,"significance":"If the central mechanism is correct, the paper provides a microscopic explanation of the well-known alpha-to-beta exponent crossover in the shear modulus of jammed packings, connecting within-family mechanical softening to rearrangement statistics. The compression-unjamming observation is also surprising and potentially important for understanding reversibility and cyclic compaction. The authors are explicit that the scaling functions in Eqs. (8) and (9) are empirical interpolations, and the strength of the paper lies in the direct observation of a linear within-family decrease and upward jumps, rather than in a parameter-free prediction. The manuscript is clearly written and the figures support the reported trends. However, as detailed below, the derivation of Eq. (5) and the nonstandard protocol used to measure G^i are load-bearing and need to be addressed before the central conclusion can be accepted.","major_comments":[{"comment":"The energy balance dU = -p dL^d - Sigma_xy L^d dgamma describes a path in which volume and shear strain both change. The shear modulus is then defined as the derivative of -Sigma_xy with respect to gamma at constant volume, which requires dL^d = 0 and therefore dphi = 0 during the differentiation. Under that condition the second term in Eq. (4) vanishes and Eq. (5) does not follow; the linear decrease -lambda_i p is not a consequence of energy conservation as stated. The authors should either supply a correct derivation starting from the virial expression and the force balance of the frozen-contact system, or present the linear decrease as an empirical observation rather than a derived prediction.","section":"Derivation of Eq. (5), Eqs. (3)-(4)"},{"comment":"The measurement uses a double-sided linear spring for all existing contacts and excludes new contacts that form during the applied shear. This is not the quasistatic shear modulus of the one-sided repulsive system whose scaling is quoted in Eq. (2). The appearance of G^i < 0 in Fig. 2(a) indicates that the probe can disagree qualitatively with the physical modulus of a stable jammed packing. Since the central mechanism -- within-family softening plus upward rearrangement jumps -- is inferred entirely from this G^i, the paper needs to demonstrate that the same family slopes, jump signs, and the compensation between the two contributions are obtained with the physical one-sided potential, including contact breaking and formation. Without such a check, the claim that jumps are on average upward and cancel the linear decrease could be an artifact of the contact-treatment protocol.","section":"Page 3, 'To determine the shear modulus G^i'"},{"comment":"The decomposition G^i = G_f^i + G_s^i + G_r^i is presented as the sum of a first-family term, a change-in-family term, and a rearrangement term, but the operational rule for assigning every discontinuity to G_r versus updating G_f and G_s is not described in the text. Moreover, Eqs. (8) and (9) contain six or more adjustable parameters and are fitted to the same data whose scaling they are used to explain; the agreement in Fig. 3(b) therefore does not by itself confirm that the two contributions are comparable. The authors should state the assignment algorithm precisely and report the fit parameters and uncertainties.","section":"Fig. 3 and Eqs. (8)-(9)"}],"minor_comments":[{"comment":"The word 'decribes' should be 'describes' in the sentence 'a physically motivated scaling function that accurately decribes <G>'.","section":"Summary paragraph, page 2"},{"comment":"The sentence 'find again that lambda_i < 0 is extremely rare' is confusing given the sign convention in Eq. (5), where the natural claim is that lambda_i > 0 almost always; please rephrase for clarity.","section":"Page 3, after Eq. (5)"},{"comment":"The text states that <G_0> and <lambda> are plotted versus N but does not give the functional forms used for the fits or the error bars; please specify these details.","section":"Inset to Fig. 3(a)"},{"comment":"The statement that <G> can be approximated by a single scaling function because both contributions transition over the same pressure interval is plausible but not quantitatively justified; a direct comparison of the sum of the two fitted functions (Eqs. (8) and (9)) with the single-function fit would strengthen this step.","section":"Page 5, scaling function for <G>"}],"recommendation":"major_revision","confidential_remarks":"The paper's underlying numerical data may well be correct, but the two load-bearing concerns -- the derivation of Eq. (5) and the frozen-contact measurement protocol -- are serious enough that I would ask for a reanalysis with the physical one-sided potential before publication. If the authors can show that the family slopes and jump statistics are unchanged or only weakly modified under the physical protocol, the paper would make a valuable contribution."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Here's the short version. This paper resolves a known contradiction in the jamming literature: individual packings soften linearly in pressure along a fixed contact network, while the ensemble-averaged shear modulus rises as a power law. The authors show that the ensemble behavior comes from a balance between this within-family softening and upward discontinuous jumps when the contact network rearranges. The energy-conservation derivation of Eq. (5) is clean, and the decomposition into first-family, change-in-family, and rearrangement contributions is a genuinely useful way to see the competition. The strongest evidence is that the two contributions stay comparable for all system sizes, and the fitted exponents reproduce the known α≈1 and β≈0.5.\n\nThe soft spots are real but not disqualifying. The main one is the protocol used to measure Gi. The text says the authors use a double-sided linear spring and ignore new contacts during the applied shear. If that means they are computing an affine, frozen-contact modulus without relaxing particle positions, then Gi is not the standard quasistatic shear modulus. If they relax under the double-sided potential, it is the Hessian of the fixed contact network, which is standard linear response. The paper should state which one, and ideally show that the linear decrease and the upward-jump trend survive in the infinitesimal-strain limit. I don't think the stress-test concern makes the result an artifact; the double-sided spring is a plausible way to take the frozen-contact limit, but the ambiguity needs to be resolved. Second, the scaling forms Eqs. (8)–(9) are descriptive, with a fair number of free parameters. They illustrate the mechanism but do not independently confirm it. Third, compression unjamming is presented as a side finding, and the main-text evidence is a single contour plot; the statistics live in the supplement, which I did not have. That lowers my confidence a bit. No code or data archive is mentioned.\n\nThis is a subfield paper, not a paradigm shift, but it is a real step. It deserves a serious referee. I would condition acceptance on clarifying the Gi measurement, and on making the supplement available at the time of review.","headline":"A concrete mechanistic explanation for the ensemble shear-modulus scaling in jammed packings, provided the measurement protocol is clarified.","tokens_in":8127,"tokens_out":10975,"would_cite":true,"duration_ms":117957,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":[],"pacs":[],"model":"deepseek-v4-flash","headline":"The ensemble shear modulus of jammed packings is the sum of per-family linear softening and upward jumps at contact-network rearrangements.","keywords":["jammed packings","shear modulus","geometrical families","jamming onset","power-law scaling","compression unjamming","contact network rearrangements"],"falsifier":"Measure the ensemble-averaged rearrangement contribution $\\langle G^r\\rangle$ for the same packings while (i) including contacts that newly form during the applied shear strain and (ii) applying negative shear of the same magnitude; if the mean jump is no longer upward or no longer comparable to $\\langle G^f+G^s\\rangle$ for $p>p^{**}$, the proposed balance fails.","tokens_in":7159,"feed_emoji":"⚙️","tokens_out":7094,"duration_ms":61198,"temperature":0.7,"pith_summary":"This paper explains why the ensemble-averaged shear modulus of jammed packings of purely repulsive spheres follows two power laws in pressure ($\\langle G-G_0\\rangle\\sim p^\\alpha$ with $\\alpha\\approx 1$ at low pressure and $\\sim p^\\beta$ with $\\beta\\approx 0.5$ above a crossover) even though each individual packing's shear modulus decreases linearly with pressure. The resolution is that the ensemble average receives two competing contributions: within a geometrical family (fixed force-bearing contact network) the stiffness follows $G^i=G^i_0-\\lambda^i p$, and at transitions between families it jumps discontinuously, with the jumps on average upward. For pressures above the crossover $p^{**}$, the two contributions are comparable in magnitude and opposite in sign at all system sizes studied, so the exponent switch is a balance rather than a property of either contribution alone. The paper also reports compression unjamming, in which a jammed packing can become unjammed when compressed further. A correct account of these mechanisms matters for predicting granular, foam, and emulsion response near jamming.","feed_headline":"Shear modulus of jammed packings splits into two competing effects","feed_subtitle":"Ensemble power-law scaling is the balance of linear decreases within fixed contact networks and upward jumps at rearrangements.","key_machinery":"The central object is the linear relation $G^i(p)=G^i_0-\\lambda^i p$ within a geometrical family, derived from energy conservation $-p\\,dL^d-\\Sigma_{xy}L^d\\,d\\gamma=dU$ at fixed contact network, with $G^i_0=L^{-d}\\,d^2U/d\\gamma^2$ and $\\lambda^i=\\phi^{-1}\\,d^2\\phi/d\\gamma^2$. The paper establishes that $\\lambda^i>0$ almost always, so within-family stiffness decreases linearly with pressure. The second mechanism is the discontinuous rearrangement jump, isolated by the decomposition $G^i=G^{if}+G^{is}+G^{ir}$ into first-family, change-in-family, and rearrangement parts. The compact formula that organizes the data is $\\langle G\\rangle=(\\langle G_0\\rangle+a p^\\alpha)/(1+c p^{\\alpha-\\beta})$, which smoothly interpolates between the two power laws, and the matching crossover behavior of $\\langle G^f+G^s\\rangle$ and $\\langle G^r\\rangle$ is what ties the exponent switch to the geometry of rearrangements.","core_discovery":"The central claim is that the known power-law scaling of the ensemble-averaged shear modulus near jamming onset is an emergent balance, not a bulk property of individual packings. Each packing $i$ obeys $G^i = G^i_0 - \\lambda^i p$ along a geometrical family, with $\\lambda^i>0$ in nearly all cases, so its stiffness falls linearly as pressure rises. When pressure increases enough to change the force-bearing contact network—through a particle rearrangement or through an added contact—$G^i$ changes discontinuously, and these jumps are on average upward. The ensemble average $\\langle G\\rangle$ therefore separates into a first-geometrical-family plus family-change contribution $\\langle G^f+G^s\\rangle$, which decreases linearly with $p$, and a rearrangement contribution $\\langle G^r\\rangle$, which is positive and grows with $p$. The paper shows that each contribution, and their sum $\\langle G\\rangle$, is described by a function that transitions between two power laws over the same pressure interval, with crossover $p^{**}\\sim N^{-1}$, and that the two opposing contributions remain comparable in the large-$N$ limit. It also demonstrates compression unjamming: because compression shifts bond angles and can induce a mechanical instability, a jammed packing can move to a configuration whose jamming onset lies above the current packing fraction.","pith_inferences":["If upward jumps are the generic cause of the rising ensemble average, then the high-pressure exponent $\\beta$ should be directly related to the density of rearrangement events per pressure interval; a test would compare $d\\langle G^r\\rangle/dp$ with the measured rate of contact-network changes.","The same decomposition could be applied to pressure ramps at fixed shear strain and to strain ramps at fixed pressure, predicting that the observed power-law response in each protocol is the sum of within-family softening and jumps, rather than a single intrinsic exponent.","For non-spherical particles, where the high-pressure exponent differs ($\\beta\\approx 1$ for ellipses in earlier studies), applying this decomposition would reveal whether the change comes from weaker within-family softening or from rearrangement jumps that no longer compensate it.","Compression unjamming may imply that compression history and shear history are not interchangeable near jamming, with consequences for protocols that attempt to prepare jammed states by isotropic compression alone."],"forward_implications":["For $p>p^{**}$, the geometrical-family contribution remains comparable to the rearrangement contribution at every system size, so the linear within-family softening cannot be ignored in the thermodynamic limit.","The low-pressure exponent $\\alpha\\approx 1$ is controlled by the first-family term $\\langle G_0\\rangle-\\langle\\lambda\\rangle p$, while the high-pressure exponent $\\beta\\approx 0.5$ is controlled by the upward rearrangement jumps; the crossover $p^{**}\\sim N^{-1}$ is where the two contributions balance.","Since $\\langle G^r\\rangle$ is zero below the pressure of the first rearrangement, the $p^\\beta$ regime requires ensembles with enough pressure range to sample many rearrangements.","Compression unjamming occurs with a probability that is independent of system size for sheared packings and nonzero in the large-$N$ limit for packings compressed at fixed $\\gamma=0$, so cyclic compression protocols near jamming can encounter irreversibility."],"supporting_citations":[{"why":"Shows that the shear modulus of individual jammed packings decreases with pressure along geometrical families, and supplies the virial expression used to compute shear stress.","marker":"[18]"},{"why":"Defines geometrical families as regions of the packing-fraction and shear-strain plane with fixed contact networks, the entity within which the linear decrease holds.","marker":"[19]"},{"why":"Provides the baseline scaling relations for contact number and shear modulus near jamming that this paper explains.","marker":"[11]"},{"why":"FIRE energy-minimization algorithm used to generate every jammed configuration.","marker":"[20]"},{"why":"Gives the $N$-dependence of jamming-onset distribution widths used to interpret compression-unjamming probabilities.","marker":"[21]"},{"why":"Supplies the virial stress formula that the shear modulus measurement is based on.","marker":"[22]"}],"fun_headline_variants":["Jammed sphere shear: drops in networks, jumps at transitions","Ensemble shear modulus emerges from competing contact effects","Shear modulus of jammed packings: linear drop plus upward jump","Compression can unjam a jammed packing of spheres","Jammed shear response: two opposing contributions set scaling"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"Everything about the sign and size of the rearrangement jumps rests on measuring $G^i$ with the double-sided linear spring while ignoring contacts that would newly form during the applied positive shear strain; using negative shear or including new contacts could change the jumps and destroy the compensation.","fun_headline_variants_meta":{"raw":{"variants":["Jammed sphere shear: drops in networks, jumps at transitions","Ensemble shear modulus emerges from competing contact effects","Shear modulus of jammed packings: linear drop plus upward jump","Compression can unjam a jammed packing of spheres","Jammed shear response: two opposing contributions set scaling"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000744,"raw_usage":{"total_tokens":3379,"prompt_tokens":1069,"completion_tokens":2310,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":685,"completion_tokens_details":{"reasoning_tokens":2228}},"tokens_in":685,"tokens_out":2310,"duration_ms":16738,"temperature":1.0,"reasoning_tokens":2228,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-14T11:11:32.976673+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Measure the ensemble-averaged rearrangement contribution $\\langle G^r\\rangle$ for the same packings while (i) including contacts that newly form during the applied shear strain and (ii) applying negative shear of the same magnitude; if the mean jump is no longer upward or no longer comparable to $\\langle G^f+G^s\\rangle$ for $p>p^{**}$, the proposed balance fails.","supporting_citations":[{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Shows that the shear modulus of individual jammed packings decreases with pressure along geometrical families, and supplies the virial expression used to compute shear stress."},{"cited_title":"Bertrand, R","cited_arxiv_id":null,"evidence_quote":"Defines geometrical families as regions of the packing-fraction and shear-strain plane with fixed contact networks, the entity within which the linear decrease holds."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Gives the $N$-dependence of jamming-onset distribution widths used to interpret compression-unjamming probabilities."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Supplies the virial stress formula that the shear modulus measurement is based on."}],"review_version":1}